The Caﬁarelli-Kohn-Nirenberg Theorem - A Direct Proof by...

The Caffarelli-Kohn-Nirenberg Theorem -A Direct Proof by Campanato’s Method

Jorg Wolf

Abstract. In the present paper we give a generalized notion of a suitableweak solution to the Navier-Stokes equations in any domain by the aid of a

pressure representation of the form p = p0 +∂ph

∂t(p ∈ L4/3(0,∞, L2(Ω));

ph is harmonic with respect to the spatial variable x ∈ Ω). Then we provethat for any initial velocity a ∈ L2

σ(Ω) there exists a (generalized) suitableweak solution (u, p) in Ω×]0,∞[ satisfying the following property of partialregularity: The velocity field u is continuous in a neighbourhood of a point(x0, t0) ∈ Ω×]0,∞[ if

lim supR→0+

1

R

∫

QR(x0,t0)

∣∣∣rotu× u

|u|∣∣∣2

dx dt ≤ ε?

for a sufficiently small ε? > 0.

Mathematics Subject Classification (2000). 35Q30, 35D10.

Keywords. Navier-Stokes equations, partial regularity.

1. Introduction. Statement of the Main Result

The aim of the present paper is the study of the local regularity of weak solutionsto an incompressible Newtonian fluid in a general three-dimensional domain Ωwhich is governed by the following two equations, first the by the conservation ofvolume

div u = 0 in Ω×]0,∞[ 1 , (1.1)

and second by the conservation of momentum

∂u∂t

+ (u · ∇)u−∆u = −∇p in Ω×]0,∞[, (1.2)

2 J. Wolf

where u = (u1, u2, u3) denotes the unknown velocity field and p the unknownpressure. This two equations are completed by the condition of adherence

u = 0 on ∂Ω×]0,∞[, (1.3)

and the initial condition

u(0) = a on Ω× 0, (1.4)

where a = (a1, a2, a3) denotes a given distribution of the velocity at the initialtime t = 0.

The equations (1.1)-(1.4) are called the Navier-Stokes equations which whereintroduced first by Navier [14] and later rederived by Stokes [20]. The rigorousmathematical treatment started with Leray [11], where he studied the existenceof weak solutions to (1.1)-(1.4) for the case Ω = R3 (for the notion of a weaksolutions see below). Later Hopf [8] proved the existence of weak solutions in ageneral domain by using a Galerkin approximation. For more details and furtherapproaches we refer to the monographs of Temam [23] and Sohr [18].

However despite of many efforts until now one is unable to construct a classi-cal solution to the Navier-Stokes equations for an arbitrarily given smooth initialvelocity, which has been considered as one of the seven Millennium problems in-troduced by the Clay institute. Concerning the issue of regularity the best resultwhich is known is the so-called Caffarelli-Kohn-Nirenberg theorem (cf. [4]), whichstates that the velocity field u of a suitable weak solution to the Navier-Stokesequations is bounded on a neighbourhood of (x0, t0) ∈ Q, where Q := Ω×]0,∞[, if

lim supR→0+

1R

∫

QR(x0,t0)

|∇u|2 dxdt ≤ ε0 (1.5)

for a sufficiently small ε0 > 0. This shows that the singular set Σ of all points(x0, t0) ∈ Q, where (1.5) fails is closed such that

P1(Σ) = 0. (1.6)

In fact, the first result of partial regularity is due to Scheffer. In his pioneeringpaper [15] he has introduced the notion of a suitable weak solution to (1.1)-(1.4)that is a pair (u, p) with

u ∈ L∞(0,∞;L2(Ω)3) ∩ L2(0,∞; W 1, 2σ (Ω)) 2 ,

p ∈ L3/2(Ω×]0,∞[)

satisfying (1.2) in sense of distribution, such that the local energy inequality∫

Ω

|u(t)|2φ dx + 2∫ t

0

∫

Ω

|∇u|2φ dxdt

≤∫ t

0

∫

Ω

|u|2∂φ

∂t+ ∆φ

dxdt +

∫ t

0

∫

Ω

(|u|2 + 2p)u · ∇φdxdt (1.7)

1 Throughout by bold letters we denote vector valued functions.2 For the definition of W 1, 2

σ (Ω) see below.

Partial Regularity 3

holds for all φ ∈ C∞0 (Q), for almost all t ∈]0,∞[. For such suitable weak solutionsthe author defined a set Σ of all possible singular points by a condition differentthen (1.5), proving that

H5/3(Σ) = 0. (1.8)(for details see [15], [16]).

Later Lin [13] reproved the Caffarelli-Kohn-Nirenberg theorem by using pres-sure estimates obtained by Sohr and vonWahl [19]. Moreover, Ladyzenskaya andSeregin [9] have given a more detailed proof of the partial regularity of a suitableweak solution to the Navier-Stokes equations including the case when a force fis added to the right of (1.2). Recently, Vasseur [24] established a new proof ofthe Caffarelli-Kohn-Nirenberg theorem using a direct approach based on a Moseriteration.

In contrast to the results mentioned above the partial regularity of weaksolutions to the Navier-Stokes equations in a general domain Ω is not known.Since in this case the pressure does exist only in a distributional sense one cannotuse the classical concept of a suitable weak solution. However as we will see belowwe are able to treat even this case using a pressure representation of the form

p = p0 +∂ph

∂t, where p0 denotes a Lebesgue measurable function in Q, while

ph is a harmonic function with respect to x ∈ Ω. This representation has beenintroduced first in [25].

Weak solutions. First let us introduce the function spaces and some notationswhich we use in what follows. By Wm, q(Ω), Wm, q

0 (Ω) (m = 1, 2, . . . ; 1 ≤ q < ∞)we denote the usual Sobolev space. By Dσ(Ω) we denote the vector space of allϕ ∈ C∞0 (Ω)3 with div ϕ = 0. Then define

L2σ(Ω) := closure of Dσ(Ω) in L2(Ω)3,

W 1, 2σ (Ω) := closure of Dσ(Ω) in W 1,2(Ω)3.

Given a normed vector space X with norm ‖ · ‖, we denote by Ls(a, b; X) (1 ≤s ≤ ∞) (−∞ ≤ a < b ≤ ∞) the vector space of all Bochner measurable functionsz :]a, b[→ X such that

∫ b

a

‖z(t)‖s dt < ∞ if 1 ≤ s < ∞, ess sup]a,b[

‖z(t)‖ < ∞ if s = ∞.

(see, e.g., [18; Chap. IV,1] for details).

Definition 1.1 Let a ∈ L2σ(Ω). A vector function u : Q → R3 is called a weak

solution to (1.1) -(1.4) if

u ∈ L2(0,∞; W 1, 2σ (Ω)) ∩ L∞(0,∞; L2

σ(Ω))

and there holds the following integral identity∫

Q

− u · ∂tϕ + (u · ∇)u ·ϕ +∇u : ∇ϕ

dxdt =

∫

Ω

a ·ϕ(0) dx (1.9)

4 J. Wolf

for all ϕ ∈ C∞0 ([0,∞[;Dσ(Ω))3 .

Remark 1.1. Let u : Q → R3 be a weak solution to (1.1) -(1.4). Then u can beredefined on a set of Lebesgue measure 0 such that u ∈ Cw([0,∞[;L2

σ(Ω)), i.e. forall t > 0 there holds u(t) ∈ L2

σ(Ω) such that

lims→t

∫

Ω

u(s) · ξ dx =∫

Ω

u(t) · ξ dx ∀ ξ ∈ L2σ(Ω).

In addition, we have

u(t) → a in L2σ(Ω) as t → 0+.

By the following definition we introduce the notion of a suitable weak solutionwhich differs from the one introduced in [15], [4], using the distributive pressureonly.

Definition 1.2 A weak solution u : Q → R3 to (1.1)-(1.4) is called a suitableweak solution if there exist p0 ∈ L4/3(0,∞; L2(Ω)) and ph ∈ C(Q) with ∆ph = 0,such that the function v := u +∇ph satisfies the identity

∫

Q

− v · ∂ϕ

∂t+ (u · ∇)u ·ϕ +∇v : ∇ϕ

dxdt =

∫

Q

p0 div ϕdx dt (1.10)

for all ϕ ∈ C∞0 (Q)3 together with the local energy inequality∫

Ω

|v(t)|2φ(t) dx + 2∫ t

0

∫

Ω

|∇v|2φdx ds

≤∫ t

0

∫

Ω

|v|2∂φ

∂t+ ∆φ

dxds +

∫ t

0

∫

Ω

(|u|2 + 2p0)v · ∇φ dxds (1.11)

+ 2∫ t

0

∫

Ω

∇ph × rotv · vφ dxds

for almost all 0 < t < ∞ and for all nonnegative functions φ ∈ C∞0 (Q).

Remark 1.2. As we will see in Section 2 for every weak solution u to the Navier-

Stokes equations there exists a pressure p0 +∂ph

∂t, where p0 ∈ L4/3(0,∞;L2(Ω))

and ph ∈ C(Q) being harmonic with respect to x ∈ Ω. Setting p :=|u|22

+p0 using

(u · ∇)u =12∇|u|2 − rotu× u the identity (1.10) turns into

∫

Q

− v · ∂ϕ

∂t+ (u · ∇)u ·ϕ +∇v : ∇ϕ

dx dt

=∫

Q

p div ϕ + rotu× u

dxdt (1.12)


for all ϕ ∈ C∞0 (Q)3. In addition, by virtue of Sobolev’s embedding theorem alongwith multiplicative inequalities one finds

L2(0,∞; W 1, 2σ (Ω)) ∩ L∞(0,∞;L2

σ(Ω)) ⊂ L8/3(0,∞;L4(Ω)3).

Consequently,

p ∈ L4/3(0,∞; L2(Ω)).

Our first main result is related to the existence of a suitable weak solutionto (1.1)-(1.4).

Theorem 1.3 (Main Theorem 1). For every initial velocity a ∈ L2σ(Ω) there exists a

suitable weak solution u to the Navier-Stokes equations (1.1)-(1.4) with a pressure

p = p0 +∂ph

∂taccording to Definition 1.2.

