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### Transcript of Interface dynamics for incompressible flows in 2Dweb.math. 2009. 5. 6.آ  Interface dynamics for...

• Interface dynamics for incompressible flows in 2D

Diego Córdoba

Interface dynamics for incompressible flows in 2D

• Equation

ρt + u · ∇ρ = 0, ∇ · u = 0, ρ(x1, x2, t) =

{ ρ1, x ∈ Ω1(t) ρ2, x ∈ Ω2(t),

with ρ(x , t) an active scalar, (x , t) ∈ R2 × R+, ρ1 6= ρ2 are constants, and Ω2(t) = R2 r Ω1(t).

1 SQG sharp front: ρ temperature,

u = (−R2ρ,R1ρ), R̂j = iξj/|ξ|.

2 Muskat: ρ density, µ viscosity

µ

κ u = −∇p − (0, gρ), Darcy’s law.

3 Water wave: ρ density,

ρ(ut + (u · ∇)u) = −∇p − (0, gρ), Euler.

Interface dynamics for incompressible flows in 2D

• The SQG equation

Equation

θt + u · ∇θ = 0,

u = ∇⊥ψ, θ = −(−∆)1/2ψ,

with θ(x , t) the temperature, (x , t) ∈ R2 × R+.

Constantin, Majda, and Tabak (1994)

Interface dynamics for incompressible flows in 2D

• Sharp fronts

We consider weak solutions given by

θ(x1, x2, t) = { θ1, Ω(t) θ2, R2 r Ω(t).

Interface dynamics for incompressible flows in 2D

• The 2-D vortex patch problem

Contour equation

wt + u · ∇w = 0,

u = ∇⊥ψ, w = ∆ψ,

where the vorticity is given by

w(x1, x2, t) = {

w0, Ω(t) 0, R2 r Ω(t).

Chemin (1993) Bertozzi and Constantin (1993)

Interface dynamics for incompressible flows in 2D

• The 2-D patch problem for SQG

Rodrigo (2005), for a periodic C∞ front, i.e.

θ(x1, x2, t) = { θ1, {f (x1, t) > x2} θ2, {f (x1, t) ≤ x2}.

Interface dynamics for incompressible flows in 2D

• The α-patch model

Córdoba, Fontelos, Mancho and Rodrigo (2005)

Contour equation

θt + u · ∇θ = 0,

u = ∇⊥ψ, θ = −(−∆)1−α/2ψ, 0 < α ≤ 1,

where the active scalar θ(x , t) satisfies

θ(x1, x2, t) = { θ1, Ω(t) θ2, R2 r Ω(t).

Interface dynamics for incompressible flows in 2D

• The contour equation

∂Ω(t) = {x(γ, t) = (x1(γ, t), x2(γ, t)) : γ ∈ [−π, π] = T},

where x(γ, t) is one to one. We have

∇⊥θ = (θ1 − θ2) ∂γx(γ, t) δ(x − x(γ, t)),

u = −(−∆)α/2−1∇⊥θ,

and it gives

u(x , t) = − Θα 2π

∫ T

∂γx(γ − η, t) |x − x(γ − η, t)|α

dη.

Interface dynamics for incompressible flows in 2D

• The normal velocity of the systems reads

u(x(γ, t), t) · ∂⊥γ x(γ, t) = − Θα 2π

∫ T

∂γx(γ − η, t) · ∂⊥γ x(γ, t) |x(γ, t)− x(γ − η, t)|α

dη.

The tangential velocity does not change the shape of the boundary, so that we fix the contour α-patch equations as follows:

Contour equation

xt (γ, t) = Θα 2π

∫ T

∂γx(γ, t)− ∂γx(γ − η, t) |x(γ, t)− x(γ − η, t)|α

dη, 0 < α ≤ 1,

x(γ,0) = x0(γ).

Interface dynamics for incompressible flows in 2D

• Numerical simulations

Close caption of the corner region at t=16.515 for α = 0.5 (a) and t= 4.464 for α = 1 (b). Observe that the singularity is point-like in both cases.

Interface dynamics for incompressible flows in 2D

• Numerical simulations

Evidence of finite time singularity. (a) Evolution of the inverse of the maximum curvature with time. (b) Evolution of the minimum distance between patches with time. (Insets) Here, we represent the latest stage of the evolution together with its linear interpolation.

Interface dynamics for incompressible flows in 2D

• Numerical simulations

Rescaled profiles at 20 different times in the interval (3.46,4.46) for the case α = 1.

Interface dynamics for incompressible flows in 2D

• Local well-posedness in Hs for 0 < α ≤ 1

We define

F (x)(γ, η, t) = |η|

|x(γ, t)− x(γ − η, t)| ∀ γ, η ∈ [−π, π],

with F (x)(γ,0, t) = |∂γx(γ, t)|−1.

Gancedo (2008)

Theorem

Let x0(γ) ∈ Hk (T) for k ≥ 3 with F (x0)(γ, η) 0 so that there is a unique solution to the α-patch model for 0 < α ≤ 1 in C1([0,T ]; Hk (T)), with x(γ,0) = x0(γ).

