Global in time, shift of stability and mixing solutions for the - - PowerPoint PPT Presentation

Global in time, shift of stability and mixing solutions for the Muskat problem.

Diego Córdoba ICMat-CSIC (Madrid) Porquerolles September, 2018

The Muskat problem (Muskat (1934), Saffman & Taylor (1958) )

In this talk:

◮ Scenario in R2 ◮ Finite energy ◮ No surface tension ◮ µ1 = µ2 We consider:

• 1. Open curves vanishing at infinity

lim

α→∞(z(α, t) − (α, 0)) = 0,

• 2. Periodic curves in the space variable

z(α + 2kπ, t) = z(α, t) + 2kπ(1, 0).

• 3. Closed curves ⇒ Unstable regime.

Recent Results

◮ Stability shifting: Stable to Unstable and back to Stable. with J. Gómez-Serrano and A. Zlatos. 2017 ◮ Mixing solutions. with A. Castro and D. Faraco. arXiv:1605.04822 ◮ Global existence with arbitrarily large slope. with O. Lazar. arXiv 2018

Incompressible porous media equation in R2

Two-dimensional mass balance equation in porous media (2D IPM)

• ρt + u · ∇ρ = 0

µ κu = −∇p − (0, gρ)

div u = 0

Incompressible porous media equation in R2

Two-dimensional mass balance equation in porous media (2D IPM)

• ρt + u · ∇ρ = 0

µ κu = −∇p − (0, gρ)

div u = 0 Remark: let µ = κ = g = 1 ◮ u(x) =

1 2πPV

• R2(−2 y1y2

|y|4 , y2

1−y2 2

|y|4 ) ρ(x − y)dy − 1 2 (0, ρ(x)) ,

◮ ||ρ||Lp(t) = ||ρ||Lp(0) p ∈ [1, ∞] = ⇒ ||u||Lp(t) ≤ C p ∈ (1, ∞) ◮ (∂t + u · ∇)∇⊥ρ = (∇u)∇⊥ρ.

Muskat: Contour equation

We consider ρ(x, t) = ρ1 x ∈ Ω1(t) ρ2 x ∈ Ω2(t) with ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α ∈ R}.

Muskat: Contour equation

We consider ρ(x, t) = ρ1 x ∈ Ω1(t) ρ2 x ∈ Ω2(t) with ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α ∈ R}. Darcy’s law: u = −∇p − (0, ρ) ⇒ ∇⊥ · u = −∂x1ρ. ∇⊥ · u(x, t) = −(ρ2 − ρ1)∂αz2(α, t)δ(x − z(α, t)).

Muskat: Contour equation

We consider ρ(x, t) = ρ1 x ∈ Ω1(t) ρ2 x ∈ Ω2(t) with ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α ∈ R}. Darcy’s law: u = −∇p − (0, ρ) ⇒ ∇⊥ · u = −∂x1ρ. ∇⊥ · u(x, t) = −(ρ2 − ρ1)∂αz2(α, t)δ(x − z(α, t)). Biot-Savart: u(x, t) = −ρ2 − ρ1 2π PV

• R

(x − z(β, t))⊥ |x − z(β, t)|2 ∂αz2(β, t)dβ, for x = z(α, t).

Muskat: Contour equation

We consider ρ(x, t) = ρ1 x ∈ Ω1(t) ρ2 x ∈ Ω2(t) with ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α ∈ R}. Darcy’s law: u = −∇p − (0, ρ) ⇒ ∇⊥ · u = −∂x1ρ. ∇⊥ · u(x, t) = −(ρ2 − ρ1)∂αz2(α, t)δ(x − z(α, t)). Biot-Savart: u(x, t) = −ρ2 − ρ1 2π PV

• R

(x − z(β, t))⊥ |x − z(β, t)|2 ∂αz2(β, t)dβ, for x = z(α, t). uL2(t) < ∞.