The proof Theorem 1.3 is given in Section 7 below.

Notations Let x0 ∈ R3, t0 ∈ R and 0 < R < ∞. We define the ball

BR(x0) :=

x ∈ R3∣∣∣ |x− x0| < R

and the parabolic cylinder

QR(x0, t0) := BR(x0)×]t0 −R2, t0[.

Our second main result concerns the partial regularity of suitable weak solu-tions to the Navier-Stokes equations.

Theorem 1.4 (Main Theorem 2). There exists an absolute constant ε? > 0, suchthat every suitable weak solution u to (1.1)-(1.4) is continuous in Q \ sing (u),where

sing (u) :=

(x0, t0) ∈ Q

∣∣∣∣ lim supR→0+

1R

∫

QR(x0,t0)

∣∣∣rotu× u|u|

∣∣∣2

dxdt > ε?

is a closed subset of Q with P1(sing (u)) = 0.

The proof of Theorem1.4 essentially relies on the following two propositions

Proposition 1.5. There exists an absolute constant k? > 0 such that if u is a

suitable weak solution to (1.1)-(1.4) with corresponding pressure p = p0 +∂ph

∂tthen there holds

Ξ(x0, t0; τR)

≤ k?(1 + R2M(x0, t0; R))[τ2 + τ−3Θ(x0, t0; R)

]Ξ(x0, t0; R) (1.13)

6 J. Wolf

for every 0 < τ <14, for all (x0, t0) ∈ Q and 0 < R < mindist(x0, ∂Ω),

√t0,

where

Ξ(x0, t0; ρ) :=

:=1ρ

(‖∇v‖2L2(Qρ(x0,t0))

+ ‖v‖2L10/3(Qρ(x0,t0))+ ‖v‖2L∞(t0−ρ2,t0;L2(Bρ(x0)))

)

+1ρ2‖p− pBρ

‖2L4/3(t0−ρ2,t0;L2(Bρ(x0))),

M(x0, t0; ρ) := max(x,t)∈Qρ

|∇ph(x, t)|2,

Θ(x0, t0; ρ) := ρ2M(x0, t0; ρ) +1ρ

∫

Qρ(x0,t0)


∣∣∣2

dxdt,

0 < ρ ≤ mindist(x0, ∂Ω),√

t0.Proposition 1.6. There exists an absolute constant k? > 0 such that if u is a

suitable weak solution to (1.1)-(1.4) with pressure p = p0 +∂ph

∂tthen there holds

Ξ(x0, t0; τR) ≤ k?

[τ3 + τ−3O(x0, t0; R)

]Ξ(x0, t0; R) (1.14)

for every 0 < τ <14, for all (x0, t0) ∈ Q and 0 < R < mindist(x0, ∂Ω),

√t0,

where

Ξ(x0, t0; ρ) :=1ρ‖∇v‖2L2(Qρ(x0,t0))

+1ρ2‖p− pBρ‖2L4/3(t0−ρ2,t0;L2(Bρ(x0)))

,

O(x0, t0; ρ) := ρ2M(x0, t0; ρ) +1ρ‖v‖2L4(t0−ρ2,t0;L3(Bρ(x0)))

,

0 < ρ ≤ mindist(x0, ∂Ω),√

t0.Remark 1.7. 1. By Main Theorem 1 and Main Theorem 2 we improve the theknown results concerning the interior regularity of weak solutions to the Navier-Stokes equations in the following sense

• Existence of a weak solution to the Navier-Stokes equations in an arbitrar-ily given domain Ω satisfying an analogous property of partial regularity asstated in the Caffarelli-Kohn-Nirenberg theorem;

• The boundeness of u in a neighbourhood of a regular point (x0, t0) can bereplaced by the continuity of u, which seems to be the best possible regularitywith respect to the time variable;

• Replacing the condition (1.5) by the more physical condition

lim supR→0+

1R

∫

QR(x0,t0)


∣∣∣2

dxdt < ε? (1.15)

improves the estimate of the singular set;


• The direct method based on Prop. 1.5 and Prop. 1.6 simplifies the proof andin addition enables us to specify the numerical value ε? > 0 for the definitionof the singular set (see (1.15) above).

2. The statement of Theorem 1.4 continues to hold if the right hand side of(1.2) is modified by adding a body force − div f with

f ∈ L2,λ(Q)9, λ > 3 3 .

The paper is organized as follows. In Section 2 we prove the existence of

a pressure p = p0 +∂ph

∂t, which forms the basis for proving the existence of a

(generalized) suitable weak solution according to Definition 1.2. In Section 3 weestablish fundamental estimates for functions being almost caloric which will beused for both the proof of Prop. 1.5 and Prop. 1.6. In Section 4 we provide presentthe appropriate model system together with a-priori estimates for weak solutionsto such systems in a given cylinder QR(x0, t0). The subject of Section 5 is theproof of Proposition 1.5. and the proof of Prop. 1.6. These fundamental estimateswill be achieved after having established a Caccioppoli-type inequality combinedwith a Campanato-type estimates for the pressure p along with the Campanato-type estimates for semi caloric functions and the estimates established for themodel system. The proof of the Main Theorem 2 will be completed in Section 6using a standard iteration procedure. In Section 7 we construct a suitable weaksolution to the Navier-Stokes equations according to Definition 1.2 by first solvingan approximated Navier-Stokes system and afterwards passing to limit with theparameter of approximation. Finally, in the appendix of the paper we list severallemmas which we have used throughout the paper.

2. Existence of pressure

In the present section we provide the main tool for introducing a pressure written

in the form p = p0 +∂ph

∂tfor a weak solutions to the Navier-Stokes equations and

related systems. Since this method can be applied for more general systems it isfor interest itself.

For a given domain G ⊂ R3 define

A2(G) :=

∆u∣∣∣ u ∈ W 2, 2

0 (G)

, B2(G) :=

p ∈ L2(G)∣∣∣ ∆p = 0 in G

.

Hereby W 2, 20 (G) denotes the closure of C∞0 (G) with respect to the norm

‖φ‖W 2, 20 (G) :=

( ∫

G

|∆φ|2 dx

) 12

.

3 Here L2,λ(Q) stands for the usual parabolic Campanato space (cf. [6]).

8 J. Wolf

Clearly, both A2(G) and B2(G) are closed subspaces of L2(G). Using Weyl’slemma one sees that B2(G) is the orthogonal complement of A2(G) in L2(G).Hence we have

L2(Ω) = A2(Ω) ⊕ B2(Ω). (2.1)Consequently, for every p ∈ L2(G) there exist unique p0 ∈ A2(G) and ph ∈ B2(G)such that p = p0 + ph and

‖p0‖2L2(G) + ‖ph‖2L2(G) = ‖p‖2L2(G). (2.2)

Next, noticing that

‖∇2φ‖L2(G) = ‖φ‖W 2, 20 (G) ∀φ ∈ W 2, 2

0 (G)

there holds the following

Lemma 2.1. Let v∗ ∈ W−1, 2(Ω)3 and p0 ∈ A2(G) such that

〈v∗,∇φ〉 =∫

G

p0∆φdx ∀φ ∈ C∞0 (G).

Then‖p0‖L2(G) ≤ ‖v∗‖W−1, 2(Ω). (2.3)

¤Our main objective of this section is the construction of a representation of

the non-stationary pressure given by the following theorem.

Theorem 2.2. Let u ∈ Cw([0,∞[;L2σ(Ω)) and h ∈ L1

loc([0,∞[;L2(Ω)), such that∫

Q

−u · ∂tϕ + h : ∇ϕdxdt = 0 ∀ϕ ∈ C∞0 (Q)3 with div ϕ = 0. (2.4)

Let B ⊂⊂ Ω. Then there exist functions

p0 ∈ L1loc([0,∞[; A2(Ω)),

ph ∈ C(Ω× [0,∞[) with ∆ph = 0 and (ph)B = 0 4 ,

such that∫

Q

(−u · ∂tϕ + h : ∇ϕ) dx dt

=∫

Q

(p0 div ϕ +∇ph · ∂tϕ) dxdt +∫

Ω

u(0) ·ϕ(0) dx (2.5)

for all ϕ ∈ C∞(Q)3 with supp(ϕ) ⊂⊂ Ω× [0,∞[. In addition, there holds

‖p0(t)‖L2(Ω) ≤ ‖h(t)‖L2(Ω) (2.6)

4 Let A ⊂ R3 be a Lebesgue measurable set having finite measure. Then for f ∈ L1(A) we definethe mean value

(f)A =

∫−A

f dx :=1

|A|∫

Af dx.


for almost all t ∈]0,∞[ and for each Lipschitz domain G ⊂⊂ Ω there exists aconstant cG, such that

‖ph(t)‖L2(G) ≤ cG

‖u(t)− u(0)‖L2(Ω) +

∥∥∥∥∫ t

0

h(s) ds

∥∥∥∥L2(Ω)

(2.7)

for almost all t ∈]0,∞[.

The proof of Theorem 2.2 relies essentially on the following lemma

Lemma 2.3. Let B ⊂⊂ Ω be a fixed domain. Let u ∈ L2σ(Ω) and v∗ ∈ W−1, 2(Ω)3

such that ∫

Ω

u ·ϕdx + 〈v∗, ϕ〉 = 0 ∀ϕ ∈ Dσ(Ω). (2.8)

Then there exist unique functions p0 ∈ A2(Ω) and ph ∈ C∞(Ω) with ∆ph = 0 and(ph)B = 0, such that∫

Ω

u ·ϕdx + 〈v∗, ϕ〉 =∫

Ω

(p0 + ph) div ϕdx ∀ϕ ∈ C∞0 (Ω)3. (2.9)

In addition, we have‖p0‖L2(Ω) ≤ ‖v∗‖W−1, 2(Ω). (2.10)

and for every Lipschitz domain G ⊂⊂ Ω there exists a constant cG such that

‖ph‖L2(Ω) ≤ cG(‖u‖L2(Ω) + ‖v∗‖W−1, 2(Ω)). (2.11)

Proof. Let B ⊂ Ω1 ⊂ Ω2 ⊂ . . . ⊂ Ωk ⊂⊂ Ω be sequence of open subsets withLipschitz boundary, such that

∞⋃k=1

Ωk = Ω.

