Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and...

121
Structural Stability of Nonlinear Flows Augusto Visintin - Trento Workshop “Diffuse Interface Models” Levico Terme – September 10-13, 2013 Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Transcript of Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and...

Page 1: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stabilityof Nonlinear Flows

Augusto Visintin - Trento

Workshop “Diffuse Interface Models”Levico Terme – September 10-13, 2013

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 2: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Let V be a Banach space, and αn : V → P(V ′) be an equi-coercive sequence.

Claims:

Operator Compactness: there exists a topology τ such that

∃α : V → P(V ′) : αn→τ α, (1)

and

Structural Stability:

(un, u∗n ) ∈ graph(αn) ∀n, (un, u

∗n ) (u, u∗) in V×V ′, αn→τ α (2)

⇒ (u, u∗) ∈ graph(α). (3)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 3: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Let V be a Banach space, and αn : V → P(V ′) be an equi-coercive sequence.

Claims:

Operator Compactness: there exists a topology τ such that

∃α : V → P(V ′) : αn→τ α, (1)

and

Structural Stability:

(un, u∗n ) ∈ graph(αn) ∀n, (un, u

∗n ) (u, u∗) in V×V ′, αn→τ α (2)

⇒ (u, u∗) ∈ graph(α). (3)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 4: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Let V be a Banach space, and αn : V → P(V ′) be an equi-coercive sequence.

Claims:

Operator Compactness: there exists a topology τ such that

∃α : V → P(V ′) : αn→τ α, (1)

and

Structural Stability:

(un, u∗n ) ∈ graph(αn) ∀n, (un, u

∗n ) (u, u∗) in V×V ′, αn→τ α (2)

⇒ (u, u∗) ∈ graph(α). (3)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 5: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Let V be a Banach space, and αn : V → P(V ′) be an equi-coercive sequence.

Claims:

Operator Compactness: there exists a topology τ such that

∃α : V → P(V ′) : αn→τ α, (1)

and

Structural Stability:

(un, u∗n ) ∈ graph(αn) ∀n, (un, u

∗n ) (u, u∗) in V×V ′, αn→τ α (2)

⇒ (u, u∗) ∈ graph(α). (3)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 6: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. Any max-monotone operator may be given a variational representation.

2. For max-monotone equations, operator compactness and structural stabilitymay then be studied via De Giorgi’s Γ-convergence.

3. This also applies e.g. to Dtu + α(u) 3 u∗ (α: max-monotone).

4. This may be extended to α pseudo-monotone (in the sense of Brezis).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 7: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. Any max-monotone operator may be given a variational representation.

2. For max-monotone equations, operator compactness and structural stabilitymay then be studied via De Giorgi’s Γ-convergence.

3. This also applies e.g. to Dtu + α(u) 3 u∗ (α: max-monotone).

4. This may be extended to α pseudo-monotone (in the sense of Brezis).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 8: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. Any max-monotone operator may be given a variational representation.

2. For max-monotone equations, operator compactness and structural stabilitymay then be studied via De Giorgi’s Γ-convergence.

3. This also applies e.g. to Dtu + α(u) 3 u∗ (α: max-monotone).

4. This may be extended to α pseudo-monotone (in the sense of Brezis).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 9: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. Any max-monotone operator may be given a variational representation.

2. For max-monotone equations, operator compactness and structural stabilitymay then be studied via De Giorgi’s Γ-convergence.

3. This also applies e.g. to Dtu + α(u) 3 u∗ (α: max-monotone).

4. This may be extended to α pseudo-monotone (in the sense of Brezis).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 10: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Trivial Example

In the space L2(0,T ), the ordinary derivative Dt with domainV := v ∈ H1(0,T ) : v(0) = 0 is monotone, i.e.,

(Dtu1 − Dtu2, u1 − u2) ≥ 0 ∀u1, u2 ∈ V .

Indeed:

(Dtu, u) =

∫ T

0

[Dtu(t)] u(t) dt =1

2u(T )2 − 1

2u(0)2 =

1

2u(T )2 ≥ 0 ∀u ∈ V .