Muskat: Contour equation

Taking limits uj(z(α, t), t) = − (ρ2−ρ1)BR(z, ∂αz2)(α, t) ∓ ∂αz2(α, t) 2|∂αz(α, t)|2 ∂αz(α, t), where BR(z, f)(α, t) = 1 2π PV

• R

(z(α, t) − z(β, t))⊥ |z(α, t) − z(β, t)|2 f(β, t)dβ.

Muskat: Contour equation

It yields zt(α, t) = −(ρ2 − ρ1)BR(z, ∂αz2)(α, t) + c(α, t)∂αz(α, t), for c parametrization freedom.

Muskat: Contour equation

It yields zt(α, t) = −(ρ2 − ρ1)BR(z, ∂αz2)(α, t) + c(α, t)∂αz(α, t), for c parametrization freedom. zt(α) = ρ2 − ρ1 2π PV

• R

z1(α) − z1(β) |z(α) − z(β)|2 (∂αz(α) − ∂βz(β))dβ. ◮ SOLUTIONS OF THE MUSKAT PROBLEM =

⇒ WEAK SOLUTIONS OF IPM

Contour equation as a graph

◮ The equation for a graph z(α, t) = (α, f(α, t)). αt =ρ2 − ρ1 2π

• R

(α − β)(∂αα − ∂ββ) (α − β)2 + (f(α) − f(β))2 dβ (0 = 0) ft(α) = ρ2 − ρ1 2π

• R

(α − β)(∂αf(α) − ∂βf(β)) (α − β)2 + (f(α) − f(β))2 dβ with initial data z1(α, 0) =α z2(α, 0) =f(α, 0) = f0(α).

The linearized equation

f L

t (α, t) = −ρ2 − ρ1

2 Λ(f L)(α, t), Λ = (−∆)1/2. Fourier transform:

• f L(ξ, t) =

f0(ξ, t) exp

• − ρ2 − ρ1

2 |ξ|t

• .

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

Local existence theory

For a general interface ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)), α ∈ R} after taking k derivatives (k ≥ 3) it can be shown that ∂t∂k

αz(α, t) = − (ρ2 − ρ1) ∂αz1(α, t)

|∂αz(α, t)|2

• σ(α,t)≡R−T

Λ∂k

αz(α, t) + l.o.t.

Thus we can distinguish three regimes: ◮ Stable regime: σ > 0 = ⇒ the denser fluid is always below. The Muskat problem is locally well-posed in time in Sobolev’s spaces. ◮ Fully unstable regime: σ < 0 = ⇒ the denser fluid is always above. The Muskat problem is ill-posed in Sobolev’s spaces. ◮ Partial unstable regime: σ has not a defined sign = ⇒ there is a part of the interface where the denser fluid is above.

Energy estimates for the stable regime ρ2 > ρ1

For k = 3: d dt||f||2

H3 = −

• σ(α)∂3

αf(α)Λ∂3 αf(α)dα + Controlled Quantities

Then, since σ > 0, yields −

• σ(α)∂3

αf(α)Λ∂3 αf(α) ≤ −1

2

• σ(α)Λ
• ∂3

αf(α)

2 dα ≤ −1 2

• Λσ(α)
• ∂3

αf(α)

2 Finally we obtain d dt||f||2

H3 ≤ C||f||m H3

Local existence results in the stable regime

◮ D.C. and F. Gancedo (2007). Local existence in H3 (and ill-posedness for ρ2 < ρ1). ◮ A. Cheng, R. Granero and S. Shkoller (2016). Local existence in H2. ◮ P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol (2017). Local existence in W2,p for p>1. ◮ B-V. Matioc (arxiv). Local existence in H

3 2 +ǫ.