For each k ∈ N we define w∗k ∈ W−1, 2(Ωk) by

〈w∗k,ϕ〉 :=

∫

Ωk

u ·ϕdx + 〈v∗, ϕ〉, ϕ ∈ W 1, 2(Ωk)3.

Clearly, by (2.8) one easily verifies w∗k ∈ W 1, 2

σ (Ωk)⊥. According to [7; Chap. III,Th. 3.1, Th. 5.2] there exist pk ∈ L2(Ωk), such that

〈w∗k, ϕ〉 =

∫

Ωk

pk div ϕdx ∀ϕ ∈ C∞0 (Ωk). (2.12)

Once more consulting [7] shows that for every ` ∈ N, ` ≤ k there holds

‖pk − (pk)Ω`‖L2(Ω`) ≤ c` sup

0 6=ϕ∈C∞0 (Ω`)

〈w∗k, ϕ〉

‖∇ϕ‖L2(Ω`)

≤ c` (‖u‖L2(Ω) + ‖v∗‖W−1, 2(Ω)) 5. (2.13)

5 Remark that for any bounded domain G ⊂ R3 with Lipschitz boundary there holds

‖ϕ‖L2(G) ≤ c‖∇ϕ‖L2(G) ∀ϕ ∈ C∞0 (G),

with a constant c > 0 depending on G only.

10 J. Wolf

Next, one finds unique functions p0,k ∈ A2(Ωk) and ph,k ∈ B2(Ωk) such that

pk = p0,k + ph,k in Ωk.

In what follows let us extend pk, p0,k and ph,k outside of Ωk by 0 without changing

the notations. In particular, we have p0,k ∈ A2(Ω). Noticing that∫

Ωk

u ·∇φ dx = 0

for all φ ∈ C∞0 (Ωk) from the identity (2.12) it follows

〈v∗,∇φ〉 =∫

Ωk

p0,k∆φ dx ∀φ ∈ C∞0 (Ωk).

Then by the aid of Lemma 2.1 one infers

‖p0,k‖L2(Ω) ≤ ‖v∗‖W−1, 2(Ω). (2.14)

Let ` ∈ N be fixed. Then using Jensen’s inequality along with (2.13) and(2.14) one gets for every k ≥ `

‖ph,k − (ph,k)B‖L2(Ω`) ≤ 2|Ω`||B| ‖ph,k − (ph,k)Ω`

‖L2(Ω`)

≤ c|Ω`||B| (‖p0,k‖L2(Ω) + ‖pk − (pk)Ω`

‖L2(Ω`))

≤ c|Ω`||B| (‖u‖L2(Ω) + ‖v∗‖W−1, 2(Ω)), (2.15)

where c = const independent on k ∈ N. Hence, observing (2.14) and (2.15) bymeans of reflexivity using Cantor’s diagonalization principle one can choose asequence kj in N such that

p0,kj p0 in A2(Ω), (2.16)

ph,kj − (ph,kj )B ph in L2loc(Ω) as j → +∞. (2.17)

By the aid Banach-Steinhaus’ theorem from (2.14) one immediately gets (2.10). Onthe other hand, given a Lipschitz domain G ⊂⊂ Ω by means of Banach-Steinhaus’theorem taking into account (2.15) yields

‖ph‖L2(G) ≤ cG(‖u‖ + ‖v∗‖W−1, 2(Ω)) (2.18)

with a constant cG = const depending on G only. Whence (2.11).Let ϕ ∈ C∞0 (Ω)3. There exists N(ϕ) ∈ N such that supp(ϕ) ∈ Ωk for all

k ≥ N(ϕ). Then for kj ≥ N(ϕ) we have∫

Ω

u ·ϕ dx + 〈v∗, ϕ〉 = 〈w∗kj

, ϕ〉 =∫

Ω

(p0,kj + ph,kj ) div ϕdx.

Using (2.16) and (2.17) carrying out the passage to the limit j → +∞ from theidentity above one gets (2.9).

Uniqueness. Let p0 ∈ A2(Ω) and ph ∈ L2loc(Ω) with ∆ph = 0 and (ph)B = 0,

such that ∫

Ω

(p0 + ph) div ϕdx = 0 ∀ϕ ∈ C∞0 (Ω)3.


Then, p0 + ph = C0 = const in Ω. Hence, p0 = C0 − ph ∈ A2(Ω) ∩ B2(Ω), whichimplies 0 = p0 = C0 − ph. Finally, by virtue of (ph)B = 0 one finds C0 = 0 andtherefore p0 + ph = 0. ¤

Proof of Theorem2.2 Arguing as in [25] one finds∫

Ω

(u(t)− u(0)) ·ψ + h(t) : ∇ψ dx = 0 ∀ψ ∈ C∞0,σ(Ω), (2.19)

where

h(t) :=∫ t

0

h(s) ds, t ∈ [0,∞[.

According to Lemma2.2 there exists unique functions p0(t) ∈ A2(Ω) andph(t) ∈ C∞(Ω) with ∆ph(t) = 0 and (ph(t))B = 0, such that

∫

Ω

(u(t)− u(0)) ·ψ + h(t) : ∇ψ dx =∫

Ω

(p0(t) + ph(t)) div ψ dx (2.20)

for all ψ ∈ C∞0 (Ω)3.Now into (2.20) replacing ψ by ∇φ, where φ ∈ C∞0 (Ω) one arrives at

∫

Ω

p0(t)∆φdx =∫

Ω

h(t) : ∇2φdx ∀φ ∈ W 2, 20 (Ω). (2.21)

Next, let 0 ≤ t < s ≤ ∞ be arbitrarily chosen. Then from (2.21) by the aidof Lemma 2.1 and Minkowski’s inequality one estimates

‖p0(t)− p0(s)‖L2(Ω) ≤ ‖h(t)− h(s)‖L2(Ω) ≤∫ s

t

‖h(%)‖L2(Ω) d%. (2.22)

By means of the absolutely continuity of the Lebesgue integral it follows thatp0 ∈ C([0,∞[;L2(Ω)).

Let G ⊂⊂ Ω be arbitrarily chosen. Define

pG(t) := p0(t) + ph(t)− (p0(t) + ph(t))G in G.

Then (2.20) turns into∫

Ω

(u(t)− u(0)) ·ψ + h(t) : ∇ψ dx =∫

Ω

pG(t) div ψ dx (2.23)

for all ψ ∈ C∞0 (G)3.As in [25] one can verify that pG ∈ Cw([0,∞[;L2(G)). Notice that p0 ∈

C([0,∞[; L2(Ω)) yields p0 − (p0)G ∈ C([0,∞[;L2(G)) and therefore ph − (ph)G ∈Cw([0,∞[; L2(G)). By virtue of Lemma A.1 (cf. appendix below) the function ph−(ph)G must be continuous in G×[0,∞[. In addition, recalling pB = 0 the statementabove with G = B shows that ph is continuous in B × [0,∞[ . According toLemmaA.2 one obtains ph ∈ C(Ω × [0,∞[). Since at time t = 0 the right handside of (2.20) vanishes we have ph(0) = 0.

Since p0 ∈ W 1, 1loc ([0,∞[; A2(Ω)) (cf. [25]) the weak derivative p0 := ∂tp0

belongs to ∈ L1loc([0,∞[; A2(Ω)). Let ϕ ∈ C∞(Q)3 with supp(ϕ) ⊂⊂ Ω × [0,∞[.

12 J. Wolf

Then into (2.20) inserting ψ = ∂tϕ(t) (0 < t < ∞) integrating over ]0,∞[ andusing integration by parts implies (2.5).

To verify (2.6) we proceed as follows. Let η ∈ C∞0 (0,∞) and φ ∈ C∞0 (Ω) bearbitrarily chosen. Then from (2.5) with ϕ(x, t) = η(t)∇φ(x) using integration byparts yields

∫ ∞

0

η(t)∫

Ω

h(x, t) : ∇2(x)φdxdt =∫ ∞

0

η(t)∫

Ω

p0(x, t) : ∆φ(x) dxdt.

Now, if t ∈]0,∞[ is a Lebesgue point of both h and p0 with respect to the timevariable then from the identity above it follows

∫

Ω

h(t) : ∇2φ dx =∫

Ω

p0(t) : ∆φ dx ∀φ ∈ C∞0 (Ω).

Owing to Lemma 2.1 one gets the desired inequality (2.6). On the other hand,from (2.20) using (2.11) (cf. Lemma 2.2) one easily verifies (2.7). This completesthe proof of the lemma. ¤

Remark 2.4. The representation of the pressure introduced in Theorem 2.2 isunique in the following sense. If q0 ∈ L1

loc([0,∞[;A2(Ω)) and qh ∈ L1loc(Ω× [0,∞[)

with ∆qh = 0 and (qh)B = 0 are functions fulfilling the identity (2.5) with q0 in-stead of p0 and with qh instead of ph, then q0 = p0 and qh = ph almost everywherein Q. Indeed, taking the difference between this two identities yields

∫

Q

(p0 − q0) div ϕdxdt = −∫

Q

∇(ph − qh) · ∂tϕdxdt

for all ϕ ∈ C∞(Q)3 with supp(ϕ) ⊂⊂ Ω × [0,∞[. Arguing as above recalling thedefinition of the space A2(Ω) shows that p0 = 0. Thus, the identity above turnsinto∫ ∞

0

η′(t)∫

Ω

∇(ph(t)− qh(t)) ·ψ dxdt = 0 ∀ψ ∈ C∞0 (Ω), ∀ η ∈ C∞0 ([0,∞[),

which leads to ∇ph(·, t) = 0 for almost all t > 0. This shows that ph(·, t) = constfor almost all t > 0. Additionally, having (ph − qh)B = 0 implies ph = 0 almosteverywhere in Q.

3. Fundamental estimates for semi caloric functions

Definition 3.1 A function U ∈ L2(Q1)3 is said to be semi caloric if there existsq ∈ L1(−1, 0; B2(B1)), such that

∂U∂t

−∆U = −∇q in Q1 (3.1)

in sense of distributions. Then q is called the caloric pressure related to U.