May Dtu = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 11: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Trivial Example

In the space L2(0,T ), the ordinary derivative Dt with domainV := v ∈ H1(0,T ) : v(0) = 0 is monotone, i.e.,

(Dtu1 − Dtu2, u1 − u2) ≥ 0 ∀u1, u2 ∈ V .

Indeed:

(Dtu, u) =

∫ T

0

[Dtu(t)] u(t) dt =1

2u(T )2 − 1

2u(0)2 =

1

2u(T )2 ≥ 0 ∀u ∈ V .

May Dtu = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 12: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Trivial Example

In the space L2(0,T ), the ordinary derivative Dt with domainV := v ∈ H1(0,T ) : v(0) = 0 is monotone, i.e.,

(Dtu1 − Dtu2, u1 − u2) ≥ 0 ∀u1, u2 ∈ V .

Indeed:

(Dtu, u) =

∫ T

0

[Dtu(t)] u(t) dt =1

2u(T )2 − 1

2u(0)2 =

1

2u(T )2 ≥ 0 ∀u ∈ V .

May Dtu = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 13: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Trivial Example

In the space L2(0,T ), the ordinary derivative Dt with domainV := v ∈ H1(0,T ) : v(0) = 0 is monotone, i.e.,

(Dtu1 − Dtu2, u1 − u2) ≥ 0 ∀u1, u2 ∈ V .

Indeed:

(Dtu, u) =

∫ T

0

[Dtu(t)] u(t) dt =1

2u(T )2 − 1

2u(0)2 =

1

2u(T )2 ≥ 0 ∀u ∈ V .

May Dtu = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 14: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

More generally:

α : W →W ′ is monotone ⇒ Dt + α is also monotone

in V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0.

(4)

E.g., for any quasilinear elliptic operator A : W →W ′,

Dt + A is monotone (in space-time).

May Dtu + Au = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 15: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

More generally:

α : W →W ′ is monotone ⇒ Dt + α is also monotone

in V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0.

(4)

E.g., for any quasilinear elliptic operator A : W →W ′,

Dt + A is monotone (in space-time).

May Dtu + Au = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 16: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

More generally:

α : W →W ′ is monotone ⇒ Dt + α is also monotone

in V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0.

(4)

E.g., for any quasilinear elliptic operator A : W →W ′,

Dt + A is monotone (in space-time).

May Dtu + Au = f be represented as a minimum problem?

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 17: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 18: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 19: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 20: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 21: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 22: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

1. The Fitzpatrick Theory

Let V be a (real) Banach space with V ′ separable, and α : V → P(V ′).

1n 1988 Fitzpatrick introduced the function

fα : V×V ′ → R ∪ +∞

fα(v , v∗) := 〈v∗, v〉+ sup〈v∗ − v∗0 , v0 − v〉 : (v0, v

∗0 ) ∈ graph(α)

= sup

〈v∗, v0〉+ 〈v∗0 , v〉 − 〈v∗0 , v0〉 : (v0, v

∗0 ) ∈ graph(α)

.

(5)

This function is clearly convex and l.s.c..

Fitzpatrick’s Theorem. If α is max-monotone, then

fα(v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′,

fα(v , v∗) = 〈v∗, v〉 ⇔ v∗ ∈ α(v).(6)

Conversely, (6) entails that α is monotone (possibly not maximal).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 23: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 24: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 25: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 26: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 27: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 28: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization Principle

For any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 29: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 30: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 31: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Representative functions

Let F(V ) be the class of the functions f such that

f : V×V ′ → R ∪ +∞ is convex and l.s.c.,

f (v , v∗) ≥ 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′.(7)

To any f ∈ F(V ) we associate the operator αf : V → P(V ′) such that

v∗ ∈ αf (v) ⇔ f (v , v∗) = 〈v∗, v〉, (8)

and say that f (variationally) represents αf .

max-monotone ⇒ representable ⇒ monotone.