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

◮ Maximum principle for the L2−norm ||f(·, t)||2

L2(R) +

T

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ = ||f0||2

L2(R)

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

◮ Maximum principle for the L2−norm ||f(·, t)||2

L2(R) +

T

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ = ||f0||2

L2(R)

Compare with the linear case ||f(·, t)||L2(R) + T

• R

f(α) − f(β) α − β 2 dαdβ

• =
• R f(x)Λf(x)dx=||Λ

1 2 f(·,t)||2 L2(R)

dt = ||f0||2

L2(R)

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

◮ Maximum principle for the L2−norm ||f(·, t)||2

L2(R) +

T

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ = ||f0||2

L2(R)

Compare with the linear case ||f(·, t)||L2(R) + T

• R

f(α) − f(β) α − β 2 dαdβ

• =
• R f(x)Λf(x)dx=||Λ

1 2 f(·,t)||2 L2(R)

dt = ||f0||2

L2(R)

But 1 2

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ ≤ C||f(·, t)||L1

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

◮ Maximum principle for the L2−norm ||f(·, t)||2

L2(R) +

T

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ = ||f0||2

L2(R)

◮ Maximum principle: fL∞(t) ≤ fL∞(0). Periodic case: f − 1 2π

• T

f0dαL∞(t) ≤ f0− 1 2π

• T

f0dαL∞e−Ct. Flat at infinity: fL∞(t) ≤ f0L∞ 1 + Ct .

Conserved quantities in the stable regime: (z1, z2) = (α, f(α, t))

◮

• f(α, t) dα =
• f0(α) dα.

◮ Maximum principle for the L2−norm ||f(·, t)||2

L2(R) +

T

• R
• R

log

• 1 +

f(α) − f(β) α − β 2 dαdβ = ||f0||2

L2(R)

◮ Maximum principle: fL∞(t) ≤ fL∞(0). Periodic case: f − 1 2π

• T

f0dαL∞(t) ≤ f0− 1 2π

• T

f0dαL∞e−Ct. Flat at infinity: fL∞(t) ≤ f0L∞ 1 + Ct . ◮ Maximum principle: If fαL∞(0) < 1 then fαL∞(t) ≤ fαL∞(0).

Global existence for ||∂αf0||L∞(R) < 1

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface.

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface. ◮ D. C., P. Constantin, F. Gancedo and R. Strain (2013). Global Lispschitz solutions if ||∂αf0||L∞(R) < 1.

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface. ◮ D. C., P. Constantin, F. Gancedo and R. Strain (2013). Global Lispschitz solutions if ||∂αf0||L∞(R) < 1. ◮ D. C., P. Constantin, F. Gancedo, L. Piazza and R. Strain (2016). Global existence in H3 if

• R

|ξ||ˆ f0(ξ)|dξ < 1 3 (⇒ ||∂αf0||L∞(R) < 1)

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface. ◮ D. C., P. Constantin, F. Gancedo and R. Strain (2013). Global Lispschitz solutions if ||∂αf0||L∞(R) < 1. ◮ D. C., P. Constantin, F. Gancedo, L. Piazza and R. Strain (2016). Global existence in H3 if

• R

|ξ||ˆ f0(ξ)|dξ < 1 3 (⇒ ||∂αf0||L∞(R) < 1) ◮ P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol (2017). (1) Criterion for blow-up with ||∂αf||L∞(R) ≤ C. (2) Global existence in W2,p with initial data f0 ∈ W2,p and ||∂αf0||L∞(R) ≤ ǫ.

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface. ◮ D. C., P. Constantin, F. Gancedo and R. Strain (2013). Global Lispschitz solutions if ||∂αf0||L∞(R) < 1. ◮ D. C., P. Constantin, F. Gancedo, L. Piazza and R. Strain (2016). Global existence in H3 if

• R

|ξ||ˆ f0(ξ)|dξ < 1 3 (⇒ ||∂αf0||L∞(R) < 1) ◮ P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol (2017). (1) Criterion for blow-up with ||∂αf||L∞(R) ≤ C. (2) Global existence in W2,p with initial data f0 ∈ W2,p and ||∂αf0||L∞(R) ≤ ǫ. ◮ B-V. Matioc (arxiv2016). Global existence in H

3 2 +ǫ with small initial data in

H

3 2 +ǫ.