Let us start with the following Caccioppoli-type inequality.


Lemma 3.1. Let U ∈ L2(Q1)3×L1(−1, 0; B2(B1)) be a semi caloric function withits caloric pressure q. Then ∇mU ∈ L2

loc(B1×] − 1, 0])3 for all m ∈ N and thereexists a constant c1 depending on m ∈ N only such that

‖∇mU‖2L∞(−1/4,0;L2(B1/2))≤ c1

‖U‖2L2(Q1)

+ ‖q‖2L1(−1,0;L2(B1))

. (3.2)

Proof. By using a standard mollification argument one easily proves

U ∈ L∞(−ρ2, 0; Wm, 2(Bρ)3),∂U∂t

∈ L1(−ρ2, 0; Wm, 2(Bρ)3), m = 1, 2, . . .

for all 0 < ρ < 1. Furthermore for each multi-index α = (α1, α2, α3) there holds

∂DαU∂t

−∆DαU = −∇Dαq in Q1. (3.3)

Fix m ∈ N. Let φ ∈ C∞0 (B1×] − 1, 0]) be a cut-off function with 0 ≤ φ ≤ 1in Q1 and φ ≡ 1 on Q1/2. Multiplying (3.3) by φ2m+2DαU summing up over|α| = m (m ∈ N) integrating over B1×]− 1, t[ (t ∈]− 1, 0[) applying integration byparts along with Cauchy-Schwartz’s and Young’s inequality yields

∫

B1

φ2m+2(x, t)|∇mU(x, t)|2 dx + 2∫ t

−1

∫

B1

φ2m+2|∇m+1U|2 dxds

≤∫ t

−1

∫

B1

( ∂

∂t+ ∆

)(φ2m+2)|∇mU|2 dxds

+ 2

∫ t

−1

( ∫

B1

φ2m+2|∇m+1q|2 dx

)1/2

ds

2

+12

ess sups∈]−1,0[

∫

B1

φ2m+2(x, s)|∇mU(x, s)|2 dx.

On the other hand, recalling that ∆q = 0 using integration by parts implies∫

B1

φ2m+2(x, s)|∇m+1q(x, s)|2 dx = 2−(m+1)

∫

B1

(∆m+1φ2m+2)(x, s)q2(x, s) dx

f.a.a. s ∈]− 1, 0[. Combining the last two statements shows that

ess supt∈]−1,0[

∫

B1

φ2m+2(x, t)|∇mU(x, t)|2 dx +∫

Q1

φ2m+2|∇m+1U|2 dxdt

≤ c

∫

Q1

φ2m|∇mU|2 dxdt + c

∫ 0

−1

( ∫

B1

q2 dx

)1/2

dt

2

,

14 J. Wolf

where c = const depending on m only. Iterating this inequality m-times gives

ess supt∈]−1,0[

∫

B1

φ2m+2(x, t)|∇mU(x, t)|2 dx

≤ c

∫

Q1

|U|2 dxdt + c

∫ 0

−1

( ∫

B1

q2 dx

)1/2

dt

2

,

with c = const depending only on m. This completes the proof of (3.2) ¤

We are now in a position to prove the following fundamental estimate

Theorem 3.2. Let U ∈ L2(Q1)3 be a semi caloric function with its caloric pressureq ∈ L1(−1, 0; L2(B1)). Then

‖U‖2L2(Qτ ) ≤ c2τ5‖U‖2L2(Q1)

+ ‖q‖2L1(−1,0;L2(B1))

(3.4)

for every 0 < τ < 1 with an absolute constant c2 > 0.Additionally, if q ∈ L4/3(−1, 0; L2(B1)) then

‖U−UQτ‖2L2(Qτ ) ≤ c3τ

6‖U−UQ1‖2L2(Q1)

+ ‖q‖2L4/3(−1,0;L2(B1))

(3.5)

for every 0 < τ < 1 with an absolute constant c3 > 0.

Proof. Clearly both fundamental estimates are trivially fulfilled for12≤ τ < 1.

Now, fix 0 < τ <12. Then using Sobolev’s embedding theorem together with (3.2)

(with m = 2) yields

‖U‖2L2(Qτ ) ≤ |B1| τ5‖U‖2L∞(Q1/2)

≤ cτ5 ess supt∈]−1/4,0[

∫

B1/2

|∇2U(x, t)|2 dx

≤ cτ5

∫

Q1

|U|2 dxdt + cτ5

∫ 0

−1

( ∫

B1

q2 dx

)1/2

dt

2

6 .

Whence, (3.4).In order to verify (3.5) we first make use of the Poincare-type inequality

(A.10) (cf. the appendix below) to get

‖U−UQτ ‖2L2(Qτ )

≤ cτ2‖∇UQτ ‖2L2(Qτ ) + τ2c

∫ 0

−τ2

( ∫

Bτ

|∇q|2 dx

)1/2

dt

2

. (3.6)

6 In what follows c denotes an absolute positive constant, which value may change from line toline and will be specified if necessary.


As above by using Sobolev’s embedding theorem one infers

‖∇UQτ‖2L2(Qτ ) ≤ cτ5 ess sup

t∈]−1/4,0[

∫

B1/2

|∇3U(x, t)|2 dx.

Estimating the right hand side of this inequality by the aid of Lemma 3.1 withm = 3, replacing U by U−UQ1 therein, leads to

‖∇UQτ ‖2L2(Qτ )

≤ cτ5

∫

Q1

|U−UQ1 |2 dxdt + cτ5

∫ 0

−1

( ∫

B1

q2 dx

)1/2

dt

2

. (3.7)

Next, thanks to ∆q = 0 using Caccioppoli’s inequality and Holder’s inequalityyields

∫ 0

−τ2

( ∫

Bτ

|∇q|2 dx

)1/2

dt

2

≤ cτ4

∫ 0

−1

‖∇q(t)‖4/3L∞(B1/2)

dt

3/2

≤ cτ4

∫ 0

−1

( ∫

B1

q2 dx

)2/3

dt

3/2

. (3.8)

Finally estimating the right hand side of (3.6) from above by (3.7) and (3.8)completes the proof of (3.5). ¤

4. Estimates for weak solutions to the model system

Let u be a suitable weak solution to (1.1)-(1.4) with pressure p represented by

p0 +∂ph

∂t. Set v := u +∇ph and p :=

|u|22

+ p0. As we have stated in Remark 1.2

the functions v and p satisfy the identity (1.12). In addition, there holds the localenergy inequality (1.11).

Let (x0, t0) ∈ Q and 0 < R < mindist(x0, ∂Ω),√

t0 be fixed. Recalling theorthogonal decomposition L2(BR(x0)) = A2(BR(x0))⊕B2(BR(x0)), there exist

p0,R ∈ L4/3(0,∞; A2(BR(x0))),

ph,R ∈ L4/3(0,∞; B2(BR(x0)))

such that

p(t)− pBR(x0)(t) = p0,R(t) + ph,R(t) in BR(x0) f.a.a. t ∈]0,∞[.

Clearly, for almost all t ∈]0,∞[ we have

u(t)× rotu(t) ∈ L6/5(BR(x0))3, p0,R(t) ∈ A2(BR(x0)) (4.1)

Letting t ∈]0,∞[ with (4.1) there holds∫

BR(x0)

p0,R(t)∆φ dx =∫

BR(x0)

u(t)× rotu(t) · ∇φdx (4.2)

16 J. Wolf

for all φ ∈ C∞0 (BR(x0)) (cf. Theorem 2.2 above). Then, by the aid of Lemma A.3one gets p0,R(t) ∈ W 1, 6/5(BR(x0)) together with the estimate

‖∇p0,R(t)‖4/3

L6/5(BR(x0))≤ c‖rotu(t)× u(t)‖4/3

L6/5(BR(x0)). (4.3)

In particular, the function t 7→ ‖rotu(t)×u(t)‖4/3

L6/5(BR(x0))belonging to L1(0,∞)

one gets∇p0,R ∈ L4/3(0,∞; L6/5(BR(x0))3).

Moreover, the inequality (4.3) implies

‖∇p0,R‖2L4/3(t0−R2,t0;L6/5(BR(x0)))≤ c4‖rotu× u‖2L4/3(t0−R2,t0;L6/5(BR(x0)))

(4.4)

with an absolute constant c4 > 0.In order to estimate the right hand side of (4.4) we will proceed in two

different ways.Firstly, using Holder’s inequality, recalling that u = v +∇ph yields

‖rotu(t)× u(t)‖4/3

L6/5(BR(x0))

≤( ∫

BR(x0)

∣∣∣rotu(t)× u(t)|u(t)|

∣∣∣6/5

(|v|+ |∇ph|)6/5 dx

) 109

≤ c∥∥∥rotu(t)× u(t)

|u(t)|∥∥∥

4/3

L2(BR(x0))‖v(t)‖4/3

L3(BR(x0))

+ cR2[M(x0, t0; R)]2/3 ‖∇v(t)‖4/3L2(BR(x0))

(4.5)

for almost all t ∈]t0 − R2, t0[. Integrating both sides of (4.5) over the interval]t0 −R2, t0[ once more using Holder’s inequality shows that

‖rotu× u‖2L4/3(t0−R2,t0;L6/5(BR(x0)))

≤ c5RΘ(x0, t0; R)‖v‖2L4(t0−R2,t0;L3(BR(x0)))(4.6)

+ c6R2M(x0, t0;R)‖∇v‖2L2(QR(x0,t0))

,

where c5 and c6 deonote absolute constants.Secondely, noticing rotu = rotv using Holder’s inequality gives

‖rotu× u‖2L4/3(t0−R2,t0;L6/5(BR(x0)))

≤ c7‖u‖2L4(t0−R2,t0;L3(BR(x0)))‖∇v‖2L2(QR(x0,t0))

, (4.7)

where c7 denotes an absolute constant.

In order to fix the model problem we will use the following lemma which forreaders convenience will be proved at the end of this section.