Null-Minimization PrincipleFor any f ∈ F(V ), setting

F (v , v∗) := f (v , v∗)− 〈v∗, v〉 ∀(v , v∗) ∈ V×V ′, (9)

by (7) and (8),v∗ ∈ αf (v) ⇔ F (v , v∗) = inf F = 0. (10)

The vanishing of the minimum cannot be dispensed with !

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 32: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 33: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 34: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 35: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 36: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 37: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 38: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 39: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 40: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Examples of Representative functions

Fitzpatrick’s function: fα represents the max-monotone operator α.

Fenchel’s function. For any convex and l.s.c. function ϕ : V → R ∪ +∞,the subdifferential ∂ϕ is represented by

f (v , v∗) = ϕ(v) + ϕ∗(v∗) ∀(v , v∗) ∈ V×V ′; (11)

f is self-dual, i.e., f ∗ = f in the duality between V×V ′ and V ′×V .

Extended Brezis-Ekeland-Nayroles’s Principle. Let

α : V → P(V ′) be represented by fα,

L : V → V ′ be bounded, linear and monotone(e.g., L = Dt , V =

v ∈ L2(0,T ; W ) ∩ H1(0,T ; W ′) : v(0) = 0

).

Then α + L is represented by

fα+L(v , v∗) := fα(v , v∗ − Lv) + 〈Lv , v〉 ∀(v , v∗) ∈ V×V ′; (12)

indeed fα+L(v , v∗) = 〈v∗, v〉 ⇔ [α + L](v) 3 v∗.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 41: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

For instance, ifβ : RN → P(RN) is max-monotone,

h ∈ L2(0,T ; H−1(Ω)),(13)

then the problem

u ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω))

Dtu −∇·β(∇u) 3 h in H−1(Ω), a.e. in ]0,T [

u(·, 0) = 0 in Ω

(14)

is equivalent toF (u, h) = inf F = 0, (15)

where

F (v , v∗) :=∫ T

0

dt

∫Ω

[fβ(∇v ,∇∆−1(Dtv − v∗)

)− 〈v∗, v〉

]dx +

1

2

∫Ω

|v(·,T )|2 dx

∀v ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω)), v(0) = 0,

∀v∗ ∈ L2(0,T ; H−1(Ω)).

(16)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 42: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

For instance, ifβ : RN → P(RN) is max-monotone,

h ∈ L2(0,T ; H−1(Ω)),(13)

then the problem

u ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω))

Dtu −∇·β(∇u) 3 h in H−1(Ω), a.e. in ]0,T [

u(·, 0) = 0 in Ω

(14)

is equivalent toF (u, h) = inf F = 0, (15)

where

F (v , v∗) :=∫ T

0

dt

∫Ω

[fβ(∇v ,∇∆−1(Dtv − v∗)

)− 〈v∗, v〉

]dx +

1

2

∫Ω

|v(·,T )|2 dx

∀v ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω)), v(0) = 0,

∀v∗ ∈ L2(0,T ; H−1(Ω)).

(16)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 43: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

For instance, ifβ : RN → P(RN) is max-monotone,

h ∈ L2(0,T ; H−1(Ω)),(13)

then the problem

u ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω))

Dtu −∇·β(∇u) 3 h in H−1(Ω), a.e. in ]0,T [

u(·, 0) = 0 in Ω

(14)

is equivalent toF (u, h) = inf F = 0, (15)

where

F (v , v∗) :=∫ T

0

dt

∫Ω

[fβ(∇v ,∇∆−1(Dtv − v∗)

)− 〈v∗, v〉

]dx +

1

2

∫Ω

|v(·,T )|2 dx

∀v ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω)), v(0) = 0,

∀v∗ ∈ L2(0,T ; H−1(Ω)).