Global existence for ||∂αf0||L∞(R) < 1

◮ D.C. and F. Gancedo (2007). Global existence and gain of analyticity from a perturbation of flat interface. ◮ D. C., P. Constantin, F. Gancedo and R. Strain (2013). Global Lispschitz solutions if ||∂αf0||L∞(R) < 1. ◮ D. C., P. Constantin, F. Gancedo, L. Piazza and R. Strain (2016). Global existence in H3 if

• R

|ξ||ˆ f0(ξ)|dξ < 1 3 (⇒ ||∂αf0||L∞(R) < 1) ◮ P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol (2017). (1) Criterion for blow-up with ||∂αf||L∞(R) ≤ C. (2) Global existence in W2,p with initial data f0 ∈ W2,p and ||∂αf0||L∞(R) ≤ ǫ. ◮ B-V. Matioc (arxiv2016). Global existence in H

3 2 +ǫ with small initial data in

H

3 2 +ǫ.

◮ S. Cameron (arxiv2017). Global classical solutions if ||∂αf0||L∞(R) < 1.

What happens if ||∂αf0||L∞(R) > 1 (with finite energy)?

◮ Numerical simulations of Turning (i.e. shift of stability) by Maria López-Fernández

What happens if ||∂αf0||L∞(R) > 1 (with finite energy)?

◮ Numerical simulations of Turning (i.e. shift of stability) by Maria López-Fernández ◮ Theorem (2012): ∃f0 ∈ H4 and a T∗ st limt→T∗||∂αf||L∞(R) = ∞ (joint work with A. Castro, C. Fefferman, F. Gancedo and M. López-Fernandez).

What happens if ||∂αf0||L∞(R) > 1 (with finite energy)?

◮ Numerical simulations of Turning (i.e. shift of stability) by Maria López-Fernández ◮ Theorem (2012): ∃f0 ∈ H4 and a T∗ st limt→T∗||∂αf||L∞(R) = ∞ (joint work with A. Castro, C. Fefferman, F. Gancedo and M. López-Fernandez). ◮ Numerical evidence of turning with ||∂αf0||L∞ = 22 by J. Gómez-Serrano. Is there a turning for ||∂αf0||L∞ = 1 + ǫ?

What happens after Turning?

◮ In the stable regime a solution of Muskat becomes immediately real-analytic and then passes to the unstable regime in finite time. Moreover, the Cauchy-Kowalewski theorem shows that a real-analytic Muskat solution continues to exist for a short time after the turnover.

What happens after Turning?

◮ In the stable regime a solution of Muskat becomes immediately real-analytic and then passes to the unstable regime in finite time. Moreover, the Cauchy-Kowalewski theorem shows that a real-analytic Muskat solution continues to exist for a short time after the turnover. ◮ Breakdown of smoothness (2013): There exist interfaces of the Muskat problem such that after turnover their smoothness breaks down (is not C4). Joint work with A. Castro, C. Fefferman and F. Gancedo.

What happens after Turning?

◮ In the stable regime a solution of Muskat becomes immediately real-analytic and then passes to the unstable regime in finite time. Moreover, the Cauchy-Kowalewski theorem shows that a real-analytic Muskat solution continues to exist for a short time after the turnover. ◮ Breakdown of smoothness (2013): There exist interfaces of the Muskat problem such that after turnover their smoothness breaks down (is not C4). Joint work with A. Castro, C. Fefferman and F. Gancedo. ◮ Double shift of stability (2017): Turning stable-unstable-stable (also unstable-stable-unstable). Joint work with J. Gómez-Serrano and A. Zlatos.

The transition u-s-u

−8 −6 −4 −2 2 4 6 8 x 10

−3

−2 −1 1 2 x y

t = 0, thick line; t = −1.2 · 10−2, broken line; t = −2.4 · 10−2, dotted line; t = −3.6 · 10−2, broken and dotted line; t = −4.92 · 10−2, thin line.