Lemma 4.1. Given f ∈ L1(t0 − R2, t0; Lσ(BR(x0))) (1 < σ ≤ 2), there exists aunique function w ∈ L

5σ3+σ (t0−R2, t0; W 1, 5σ

3+σ (BR(x0)))∩L5σ/3(QR(x0, t0)), suchthat ∫

QR(x0,t0)

(−wϕt +∇w · ∇ϕ) dxdt =∫

QR(x0,t0)

fϕ dxdt (4.8)


for all ϕ ∈ C∞0 (BR(x0)× [t0 −R2, t0[). In addition, there holds

‖w‖L5σ/3(QR(x0,t0)) ≤ c‖f‖L1(t0−R2,t0;Lσ(BR(x0))) (4.9)

with an constant c > 0 depending only on σ.

The model problem. Applying Lemma4.1 with f = fi = (rotu × u)i −∂p0,R

∂xi(i = 1, 2, 3) and σ =

65

one gets a unique function W ∈ L10/7(t0 −R2, t0; W 1, 10/7(BR(x0))3) ∩ L2(QR(x0, t0))3 such that

∫

QR(x0,t0)

(−W ·ϕt +∇W : ∇ϕ) dxdt

=∫

QR(x0,t0)

(rotu× u−∇p0,R) ·ϕdxdt (4.10)

for all ϕ ∈ C∞0 (BR(x0) × [t0 − R2, t0[). Clearly, using Jensen’s inequality theestimate (4.9) yields

‖W‖2L2(QR(x0,t0))≤ cR‖rotu× u−∇p0,R‖2L4/3(t0−R2,t0;L6/5(BR(x0)))

.

Now, the right hand side of this inequality can be estimated by (4.4) together with(4.6). Thus,

‖W‖2L2(QR(x0,t0))≤ c8R

4M(x0, t0;R)‖∇v‖2L2(QR(x0,t0))

+ c9R2Θ(x0, t0; R)‖v‖2L4(t0−R2,t0;L3(BR(x0)))

. (4.11)

Alternatively, using (4.7) instead of (4.6) one gets

‖W −WQR‖2L2(QR(x0,t0))

≤ c10R2O(x0, t0;R)‖∇v‖2L2(QR(x0,t0))

. (4.12)

Proof of Lemma4.1 It will be sufficient to prove the assertion of the lemma forthe case (x0, t0) = (0, 0) and R = 1. The general case can be deduced easily byintroducing an appropriate transformation of coordinates.

Let f ∈ L1(−1, 0; Lσ(B1)) be given. Clearly there exists a sequence fm inL2(Q1), such that

fm → f in L1(−1, 0; Lσ(B1)) as m →∞.

Let m ∈ N. Consulting [12], there exists a unique function

wm ∈ L2(−1, 0; W 1, 2(B1)) ∩ C([−1, 0];L2(B1)),

satisfying ∫

Q1

(−wmϕt +∇wm · ∇ϕ) dxdt =∫

Q1

fmϕ dx dt (4.13)

for all ϕ ∈ C∞0 (B1 × [−1, 0[). In particular,dwm

dt∈ L2(−1, 0; W−1, 2(B1)), such

that∫ t

−1

⟨ dwm

dt(s), ϕ(s)

⟩ds +

∫ t

−1

∫

B1

∇wm · ∇ϕdxds =∫ t

−1

∫

B1

fmϕdxds (4.14)

18 J. Wolf

for all ϕ ∈ L2(−1, 0; W 1, 2(B1)), for all t ∈] − 1, 0[. Observing (4.13) using inte-gration by parts it follows that wm(−1) = 0. Into (4.14) inserting the admissibletest function ϕ = wm|wm|σ−2 using integration by parts along with Holder’sinequality gives

1σ‖wm(t)‖σ

Lσ(B1)+ (σ − 1)

∫ t

−1

∫

B1

|∇wm|2|wm|σ−2 dx ds

≤∫ t

−1

∫

B1

|fm||wm|σ−1 dx ds

≤ ‖fm‖L1(−1,0;Lσ(B1))‖wm‖σ−1L∞(−1,0;Lσ(B1))

.

Define θm := wm|wm|σ−22 . From the inequality above one infers

θm ∈ L2(−1, 0; W 1, 2(B1)) ∩ L∞(−1, 0;L2(B1))

together with

‖θm‖2L∞(−1,0;L2(B1))+ ‖∇θm‖2L2(Q1)

≤ σ′

2‖fm‖L1(−1,0;Lσ(B1))‖θm‖2/σ′

L∞(−1,0;L2(B1)).

By means of Sobolev’s embedding theorem together with Holder’s inequality ap-plying Young’s inequality implies

‖θm‖L10/3(Q1) + ‖∇θm‖L2(Q1) ≤ c‖fm‖σ/2L1(−1,0;Lσ(B1))

,

where c = const > 0 depending only on σ. Hence, recalling the definition of θm

using Holder’s inequality shows that

‖wm‖L5σ/3(Q1) + ‖∇wm‖L

5σ3+σ (Q1)

≤ c‖fm‖L1(−1,0;Lσ(B1)).

Finally, passing to the limit m →∞ using Banach-Steinhaus’ theorem proves theassertion Lemma 3.3. ¤

5. Proof of Propositions 1.5/1.6

As in the previous section by u we denote a suitable weak solution to (1.1)-(1.4)

with pressure p represented by p0 +∂ph

∂t. As before setting v := u + ∇ph and

p :=|u|22

+ p0 there holds (1.12) together with the local energy inequality (1.11).

For the sake of simplicity we define the following quantities which are invari-ant under the natural scaling of the Navier-Stokes equations. Let (x0, t0) ∈ Q and


0 < ρ < mindist(x0, ∂Ω),√

t0. We define

V(x0, t0; ρ) :=1ρ

[‖v‖2L∞(t0−ρ2,t0;L2(Bρ(x0)))

+ ‖v‖2L10/3(Qρ(x0,t0))+ ‖∇v‖2L2(Qρ(x0,t0))

],

P(x0, t0; ρ) :=1ρ2‖p− pBρ(x0)‖2L4/3(t0−ρ2,t0;L2(Bρ(x0)))

,

Z(x0, t0; ρ|ϕ) :=1ρ3‖ϕ‖2L2(Qρ(x0,t0))

, ϕ ∈ L2(Qρ(x0, t0))3.

Throughout let (x0, t0) ∈ Q be fixed. For the sake of notational simplic-ity we will write V(ρ),P(ρ),Z(ρ|ϕ), Θ(ρ) and M(ρ) resp. in place of V(x0, t0; ρ),P(x0, t0; ρ),Z(x0, t0; ρ|ϕ), Θ(x0, t0; ρ) and M(x0, t0; ρ) resp.

First let us prove some fundamental estimates which form the bases of theproof of the Prop. 1.5. We start with a Caccioppoli-type inequality which immedi-ately follows from the local energy inequality (1.11).

Lemma 5.1. There exist absolute positive constants c11 and c12 such that

V(ρ) ≤ c11

[1 + ρ2M(2ρ)

]Z(2ρ|v) + c12P(2ρ) (5.1)

for all 0 < ρ <12

mindist(x0, ∂Ω),√

t0.

Proof. Let 0 < ρ <12


t0. Let φ ∈ C∞(R4) be a cut-off func-

tion, such that 0 ≤ φ ≤ 1 in R4. φ ≡ 0 in R4 \B2ρ(x0)×]−∞, t0 − 4ρ2[, φ ≡ 1 inBρ(x0)×]t0 − ρ2,∞[ and

|∇φ|2 + |∇2φ|+ |∂tφ| ≤ c0

ρ2in R4.

From the local energy inequality (1.11) (replacing φ by φ2 therein) usingHolder’s and Young’s inequality one easily estimates

‖φv‖2L∞(t0−4ρ2;L2(B2ρ)) + ‖φ∇v‖2L2(Q2ρ)

≤ c[ρ−2 + max

Q2ρ

|∇ph|2]‖v‖2L2(Q2ρ) + cρ−1

∫

Q2ρ

|p− pB2ρ | |v|φ dxdt.

With help of Cauchy-Schwartz’s inequality verifying∫

Q2ρ

|p− pB2ρ | |v|φdxdt

≤ ‖p− pB2ρ‖L1(t0−2ρ2,t0;L2(B2ρ))‖φv‖L∞(t0−4ρ2,t0;L2(B2ρ))

applying Young’s inequality one arrives at

‖φv‖2L∞(t0−4ρ2;L2(B2ρ)) + ‖φ∇v‖2L2(Q2ρ)

≤ cρ[1 + ρ2M(2ρ)

]Z(2ρ|v) + cρP(2ρ). (5.2)

20 J. Wolf

Finally, by the aid of multiplicative inequality and Sobolev-Poincare inequality

‖φv‖2L10/3(Q2ρ) ≤ ‖φv‖4/5L∞(t0−4ρ2;L2(B2ρ))‖φv‖6/5

L2(t0−4ρ2;L6(B2ρ))

≤ c(‖φv‖2L∞(t0−4ρ2;L2(B2ρ)) + ‖φ∇v‖2L2(Q2ρ)

)

+ cρZ(2ρ|v). (5.3)

Combining (5.2) and (5.3) gives (5.1). ¤

Next we provide a fundamental estimate for the pressure. We have the

Lemma 5.2. There exist absolute positive constants c13 and c14 such that for every

0 < τ <12

we have

P(2τR) ≤ c13τ3P(R) + c14τ

−3[R2M(R) + Θ(R)]V(R) (5.4)

for all 0 < R < mindist(x0, ∂Ω),√

t0.Proof. Let 0 < R < mindist(x0, ∂Ω),

√t0 be arbitrarily chosen. By virtue of

the orthogonal decomposition L2(BR(x0)) = A2(BR(x0)) ⊕ B2(BR(x0)) one getsunique functions

p0,R ∈ L4/3(0,∞; A2(BR(x0))),

ph,R ∈ L4/3(0,∞; B2(BR(x0)))

such thatp(t)− p(t)BR(x0) = p0,R(t) + pR,h(t) in BR(x0)

for almost all t ∈]0,∞[.