(16)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 44: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

For instance, ifβ : RN → P(RN) is max-monotone,

h ∈ L2(0,T ; H−1(Ω)),(13)

then the problem

u ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω))

Dtu −∇·β(∇u) 3 h in H−1(Ω), a.e. in ]0,T [

u(·, 0) = 0 in Ω

(14)

is equivalent toF (u, h) = inf F = 0, (15)

where

F (v , v∗) :=∫ T

0

dt

∫Ω

[fβ(∇v ,∇∆−1(Dtv − v∗)

)− 〈v∗, v〉

]dx +

1

2

∫Ω

|v(·,T )|2 dx

∀v ∈ L2(0,T ; H10 (Ω)) ∩ H1(0,T ; H−1(Ω)), v(0) = 0,

∀v∗ ∈ L2(0,T ; H−1(Ω)).

(16)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 45: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

References

H. Brezis, I. Ekeland: Un principe variationnel associe a certaines equationsparaboliques. C. R. Acad. Sci. Paris Ser. A-B 282 (1976) 971–974, and ibid.1197–1198

B. Nayroles: Deux theoremes de minimum pour certains systemes dissipatifs.C. R. Acad. Sci. Paris Ser. A-B 282 (1976) A1035–A1038

S. Fitzpatrick: Representing monotone operators by convex functions.Austral. Nat. Univ., Canberra, 1988

J.-E. Martinez-Legaz, M. Thera: A convex representation of maximal monotoneoperators. J. Nonlinear Convex Anal. 2 (2001), 243–247

R.S. Burachik, B.F. Svaiter: Maximal monotone operators, convex functions, and aspecial family of enlargements. Set-Valued 10 (2002) 297–316

A.V.: Extension of the Brezis-Ekeland-Nayroles principle to monotone operators.Adv. Math. Sci. Appl. 18 (2008) 633–650

N. Ghoussoub: Selfdual Partial Differential Systems and their Variational Principles.Springer, 2009

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 46: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 47: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 48: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 49: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 50: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 51: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 52: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 53: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

2. A Nonlinear (“Weak-Type”) Convergence

(vn, v∗n ) →

π(v , v∗) in V×V ′ ⇔

vn v in V , v∗n ∗ v∗ in V ′, 〈v∗n , vn〉 → 〈v∗, v〉.

(17)

The associated topology is metrizable on bounded subsets of V×V ′.

Γπ-Compactness

Let the sequence ψn : V×V ′ → R ∪ +∞ be equi-coercive, i.e.,

∀C ∈ R, supn

‖v‖V + ‖v∗‖V ′ : ψn(v , v∗) ≤ C

< +∞. (18)

Then: (i) ψn Γπ-converges to some function ψ, up to a subsequence.

(ii) If ψn ∈ F(V ) ∀n, then ψ ∈ F(V ).

(iii) If any ψn (ψ, resp.) represents the operator αn (α, resp.), then

lim supn→∞

graph(αn) ⊂ graph(α) (19)

in the sense of Kuratowski w.r.t. the topology π.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 54: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 55: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 56: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 57: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 58: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 59: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Compactness and Structural Stability (or G-compactness and G-convergence)

Let (S,D,O) be a triplet of topological spaces, withS: space of the admissible solutions of the problem,D: space of the admissible data,O: space of the admissible operators o : S → D : s 7→ d .

Any (d , o) ∈ D×O determines the problem

find s ∈ S such that d ∈ o(s). (20)

Definition 1. The class of problems (S,D,O) is compact if

any bounded sequence (sn, dn, on) in (S,D,O)

accumulates at some (s, d , o).(21)

Definition 2. The class of problems (S,D,O) is structurally stable if

dn ∈ on(sn) ∀n, (sn, dn, on) (s, d , o) ⇒ d ∈ o(s). (22)

This is a sort of enhanced well-posedness, as it also involves operator convergence.

Approximation of the operator is often mandatory, e.g. in numerical analysis.This may raise the issues of operator convergence and structural stability.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 60: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 61: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 62: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 63: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 64: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.

In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 65: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 66: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)

∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 67: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

3. A Monotone Flow

Set X0 =

v ∈ L2(0,T ; V ) ∩ H1(0,T ; V ′) : v(0) = 0

(a vanishing initial datum may be retrieved by a shift in the unknown function).