The transition u-s-u

Theorem

There exists a solution to the Muskat problem that starts unstable, becomes stable, and finally returns to the stable regime. All transitions occur in finite time. Main steps of the proof: ◮ We ”decouple” the influence of the regions by choosing R large enough. The center region transitions from stable to unstable at t = 0. The tail regions transition from unstable to stable at t = 0.

−25 −20 −15 −10 −5 5 10 15 20 25 −3 −2 −1 1 2 3

R R

◮ We perturb this scheme to ensure analyticity of the solution and to make the s-u transition at t1 > 0 and the u-s transition at t2 < 0.

The transition s-u-s

Theorem

There exists a solution to the Muskat problem that does a s-u-s transition. ◮ Harder, since one has to ensure that every piece which is in the unstable regime goes back to the stable one. Before, if one piece turned unstable it was enough. ◮ We will start in the unstable regime, then show that forwards and backwards in time we move onto the stable one. ◮ The important quantity to look at is ∂xz1(0, t).

The transition s-u-s

Let ε ≥ 0 and consider z1

ε(x, 0) = x − sin(x) − ε sin(x),

z2

ε(x, 0) = A(ε) sin(2x).

• 1. For any ε ∈ [0, 10−6], there exists A(ε) ∈ (1.08050, 1.08055) such that if zε is

a solution to the Muskat problem with initial data zε(x, 0), then ∂t∂xz1

ε(0, 0) = 0.

• 2. Fixing A(ε) for each ε, the solution satisfies

∂tt∂xz1

ε(0, 0) ≥ 30

(1) ◮ ∂xz1(0, t) ∼ −ε + Ct2. For small times T1 < 0, T2 > 0, we transition s-u at T1 and u-s at T2. ◮ The estimates are computer-assisted (too many terms to bound all of them by hand). ◮ One can piece together a similar construction as in the u-s-u case to show an u-s-u-s-u transition.

Global existence for arbitrarily large slope

Global existence for arbitrarily large slope

◮ F. Deng, Z. Lei and F.Lin (2017). Global existence for arbitrarily large monotonic initial data (Not in L2).

Global existence for arbitrarily large slope

◮ F. Deng, Z. Lei and F.Lin (2017). Global existence for arbitrarily large monotonic initial data (Not in L2). ◮ S. Cameron (2018 updated version of the arxives): Global existence for (supxf ′

0(x))(supy − f ′ 0(y)) < 1

Global existence for arbitrarily large slope

◮ F. Deng, Z. Lei and F.Lin (2017). Global existence for arbitrarily large monotonic initial data (Not in L2). ◮ S. Cameron (2018 updated version of the arxives): Global existence for (supxf ′

0(x))(supy − f ′ 0(y)) < 1

Theorem

Assume f0 ∈ H5/2 with f0 ˙

H3/2 small enough, then, there exists a unique strong

solution f which verifies f ∈ L∞([0, T], H3/2) ∩ L2([0, T], ˙ H5/2), for all T > 0. Joint work with O. Lazar.

Global existence for arbitrarily large slope: proof

Main steps of the proof: ◮ The proof is based on the use of a new formulation of the Muskat equation that involves oscillatory terms as well as a careful use of Besov space techniques. ◮ ft(t, x) = ρ π P.V.

• ∂x∆αf

∞ e−δ cos(δ∆αf) dδ dα f(0, x) = f0(x). where ∆αf ≡ f(x,t)−f(x−α,t)

α

.

Global existence for arbitrarily large slope: proof

◮ A priori estimates in ˙ H3/2: 1 2∂tf2

˙ H3/2

=

• Hfxx
• ∂xx∆αf

∞ e−δ cos(δ∆αf(x)) dδ dα dx −

• Hfxx
• (∂x∆αf)2

∞ δe−δ sin(δ∆αf(x)) dδ dα dx = I1 + I2 We can estimate |I2| ≤ f2

˙ H2f ˙ H3/2

and the most singular term is I1 |I1| f2

H2(f2 ˙ H3/2 + f ˙ H3/2) − πf2 ˙ H2 + π

K2 1 + K2 f2

˙ H2

where K = fxL∞L∞.