Interface dynamics for incompressible flows in 2D

• Existence for α = 1; the SQG sharp front

We modify the equation as follows:

Contour equation

xt (γ, t) = ∫

T

∂γx(γ, t)− ∂γx(γ − η, t) |x(γ, t)− x(γ − η, t)|

dη + λ(γ, t)∂γx(γ, t), (1)

with

λ(γ, t) = γ+π

∫ T

∂γx(γ, t) |∂γx(γ, t)|2

·∂γ (∫

T

∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|

dη ) dγ

− ∫ γ −π

∂γx(η, t) |∂γx(η, t)|2

·∂η (∫

T

∂γx(η, t)−∂γx(η−ξ, t) |x(η, t)−x(η−ξ, t)|

dξ ) dη.

(2)

Interface dynamics for incompressible flows in 2D

• We get |∂γx(γ, t)|2 = A(t).

Hou, Lowengrub and Shelley (1997)

Extra cancellations:

∂γx(γ, t) · ∂2γx(γ, t) = 0,

and ∂γx(γ, t) · ∂3γx(γ, t) = −|∂2γx(γ, t)|2.

Interface dynamics for incompressible flows in 2D

• Darcy’s law

Law µ

κ v = −∇p − (0,0, g ρ),

v velocity, p pressure, µ viscosity, κ permeability, ρ density, and g acceleration due to gravity.

Muskat (1937) Saffman and Taylor (1958)

Equation (Hele–Shaw)

12µ b2

v = −∇p − (0, g ρ),

b distance between the plates.

Interface dynamics for incompressible flows in 2D

• Smooth initial data with µ = const .

Equation

ρt + v · ∇ρ = 0 v = −∇p − (0, ρ) div v = 0

 Two-dimensional mass balanceequation in porous media (2DPM)

v(x) = 1

2π PV

∫ R2

(−2y1y2 |y |4

, y21 − y22 |y |4

) ρ(x − y)dy − 1 2

(0, ρ(x)) ,

(∂t + v · ∇)∇⊥ρ = (∇v)∇⊥ρ.

Interface dynamics for incompressible flows in 2D

• Local existence. Singularities with infinite energy, with a stream function given by

ψ(x1, x2, t) = x2f (x1, t) + g(x1, t).

Blow-up ⇔ ∫ T

0 ‖∇ρ‖BMO(t) dt =∞.

Geometric constraints: η = ∇⊥ρ |∇⊥ρ|

.

Numerical simulations: ‖∇ρ‖L∞(t) ∼ et .

Interface dynamics for incompressible flows in 2D

• Cordoba-Gancedo (2007)

We consider the case where the fluid has different densities, that is ρ is represented by

ρ(x1, x2, t) = { ρ1, {x2 > f (x1, t)} ρ2, {x2 < f (x1, t)}

being f the interface. Then we have

Equation

df dt

(x , t) = ρ2 − ρ1

2π PV

∫ R

(∂x f (x , t)− ∂x f (x − α, t))α α2 + (f (x , t)− f (x − α, t))2

f (x ,0) = f0(x); x ∈ R.

ρt + v · ∇ρ = 0 v = −∇p − (0, ρ) div v = 0

 ⇔ 2DPM contour equation Interface dynamics for incompressible flows in 2D

• The linearized equation

If we neglect the terms of order greater than one

ft = ρ1 − ρ2

2 (H∂x f ) =

ρ1 − ρ2 2

Λf ,

f (x ,0) = f0(x).

Applying the Fourier transform we get

f̂ (ξ) = f̂0(ξ)e ρ1−ρ2

2 |ξ|t .

Problem ρ1 < ρ2 stable case, ρ1 > ρ2 unstable case.

Interface dynamics for incompressible flows in 2D

• Local well-posedness for the stable case (ρ2 > ρ1)

The following theorem follows

Theorem

Let f0(x) ∈ Hk for k ≥ 3 and ρ2 > ρ1. Then there exists a time T > 0 so that there is a unique solution to 2DPM contour equation in C1([0,T ]; Hk ) with f (x ,0) = f0(x).

Interface dynamics for incompressible flows in 2D

• Global existence?

L∞ decays.∫ f (x , t) dx =

∫ f0(x) dx .

Global existence for small initial data: ∑ |ξ||̂f (ξ)|

• Ill-posedness for the unstable case (ρ1 > ρ2)

We define ‖g‖b =

∑ |ĝ(ξ)|eb|ξ| b ≥ 0.

If ‖g‖b 0.

Interface dynamics for incompressible flows in 2D

• We take fλ(x1, t) = λ−1f (λx1,−λt + λ1/2),

{fλ}λ>0 a family of solutions to the unstable case. Then

‖fλ‖Hs (0) = |λ|s− 3 2 ‖f‖Hs (λ

1 2 ) ≤ C|λ|s−

3 2 ‖f‖1(λ

1 2 ) ≤ C|λ|s−

3 2 e−λ

1/2 ,

and

‖fλ‖Hs (λ−1/2) = |λ|s− 3 2 ‖f‖Hs (0) ≥ |λ|s−

3 2 C‖Λ1+γ+ζ f0‖0 =∞,

for s > 3/2 and γ, ζ small enough.

Theorem Let s > 3/2, then for any ε > 0 there exists a solution f of 2DPM contour equation with ρ1 > ρ2 and 0 < δ < ε such that ‖f‖Hs (0) ≤ ε and ‖f‖Hs (δ) =∞.

Interface dynamics for incompressible flows in 2D

• The Muskat problem

Equation

ρt + u · ∇ρ = 0, ∇ · u = 0, µ

κ u = −∇p − (0, gρ),

(µ, ρ)(x , t) = {

(µ1, ρ1), x ∈ Ω1(t) (µ2, ρ2), x ∈ Ω2(t),

Siegel, Caflish & Howison (2004)