Assume that, for any n,

αn : V → P(V ′) is maximal monotone,

∃a, b > 0 : ∀n, ∀(v , v∗) ∈ graph(αn),

〈v∗, v〉 ≥ a‖v‖2V , ‖v∗‖V ′ ≤ b‖v‖V ,

hn ∈ L2(0,T ; V ′).

(23)

Let us then consider the sequence of flows

un ∈ X0, Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [. (24)

∀n, let ψn : V×V ′ → R ∪ +∞ be a representative function of αn.In X0 this inclusion is equivalent to either of the inequalities

ψn(un, hn − Dtun) ≤ 〈un, hn − Dtun〉 a.e. in ]0,T [, (25)∫ T

0

ψn(un, hn − Dtun) dt ≤∫ T

0

〈un, hn − Dtun〉 dt. (26)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 68: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 69: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 70: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 71: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

and

Structural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 72: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 73: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 74: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 75: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Pointwise-in-Time Formulation

If the injection V → V ′ is compact, then we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by ψn,

then there exists ψ : V×V ′ → R such that, up to extracting a subsequence,

ψn →Γ ψ sequentially w.r.t. the topology π of V×V ′, (27)

andStructural Stability

Let Dtun + αn(un) 3 hn in V ′, a.e. in ]0,T [, un(0) = 0 ∀n, (28)

un u in L2(0,T ; V ) ∩ H1(0,T ; V ′), hn → h in L2(0,T ; V ′). (29)

If (i) αn is represented by ψn, (ii) ψn →Γ ψ, (iii) α is represented by ψ, then

Dtu + α(u) 3 h in V ′, a.e. in ]0,T [, u(0) = 0. (30)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 76: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Global-in-Time Formulation

Let us set

Ψ(v) :=

∫ T

0

ψ(v(t)) dt ∀v ∈ V := L2(0,T ; V ).

Then, even if the injection V → V ′ is not compact, we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by Ψn,then there exists Ψ : V×V ′ → R such that, up to extracting a subsequence,

Ψn →Γ Ψ sequentially w.r.t. the topology π of V×V ′. (31)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 77: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Global-in-Time Formulation

Let us set

Ψ(v) :=

∫ T

0

ψ(v(t)) dt ∀v ∈ V := L2(0,T ; V ).

Then, even if the injection V → V ′ is not compact, we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by Ψn,then there exists Ψ : V×V ′ → R such that, up to extracting a subsequence,

Ψn →Γ Ψ sequentially w.r.t. the topology π of V×V ′. (31)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 78: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Global-in-Time Formulation

Let us set

Ψ(v) :=

∫ T

0

ψ(v(t)) dt ∀v ∈ V := L2(0,T ; V ).

Then, even if the injection V → V ′ is not compact, we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by Ψn,then there exists Ψ : V×V ′ → R such that, up to extracting a subsequence,

Ψn →Γ Ψ sequentially w.r.t. the topology π of V×V ′. (31)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 79: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Global-in-Time Formulation

Let us set

Ψ(v) :=

∫ T

0

ψ(v(t)) dt ∀v ∈ V := L2(0,T ; V ).

Then, even if the injection V → V ′ is not compact, we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by Ψn,then there exists Ψ : V×V ′ → R such that, up to extracting a subsequence,

Ψn →Γ Ψ sequentially w.r.t. the topology π of V×V ′. (31)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 80: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Global-in-Time Formulation

Let us set

Ψ(v) :=

∫ T

0

ψ(v(t)) dt ∀v ∈ V := L2(0,T ; V ).