Global existence for arbitrarily large slope: proof

◮ A priori estimates in ˙ H3/2: 1 2∂tf2

˙ H3/2

=

• Hfxx
• ∂xx∆αf

∞ e−δ cos(δ∆αf(x)) dδ dα dx −

• Hfxx
• (∂x∆αf)2

∞ δe−δ sin(δ∆αf(x)) dδ dα dx = I1 + I2 We can estimate |I2| ≤ f2

˙ H2f ˙ H3/2

and the most singular term is I1 |I1| f2

H2(f2 ˙ H3/2 + f ˙ H3/2) − πf2 ˙ H2 + π

K2 1 + K2 f2

˙ H2

where K = fxL∞L∞. ◮ Then 1 2∂tf2

˙ H3/2 +

π 1 + K2 f2

˙ H2 ≤ Cf2 ˙ H2

• f2

˙ H3/2 + f ˙ H3/2

Global existence for arbitrarily large slope: proof

◮ Similar a priori estimates in ˙ H5/2:

Lemma

Let T > 0 and f0 ∈ ˙ H5/2 ∩ ˙ H3/2 so that f0 ˙

H3/2 < C(f0,xL∞), then we have

f2

˙ H5/2(T)

+ π 1 + M2 T f2

˙ H3 ds

• f0 ˙

H5/2 +

• fL∞([0,T], ˙

H3/2) + f2 L∞([0,T], ˙ H3/2)

T f2

˙ H3 ds

where M is the space-time Lipschitz norm of f.

Fully unstable regime ρ1 > ρ2

Previous work: ◮ Ill-posedness in Sobolev spaces (D.C.-F. Gancedo 2007)

Theorem

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

Fully unstable regime ρ1 > ρ2

Previous work: ◮ Ill-posedness in Sobolev spaces (D.C.-F. Gancedo 2007)

Theorem

Let s > 3/2, then for any ε > 0 there exists a solution f of the Muskat equation with ρ1 > ρ2 and 0 < δ < ε such that fHs(0) ≤ ε and fHs(δ) = ∞. ◮ CAN WE STILL FIND WEAK SOLUTIONS FOR IPM?

Fully unstable regime ρ1 > ρ2

Previous work: ◮ Ill-posedness in Sobolev spaces (D.C.-F. Gancedo 2007)

Theorem

Let s > 3/2, then for any ε > 0 there exists a solution f of the Muskat equation with ρ1 > ρ2 and 0 < δ < ε such that fHs(0) ≤ ε and fHs(δ) = ∞. ◮ CAN WE STILL FIND WEAK SOLUTIONS FOR IPM? ◮ Mixing solutions from a flat interface (Székelyhidi 2012) ρ(x, t) =    +1 x ∈ {x2 ≥ αt} ±1 x ∈ {−αt < x2 < αt} −1 x ∈ {x2 < −αt} for α ∈ (0, 2).

Székelyhidi’s construction

Remarks: ◮ The solution starts in the fully unstable regime being flat. There exist a solution for the Muskat equations: the flat interface is a stationary solutions ◮ There is a mixing zone: {−αt < x2 < αt}. ◮ The solutions are not unique:

◮ For different values of α (the speed of the opening of the mixing zone) we have different solutions. ◮ For a fixed value of α, inside of the mixing zone there are infinitely many different densities.

2t

The definition of a mixing solution

The density ρ(x, t) and the velocity (x, t) are a "mixing solution" of the IPM system if they are a weak solution and also there exist, for every t ∈ [0, T], open simply connected domains Ω±(t) and Ωmix(t) with Ω+ ∪ Ω− ∪ Ωmix = R2 such that, for almost every (x, t) ∈ R2 × [0, T], the following holds: ρ(x, t) =

• ρ±

in Ω±(t) (ρ − ρ+)(ρ − ρ−) = 0 in Ωmix(t) . For every r > 0, x ∈ R2, 0 < t < T B((x, t), r) ⊂ ∪0<t<TΩmix(t) it holds that

• B

(ρ − ρ+)