Next, fix 0 < τ <12. With help of the Sobolev-Poincare inequality, Cacciop-

poli’s inequality and properties of harmonic functions one easily deduces

‖p(t)− p(t)B2τR‖4/3L2(B2τR)

≤ 2‖ph,R(t)− ph,R(t)B2τR‖4/3L2(B2τR) + 2‖p0,R(t)− p0,R(t)B2τR‖4/3

L2(B2τR)

≤ cτ10/3R10/3 maxx∈BR/2(x0)

|∇ph,R(x, t)|4/3 + c‖∇p0,R(t)‖4/3

L6/5(BR)

≤ cτ10/3‖ph,R(t)‖4/3L2(BR) + c‖∇p0,R(t)‖4/3

L6/5(BR)(5.5)

for almost all t ∈]t0−R2, t0[. Integrating both sides over the interval ]t0−4τ2R2, t0[it follows that

‖p− pB2τR‖2L4/3(t0−R2,t0;L2(BR(x0)))

≤ cτ5‖ph,R‖2L4/3(t0−R2,t0;L2(BR(x0)))+ c‖∇p0,R‖2L4/3(t0−R2,t0;L6/5(BR(x0)))

.

Taking into account

‖ph,R‖L4/3(t0−R2,t0;L2(BR(x0))) ≤ ‖p− pBR(x0)‖L4/3(t0−R2,t0;L2(BR(x0)))


using (4.4) along with (4.6) and dividing the result by τ2R2 applying Holder’sinequality one infers

P(2τR) ≤ cτ3P(R) + cτ−2R−1Θ(R)‖v‖2L4(t0−R2,t0;L3(BR(x0)))

+ cτ−1RM(R)‖∇v‖2L2(QR(x0,t0))

≤ cτ3P(R) + cτ−2[R2M(R) + Θ(R)]V(R).

Whence, (5.4). ¤

Remark 5.3. Repeating the proof of Lemma 5.2 while using (4.7) instead (4.6) forthe estimation of ∇p0,R one immediately gets the following alternative fundamen-tal estimate

P(2τR) ≤ c13τ3P(R) + c14τ

−3O(R)1R

∫

QR(x0,t0)

|∇v|2 dxdt (5.6)

for all 0 < R < mindist(x0, ∂Ω),√

t0, with absolute constants c13 and c14.

Now, we are in a position to complete the proof of Prop. 1.5.

Proof of Proposition 1.5 Let 0 < R < mindist(x0, ∂Ω),√

t0 be fixed. Accordingto Lemma 4.1 there exists a unique function

W ∈ L10/7(t0 −R2, t0; W 1, 10/7(BR(x0))3 ∩ L2(QR(x0, t0))3

fulfilling the identity (4.10). Furthermore, from (4.11) one easily deduces

Z(R|W) ≤ c15[R2M(R) + Θ(R)]V(R). (5.7)

Combining (1.12) and (4.10) yields∫

QR(x0,t0)

−V ·ϕt +∇V : ∇ϕ

dxdt =

∫

QR(x0,t0)

ph,R div ϕdxdt (5.8)

where V = v −W. Thus, setting

U(y, s) := V(x0 + Ry, t0 + R2s),

q(y, s) := Rph,R(x0 + Ry, t0 + R2s), (y, s) ∈ Q1

shows that the function U is semi caloric in Q1 with corresponding pressure q.Applying Theorem 3.3 one finds

‖U‖2L2(Q2τ ) ≤ cτ5‖U‖2L2(Q1)

+ ‖q‖2L4/3(−1,0;L2(B1))

.

Then using the transformation formula of the Lebesgue integral, noticing that

‖ph,R(t)‖L2(BR(x0)) ≤ ‖p(t)− pBR(x0)(t)‖L2(BR(x0)) f.a.a. t ∈]t0 −R2, t0[,

(cf. Lemma 2.1) one arrives at

Z(2τR|V) ≤ c16τ2[Z(R|V) + P(R)]. (5.9)

22 J. Wolf

Combining (5.7) and (5.9) yields

Z(2τR|v) ≤ 2Z(2τR|V) + 2Z(2τR|W)

≤ cτ2[Z(R|V) + P(R)] + τ−3Z(R|W)

≤ c[τ2 + τ−3Θ(R)

](V(R) + P(R)), (5.10)

where Θ(ρ) := R2M(ρ) + Θ(ρ), 0 < ρ < mindist(x0, ∂BR(x0)),√

t0.Now from (5.1) with ρ = τR using (5.10) together with (5.4) yields

V(τR) + P(τR)

≤ c(1 + R2M(R)Z(2τR|v) + cP(2τR)

≤ c(1 + R2M(R))[τ2 + τ−3Θ(R)

](V(R) + P(R)),

Whence, (1.13). ¤

Proof of Proposition 1.6. Given Λ ∈ R3 arbitrarily using (1.10) the identity(1.11) can be modified as follows

∫

Ω

|v(t)−Λ|2φ(t) dx + 2∫ t

0

∫

Ω

|∇v|2φdxds

≤∫ t

0

∫

Ω

|v −Λ|2∂φ

∂t+ ∆φ

dx ds +

∫ t

0

∫

Ω

2p(v −Λ) · ∇φ dxds (5.11)

+ 2∫ t

0

∫

Ω

(∇ph −Λ)× rotv · (v −Λ)φ dxds

holds for all nonnegative functions φ ∈ C∞0 (Q), for almost all 0 < t < ∞ . Hereby,the set of all t > 0, where (5.11) fails does not depend on Λ.

Let 0 < R < mindist(x0, ∂Ω),√

t0 be fixed. Arguing as in the proof ofLemma4.1 using (5.11) with Λ := vQ2ρ(x0,t0) one gets the following alternativeCaccioppoli inequality

Lemma 5.4. There exists an absolute positive constants c17 such that

V(ρ) ≤ c17

[Z(2ρ|v) + O(2ρ)V(2ρ) + P(2ρ)

](5.12)

for all 0 < ρ <12


t0, where

V(ρ) = V(x0, t0; ρ) :=1ρ‖∇v‖2L2(Qρ(x0,t0))

,

Z(ρ|ϕ) = Z(x0, t0; ρ|ϕ) :=1ρ3‖ϕ−ϕQρ(x0,t0)‖2L2(Qρ(x0,t0))

,

ϕ ∈ L2(Qρ(x0, t0))3.


Proof of Proposition 1.6 Let 0 < R < mindist(x0, ∂Ω),√

t0 be fixed. Arguing asin the proof of Prop. 1.5 instead of (4.11) using (4.12) for estimating W yields

Z(R|W) ≤ c18O(R)V(R). (5.13)

where c18 > 0 denotes an absolute constant.On the other hand, recalling that V being semi caloric in QR(x0, t0) applying

the fundamental estimate (3.5) (see Theorem3.3 above) it follows

Z(2τR|V) ≤ c19τ3[Z(R|V) + P(R)]. (5.14)

Then combining (5.7) and (5.9) yields

Z(2τR|v) ≤ 2Z(2τR|V) + 2Z(2τR|W)

≤ cτ3[Z(R|V) + P(R)] + τ−3Z(R|W)

≤ c[τ3 + τ−3O(R)

](V(R) + P(R)). (5.15)

Finally, into (5.12) putting ρ = τR, estimating the first term on the right by(5.15) and last term by (5.6) respectively gives

V(τR) + P(τR) ≤ c[τ3 + τ−3O(R)

](V(R) + P(R)).

Whence, (1.14). ¤

6. Proof of the Main Theorem

To begin with, defineε? := 0.18566 k−2.5

∗ .

Setf(τ) := k?(τ2 + τ−3ε?), τ > 0.

Let τ? > 0 such that θ? := f(τ?) = minτ>0

f(τ). In fact, by an elementary calculus

one finds

τ? = (1.5ε?)1/5,

θ? =53(1.5 · 0.18566)2/5 = 0.999 . . . < 1.

Next, let (x0, t0) ∈ Q, such that

lim supR→0+

Θ(x0, t0; R) < ε?.

Since limR→0+

R2M(x0, t0;R) = 0 we may choose 0 < R0 < mindist(x0, ∂Ω),√

t0such that

Θ(x0, t0;R) < ε?,

(1 + R2M(x0, t0;R))f(τ?) <θ? + 1

2∀ 0 < R ≤ R0.

24 J. Wolf

Then (1.13) leads to

Ξ(x0, t0; τ?R) ≤ θ? + 12

Ξ(x0, t0; R) ∀ 0 < R ≤ R0.

Consequently,lim

R→0+Ξ(x0, t0;R) = 0. (6.1)

In particular, having |rotu| ≤ 2|∇v|, there exists 0 < R1 < R0, such that

Θ(x0, t0; R1) <ε?

2, Ξ(x0, t0; R1) <

ε?

4.

By means of the absolute continuity of the Lebesgue integral we may choose 0 <ρ < mindist(x0, ∂Ω),

√t0 −R0, such that for all (y, s) ∈ Qρ(x0, t0):

Ξ(y, s;R1) <ε?

4,

Θ(y, s;R1) <ε?

2,

(1 + R2M(y, s; R1))f(τ?) <θ? + 1

2.

Now, let (y, s) ∈ Qρ(x0, t0) be fixed. Once more applying (1.13) iterativelyreplacing (x0, t0) by (y, s) and setting R = R1 therein yields

Ξ(y, s; τk? R1) ≤

(θ? + 12

)k

Ξ(y, s; R1).

Choosing 0 < α < 1 such that τα? =

θ? + 12

from the estimate above one gets

Ξ(y, s;R) ≤ C1Rα ∀ 0 < R ≤ R1, (6.2)

where C1 = const > 0 depending neither on (y, s) nor on R.

Next, fix 0 < β < 1. Choose 0 < τ <12

according to

τβ k? ≤ 12

7 .

Furthermore, by virtue of (6.2) one can select 0 < R2 ≤ R1 such that for all0 < R ≤ R2 there holds

τ−3O(y, s; R) ≤ τ3−β

2∀ (y, s) ∈ Qρ(x0, t0).