Then, even if the injection V → V ′ is not compact, we have the following results:

Compactness

If αn : V → P(V ′) is equi-bounded, equi-coercive, and is represented by Ψn,then there exists Ψ : V×V ′ → R such that, up to extracting a subsequence,

Ψn →Γ Ψ sequentially w.r.t. the topology π of V×V ′. (31)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 81: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 82: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 83: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 84: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 85: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 86: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Structural Stability

LetDtun + αn(un) 3 hn in V ′, ∀n, un(0) = 0 ∀n, (32)

un u in V, Dtun Dtu in V ′, hn → h in V ′. (33)

If αn is represented by Ψn, Ψn →Γ Ψ, α is represented by Ψ, then

Dtu + α(u) 3 h in V ′, u(0) = 0. (34)

A priori this may be associated with the onset of long memory, i.e.,

u global-in-time solution 6⇒ u pointwise-in-time solution. (35)

However this implication holds under (mild) regularity assumptions.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 87: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 88: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 89: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 90: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 91: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 92: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 93: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 94: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 95: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Tartar’s example of onset of long memory

Let an ∈ L∞(Ω), an ≥ 0 ∀n. The short-memory equation

Dtun + an(x)un = 0 a.e. in Ω×R+ (36)

has the solution

un(x , t) = u(x , 0)e−an(x)t a.e. in Ω×R+. (37)

If an ∗ a in L∞(Ω), then as n→∞ the exponential form of the solution is lost;

indeed in this case

un u 6⇒ anun au in L2loc(Ω×R+). (38)

Actually, one getsDtu + Au = 0 a.e. in Ω×R+, (39)

with A linear and with long memory: [Au](·, t) 6= A[u(·, t)] in Ω,

i.e., [Au](·, t) depends on u|Ω×]0,t[ rather than just u|Ω×t.

(the operator A has the form of a time-integral, see e.g. [Tartar].)

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 96: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 97: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 98: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 99: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 100: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 101: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 102: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 103: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 104: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

What makes the difference

between the (nonlinear) structurally stable flow

Dtu −∇·β(x ,∇u) 3 0 in H−1(Ω), a.e. in ]0,T [ (40)

and the (linear) structurally unstable flow

Dtu + a(x)u = 0 in L2(Ω), a.e. in ]0,T [ (41)

(with β(x , ·) max-monotone, a(x) ≥ 0)

? ? ? ? ?

Linearity is not the key ... the explanation stays elsewhere:

In (40) u(t) ∈ V = H10 (Ω) ⊂⊂ H = L2(Ω),

in (41) u(t) ∈ V = H = L2(Ω).

The compactness of the injection V → H makes the difference !

The classical Trotter-Kato theorem actually applies to (40) and not to (41).

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 105: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

References

V. Chiado Piat, G. Dal Maso, A. Defranceschi: G-convergence of monotoneoperators, Ann. Inst. H. Poincare, Anal. Non Lineaire 7 (1990), 123–160

A. Pankov: G-Convergence and Homogenization of Nonlinear Partial DifferentialEquations. Kluwer, Dordrecht 1997

L. Tartar: The General Theory of Homogenization. A Personalized Introduction.Springer-U.M.I., Bologna, 2009

A.V.: Variational formulation and structural stability of monotone equations.Calc. Var. Partial Differential Equations 47 (2013) 273–317

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 106: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 107: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 108: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 109: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 110: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 111: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 112: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 113: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 114: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

Conclusions

1. Any max-monotone operator may be given a variational representation.

2. Operator compactness and structural stability of max-monotone equationsmay then be proved via Γ-convergence, w.r.t. a suitable (nonlinear) topology onV×V ′.

3. This applies e.g. to Dtu + α(u) 3 h (α max-monotone).

Further Issues

1. The onset of long memory in the limit.

2. The identification of the limit of sequences of representative operators.

3. The extension to other classes of operators, including homogenization problems.

4. The extension to α pseudo-monotone.

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 115: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 116: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 117: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 118: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 119: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 120: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows

Page 121: Structural Stability of Nonlinear Flows2. For max-monotone equations, operator compactness and structural stability may then be studied via De Giorgi’s -convergence. 3. This also

A Piece of Advertisement

Symposium on Trends in Applications of Mathematics to Mechanics

Poitiers, September 8-11, 2014

in the framework of the activity of the

International Society for the Interaction of Mechanics and Mathematics (ISIMM)

Organized by Alain Miranville (local organizer), with

Ulisse Stefanelli, Lev Truskinovsky, Augusto Visintin

Augusto Visintin - Trento Structural Stability of Nonlinear Flows