Now, fix (y, s) ∈ Qρ(x0, t0) . Applying (1.14) with (y, s) instead of (x0, t0)gives

Ξ(y, s; τR) ≤ τ3−βΞ(y, s;R) ∀ 0 < R ≤ R2.

This implies ∫

QR(y,s)

|∇v|2 dx ≤ C2R4−β ∀ 0 < R ≤ R2,

7 Here k? denotes the constant, which appears in (1.14).


where the constant C2 > 0 depends neither on (y, s) ∈ Qρ(x0, t0) nor on R.Applying the Poincare-type inequality (A.10) (cf. Theorem A.1 below) gives

v ∈ L6−β(Qρ(x0, t0)),

which, by a well-known theorem of DaPrato [6], shows that v belongs to the Holderspace C(1−β)/2,(1−β)/4(Qρ(x0, t0))3. Since ∇ph is continuous in Q the velocity u iscontinuous on Qρ(x0, t0). This completes the proof of the Theorem. ¤

7. Existence of a suitable weak solution

Let Φ ∈ C∞([0,∞[) be a non-increasing function, such that 0 ≤ Φ ≤ 1 in [0,∞[,Φ ≡ 1 on [0, 1], Φ ≡ 0 in [2,∞[ and 0 ≤ −Φ′ ≤ 2. For ε > 0 we set

Φε(%) := Φ(ε%), % ∈ [0,∞[.

Let a ∈ L2σ(Ω) be arbitrarily given. Arguing as in [25] one easily can construct a

unique function uε ∈ C([0,∞[; L2σ(Ω)) ∩ L2(0,∞; W 1, 2

σ (Ω)), such that

−∫

Q

uε · ∂tϕdx dt +∫

Q

∇uε : ∇ϕdxdt

=∫

Q

uε ⊗ uεΦε(|uε|) : ∇ϕdx dt +∫

Ω

a ·ϕ(0) dx (7.1)

holds for all ϕ ∈ C∞0 ([0,∞[;Dσ(Ω)) . In addition, there holds the global energyequality

12‖uε(t)‖2L2(Ω) +

∫ t

0

∫

Ω

|∇uε|2 dxds =12‖a‖2L2(Ω) ∀ t > 0. (7.2)

From this identity one easily gets the a-priori estimate

‖uε‖L∞(0,∞;L2(Ω)) + ‖∇uε‖L2(Q) ≤ K. (7.3)

Using Sobolev’s inequality combined with multiplicative inequalities one infers

‖uε‖L8/3(0,∞;L4(Ω)) ≤ K. (7.4)

Let B ⊂⊂ Ω be a fixed domain with Lipschitz boundary. Applying Theo-rem2.2 with u = uε and hε = uε ⊗ uεΦε(|uε|) one gets two functions

p0,ε ∈ L1loc([0,∞[;A2(Ω)),

ph,ε ∈ C(Q) with ∆ph,ε = 0, and (ph,ε)B = 0

26 J. Wolf

such that

−∫

Q

uε · ∂tϕdx dt +∫

Q

∇uε : ∇ϕdxdt

=∫

Q

uε ⊗ uεΦε(|uε|) : ∇ϕdxdt

+∫

Q

(p0,ε div ϕ + ∇ph,ε · ∂tϕ) dxdt

∫

Ω

a ·ϕ(0) dx (7.5)

for all ϕ ∈ C∞(Q)3. Into (7.5) putting ϕ(x, t) = ∇φ(x)η(t) for φ ∈ C∞0 (Ω) andη ∈ C∞0 (0,∞) using integration by parts yields

∫

Ω

p0,ε(t)∆φdx =∫

Ω

uε(t)⊗ uε(t)Φε(|uε(t)|) : ∇2φdx (7.6)

for all φ ∈ C∞0 (Ω), for almost all t ∈]0,∞[. Hence, from the definition of A2(Ω) itfollows

‖p0,ε(t)‖L2(Ω) ≤ ‖uε(t)⊗ uε(t)‖L2(Ω) ≤ ‖uε(t)‖2L4(Ω).

Thus, observing (7.4) yields

‖p0,ε‖L8/3(0,∞;L2(Ω)) ≤ K. (7.7)

Furthermore, by means of (2.7) for every G ⊂⊂ Ω with Lipschitz boundarythere exists a constant cG such that

‖ph,ε(t)‖L2(G) ≤ cG

‖uε(t)− a‖L2(Ω) +

∥∥∥∥∫ t

0

(|uε(s)|2 + |∇uε(s)|) ds

∥∥∥∥L2(Ω)

.

Hence using Minkowski’s inequality along with (7.4) and (7.4) yields

‖ph,ε(t)‖L2(G)) ≤ cGK (1 +√

t). (7.8)

By virtue of reflexivity using Cantor’s diagonalization principal there exist a se-quence εj of positive numbers tending to 0 as j →∞ and functions

u ∈ L∞(0,∞;L2σ(Ω)) ∩ L2(0,∞; W 1, 2(Ω)),

q0 ∈ L4/3(0,∞;L2(Ω)),

qh ∈ L2loc(Ω× [0,∞[) with ∆ph = 0 and (ph)B = 0,

such that

uεj u in L∞(0,∞;L2σ(Ω)) ∩ L2(0,∞; W 1, 2

σ (Ω)),

p0,εj q0 in L4/3(0,∞; L2(Ω)) (7.9)

ph,εj qh in L2loc(Ω× [0,∞[) as j →∞.


Arguing as in [25] using the Lemma of Aubin-Lions we infer that for each subdomain G ⊂⊂ Ω and T > 0 there holds

uεj→ u in L2(G×]0, T [)3 as j →∞. (7.10)

Using (7.9)1 and (7.10) we are able to carry out the passage to the limit in bothsides of (7.1). Hence, u is a weak solution to (1.1)-(1.4).

Furthermore carrying out the passage to the limit in both sides of (7.5)making use of (7.9) and (7.10) gives

−∫

Q

u · ∂tϕdx dt +∫

Q

∇u : ∇ϕ dxdt

=∫

Q

u⊗ u : ∇ϕdx dt +∫

Q

(q0 div ϕ +∇qh · ∂tϕ) dx dt

∫

Ω

a ·ϕ(0) dx (7.11)

for all ϕ ∈ C∞(Q)3. On the other hand appealing to Theorem 2.2 one gets twofunctions p0 ∈ L

4/3loc ([0,∞[;A2(Ω)) and ph ∈ C(Q) with ∆ph and (ph)B = 0

fulfilling (7.11) too. Thus, by Remark 2.4 one concludes q0 = p0 and qh = ph

almost everywhere in Q.As in [25] using the properties of harmonic functions one easily proves that

for all t > 0

∇2ph,εj → ∇2ph in L2loc(Ω× [0,∞[)9 as j →∞. (7.12)

Now, setvε := uε +∇ph,ε, v := u +∇ph.

Combining (7.9)1, (7.10) and (7.12) implies

∇vεj ∇v in L2loc(Ω× [0,∞[)9, (7.13)

vεj → v in L2loc(Ω× [0,∞[)3 as j →∞. (7.14)

In addition, we have

vεj (t) → v(t) in L2loc(Ω)3 as j →∞. (7.15)

for almost all t > 0.In order to complete the proof we only need prove that u, p0 and ph satisfy

the local energy inequality (1.11) . In fact, from the identity (7.5) it follows thatdvε

dt∈ L2

loc([0,∞[;W−1, 2(Ω)3) and there holds

∫ t

0

⟨ dvε

dt(s),ϕ(s)

⟩ds +

∫

Q

∇vε : ∇ϕ dxds

=∫ t

0

∫

Ω

uε ⊗ uεΦε(|uε|)∇ϕ dxds +∫ t

0

∫

Ω

p0,ε div ϕdxds (7.16)

28 J. Wolf

for all ϕ ∈ L2(0,∞; W 1, 2σ (Ω)) and t > 0. Let φ ∈ C∞0 (Q), φ ≥ 0. Then in (7.16)

putting ϕ = uεφ using integration by parts yields

12

∫

Ω

|vε(t)|2φ(t) dx +∫ t

0

∫

Ω

|∇vε|2φ dxdt

=∫ t

0

∫

Ω

(Ψε(|uε|)uε · ∇φ + p0,εvε · ∇φ) dxdt

+∫ t

0

∫

Ω

uε ⊗ uεΦε(|uε|) : ∇(∇ph,εφ) dxdt (7.17)

for all t > 0, where

Ψε(u) := u2Φε(u)−∫ u

0

Φε(%)% d%, u ≥ 0.

Clearly,

Ψε(u) → u2

2uniformly on each interval [0, T ] as ε → 0.

Now, let t ∈]0,∞[ such that (7.15) is true. Then into (7.17) passing to thelimit with ε = εj → 0 as j →∞ using (7.12), (7.13), (7.14) and (7.15) along withthe lower semi continuity of the functional on the left of (7.17) leads to

∫

Ω

|v(t)|2φ(t) dx + 2∫ t

0

∫

Ω

|∇v|2φdxdt

≤∫ t

0

∫

Ω

(−|u|2∇ph · ∇φ + 2pv · ∇φ) dxdt

+ 2∫ t

0

∫

Ω

u⊗ u : ∇(∇phφ) dxdt (7.18)

where p =|u|22

+ p0. Using integration by parts and recalling ∆ph = 0 one easilycalculates

−∫ t

0

∫

Ω

|u|2∇ph · ∇φ dxdt + 2∫ t

0

∫

Ω

u⊗ u : ∇(∇phφ) dx dt

= 2∫ t

0

∫

Ω

(uj∂xiuj − uj∂xj ui)(∂xi ph)φdxdt

= −2∫ t

0

∫

Ω

u× rotu · ∇phφ dxdt

= 2∫ t

0

∫

Ω

∇ph × rotv · vφdxdt.

Whence (1.11). ¤


Appendix A.

Lemma A.1. Let Ω be a domain. Let p : Ω× [0,∞[→ R such that for all t ∈ [0,∞[the function x 7→ p(x, t) is harmonic in Ω. Assume that for all G ⊂⊂ Ω

t 7→∫

G

p(x, t) dx is continuous on [0,∞[; (A.1)

supt∈[0,∞[

∫

G

|p(t)| dx < +∞ ∀ 0 < T < ∞. (A.2)

Then p ∈ C(Ω× [0,∞[).

Proof. Boundeness of p. Let x0 ∈ Ω and 0 < R <12

dist(x0, ∂Ω). Let x ∈ BR(x0).

Clearly, BR(x) ⊂ B2R(x0). Thus, observing (A.2) using the well-known mean valueformula for harmonic functions one estimates

|p(x, t)| ≤∫−BR(x)

|p(y, t)| dy ≤∫−B2R(x0)

|p(y, t)| dy ≤ CR−3

for all (x, t) ∈ BR(x0)×[0, T ] (0 < T < +∞). Thus, p is bounded on BR(x0)×[0, T ]and therefore locally bounded in Ω× [0,∞[.

Next, let x ∈ BR(x0). Once more using the mean value formula and integra-tion by parts one gets

|∂xip(x, t)| =∣∣∣∣∫−BR(x)

∂xip(y, t) dy

∣∣∣∣ ≤ cR−3

∫

∂BR(x0)

|p(y, t)| dy ≤ CR−4.

(i = 1, 2, 3). Therefore, ∇p is locally bounded in Ω× [0,∞[ too.

Continuity of p. Let (x0, t0) ∈ Q. Given (x, t) ∈ BR × [0,∞[ using the trian-gular inequality together with the mean value theorem one gets

|p(x0, t0)− p(x, t)| ≤ |p(x0, t0)− p(x0, t)|+ |p(x0, t)− p(x, t)|

=∣∣∣∣∫−BR(x0)

p(y, t0)− p(y, t) dy

∣∣∣∣ + maxy∈BR(x0)

|∇p(y, t)||x0 − x|.

Observing (A.1) the estimate above immediately implies that p is continuous at(x0, t0). ¤

Lemma A.2. Let p : Ω × [0,∞[→ R such that for all t ∈ [0,∞[ the functionx 7→ p(x, t) is harmonic in Ω. Assume that

p− pG is continuous on G× [0,∞[ ∀G ⊂⊂ Ω, (A.3)

∃x0 ∈ Ω : t 7→ p(x0, t) is continuous on [0,∞[. (A.4)

Then p ∈ C(Ω× [0,∞[).

Proof. We define

A :=

x ∈ Ω∣∣∣ p(x, ·) ∈ C([0,∞[)

.

30 J. Wolf

A is open. Let x ∈ A. Let 0 < R < dist(x, ∂Ω). By means of (A.3) wehave p − pBR(x) is continuous in BR × [0,∞[. On the other hand by the meanvalue formula we have pBR(x) = p(x, ·) ∈ C([0,∞[). Thus, p is continuous inBR(x)× [0,∞[, which yields BR(x) ⊂ A.

A is closed. Let xm be a sequence in A converging to x ∈ Ω as m → +∞.Let 0 < R < dist(x, ∂Ω). Hence, there exists m ∈ N such that |xm − x| <R/2. By the triangular inequality it follows that dist(xm, ∂Ω) > R/2. Therefore,BR/2(xm) ⊂⊂ Ω. As we have proved above there holds BR/2(xm) ⊂ A and inparticular x ∈ A.

Since A 6= ∅ (cf. (A.4)) implies A = Ω. Let x ∈ Ω and 0 < R < dist(x, ∂Ω).Then p = p − pBR(x) + p(x, ·) in BR(x). By means of (A.3) taking into accountp(x, ·) ∈ C([0,∞[) proves the claim of the lemma ¤

Lemma A.3. Let g ∈ Lr(BR)3(6

5≤ r ≤ 2

)and q0 ∈ A2(BR) with

∫

BR

g · ∇φ dx =∫

BR

q0∆φ dx ∀φ ∈ C∞0 (BR). (A.5)

Then q0 ∈ W 1, r(BR) and there exists a constant Cr, such that

‖∇q0‖Lr(BR) ≤ Cr‖g‖Lr(BR). (A.6)

Proof. First let us consider the case R = 1. There exists h ∈ W 1, r0 (B1)9 such that

div h = g − gB1 with‖∇h‖Lr(B1) ≤ c‖g‖Lr(B1). (A.7)

Next, applying integration by parts, from (A.5) it follows that

−∫

B1

(h− hB1) : ∇2φdx =∫

B1

q0∆φdx ∀φ ∈ W 2, 20 (B1).

By the definition of A2(B1) there exists u ∈ W 2, 20 (B1) such that q0 = ∆u. Thus,

from the identity with φ = u above using Sobolev-Poincare’s inequality along with(A.7) yields

‖q0‖L2(B1) ≤ c‖h− hB1‖L2(B1) ≤ c‖g‖Lr(B1). (A.8)

Next, applying [Simader, Th. 9.11, p.156] with B[u, φ] =∫

B1

∆u∆φdx and

F (φ) =∫

B1

g · ∇φ dx one gets q0 ∈ W 1, r(B1) and

‖∇q0‖Lr(B1) ≤ c(‖g‖Lr(B1) + ‖u‖L2(B1)).

Finally, taking into account

‖u‖L2(B1) ≤ c‖∆u‖L2(B1) = c‖q0‖L2(B1)

making use of (A.8) shows (A.6).For the general case R > 0 the assertion easily follows from the former

case R = 1 by using an appropriate transformation of coordinates applying thetransformation formula of the Lebesgue integral. ¤


For reader’s convenience we will present a short proof of the Poincare-typeinequality we have used above.

Lemma A.4. Let u ∈ L2(QR) with ∇u ∈ L2(QR) and h ∈ L1(QR)3, such that

−∫

QR

u∂tϕdxdt =∫

QR

h · ∇ϕ dxdt ∀ϕ ∈ C∞0 (QR). (A.9)

Then∫−QR

|u− uQR|2 dx dt

≤ c0R2

∫−QR

|u− uQR|2 dx dt +

(∫−QR

|h|dxdt

)2

, (A.10)

where c0 = const > 0 depends on n only.

Proof 1) First, let us prove the assertion for the case R = 1.For a given function ζ ∈ C∞0 (B1) with 0 ≤ ζ ≤ 1 in B1 and ζ 6= 0 we define

the functional F ∈ L1(B1)∗ by means of

〈F, v〉 :=1∫

B1ζ dx

∫

B1

ζv dx, v ∈ L1(B1).

Clearly, 〈F,1〉 = 1 and

⟨F, v − 〈F, v〉

⟩= 0 ∀ v ∈ L1(B1(0)).

Since

|||v||| := ‖∇v‖L2(B1) + |〈F, v〉|

defines an equivalent norm on W 1, q(B1). Thus,

‖v − 〈F, v〉‖Lq∗ (B1(0)) ≤ c‖∇v‖Lq(B1) ∀ v ∈ W 1, q(B1). (A.11)

Now let u ∈ L2(Q1) with ∇u ∈ L2(Q1)3 and h ∈ L1(Q1)3 fulfilling (A.9).One easily calculates

u(x, t)− uQ1 = (u(x, t)− 〈F, u(t)〉) + (〈F, u(t)〉 − uQ1)

= u(x, t)− 〈F, u(t)〉 +∫−Q1

〈F, u(t)〉 − u(y, s) dy ds

32 J. Wolf

for almost all (x, t) ∈ Q1. Moreover, verifying∫−Q1

〈F, u(t)〉 − u(y, s) dy ds

=1∫

B1ζ dy

∫−Q1

∫

B1

ζ(y′)(u(y′, t)− u(y, s)) dy′ dy ds

=∫ 0

−1

〈F, u(t)− u(s)〉ds

+1

2∫

B1ζ dy

∫ 0

−1

∫−B1

∫

B1

ζ(y′)(u(y′, s)− u(y, s)) dy′ dy ds.

for almost all −1 < t− 0 gives

u(x, t)− uQ1 = u(x, t)− 〈F, u(t)〉 +∫ 0

−1

〈F, u(t)〉 − u(y, s) ds

+1

2∫

B1ζ dy

∫ 0

−1

∫−B1

∫

B1

ζ(y′)(u(y′, s)− u(y, s)) dy′ dy ds.

Hence, from the identity above one obtains

‖u(t)− uQ1‖L2(B1) ≤ ‖u(t)− 〈F, u(t)〉‖L2(B1)

+∥∥∥∥

∫ 0

−1

|〈F, u(t)− u(s)〉|ds

∥∥∥∥L2(B1)

+1∫

B1ζ dy

∥∥∥∥∫ 0

−1

∫

B1

|u(y, s)− u(s)B1 | dy ds

∥∥∥∥L2(B1)

for almost all t ∈]− 1, 0[. Then by the aid of (A.11) and Poincare’s inequality onefinds

‖u(t)− uQ1‖2L2(B1)≤ c‖∇u(t)‖2L2(B1)

+ c

∫ 0

−1

|〈F, u(t)− u(s)〉|2 ds + c‖∇u‖2L1(Q1).

for almost all t ∈] − 1, 0[. Next, integrating both sides of this inequality over theinterval ]− 1, 0[ yields

∫ 0

−1

‖u(t)− uQ1‖2L2(B1)dt

≤ c

∫ 0

−1

‖∇u(t)‖2L2(B1)dt + c

∫ 0

−1

∫ 0

−1

|〈F, u(t)− u(s)〉|2 dsdt.

Finally into (A.9) inserting ϕ(x, t) = ζ(x) using integration by parts implies

|〈F, u(t)− u(s)〉|2 =

∣∣∣∣∣1∫

B1ζ dy

∫ t

s

∫

B1

h · ∇ζ dxdt

∣∣∣∣∣

2

≤ c

( ∫

Q1

|h| dxdt

)2


for almost all s, t ∈]− 1, 0[. Whence, (A.10).2) The general case easily follows from above by means of a standard homo-

thety argument using an appropriate transformation of coordinates. ¤

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34 J. Wolf

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Jorg WolfHumboldt-Universitat zu BerlinUnter den Linden. 610099 BerlinBRDe-mail: [email protected]

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