• B

(ρ − ρ−) = 0

Main theorem: joint work with A. Castro and D. Faraco (arxiv2016)

Theorem

Let Γ(0) = {z0(s) = (z0

1(s), z0 2(s)) ∈ R2} with z0(s) − (s, 0) ∈ H5. We will assume

that Γ(0) is run from left to right and that

∂sz0

1(s)

|∂sz0(s)| > 0. Let us suppose that

ρ+ < ρ−. Then there exist infinitely many "mixing solutions" starting with the inital data of Muskat type given by Γ(0) (in the fully unstable regime) for the IPM system. New results: ◮ Piecewise constant subsolutions. C. Forster and L. Székelyhidi (arxiv2017) ◮ Linear degraded mixing solutions. A. Castro, D. Faraco and F. Mengual (arxiv2018)

Subsolution implies Mixing solutions

Definition

We will say that (ρ, u, m) is a subsolution of the IPM system if there exist open simply connected domains Ω±(t) and ΩM(t) with Ω+ ∪ Ω− ∪ ΩM = Ω and such that the following holds: ◮ The flow is incompressible ∇ · u = 0 in Ω. ◮ In Ω (ρ, u, m) satisfy the equations ∂tρ + ∇ · m =0 ∇⊥ · u = − ∂x1ρ.

◮ The density and m satisfies ρ(x, t) = ±1 m = ρu in Ω± and it is continuous in Ω. ◮ In ΩM the functions (ρ, u, m) are in the "Mixing hull"

• m − ρu + 1

2

• 0, 1 − ρ2
• 2

< 1 2

• 1 − ρ22

ρ2 < 1 Laszlo Székelyhidi 2012 proved

Theorem

Given a subsolution (ρ, u, m) there exist infinitely many mixing solutions (ρ, u) ∈ L∞ × L∞ such that (ρ, u) = (ρ, u) in Ω \ ΩM.

Example of a Mixing solution

The mixing zone ΩM = {x ∈ Ω : |x2| < αt}. then ρ(x, t) =    −1 x2 < −αt,

x2 2αt

|x2| < αt 1 x2 > αt, , m = (0, −α(1 − ρ2)) u = (0, 0) is a subsolution with α ∈ (0, 2).

Convex integration for IPM

◮ This work is based on a variant of the method of convex integration introduced for Euler equations by C. de Lellis and L. Székelyhidi Jr. Lack of uniqueness and Onsager’s conjecture (E. Wiedemann, C. Bardos, A. Choffrut, P. Isett, T. Buckmaster, V. Vicol...). ◮ D. C., D. Faraco and F. Gancedo. Lack of uniqueness for IPM. ◮ R. Shvydkoy. Non-uniqueness for active scalars with a divergence free velocity given by a Fourier multiplier operator with an even symbol. ◮ L. Székelyhidi. Lack of uniqueness for IPM (computation of the Λ-convex hull). Flat mixing solutions. ◮ P. Isett, V. Vicol. Global existence of weak solutions for active scalars with multipliers that are not odd from arbitrarily smooth initial data. And there exist nontrivial solutions, compact support in time, having any Holder regularity ρ ∈ Cα

t,x with α < 1 9.

Constructing a subsolution (ρ, u, m) from z0 ∈ Hk

We define the set ΩM ⊂ R2 ΩM = {x ∈ R2 : x = x(s, λ) for (s, λ) ∈ (−∞, ∞) × (−ε, ε)}. with x(s, λ) = z(s, t) + (0, λ) We take ρ(x) =

• ±1

in Ω±

λ ǫ

in ΩM , by Biot-Savart u(x) = 1 π ∞

−∞

∂sz(s′) 1 2ε ε

−ε

x1 − x1(s′, λ′) |x − x(s′, λ′)|2 dλ′ds′.

Constructing a subsolution (ρ, u, m) from z0 ∈ Hk

We take m = ρu − (β, α)

• 1 − ρ2

then ∂tρ + u · ∇ρ = ∇ ·

• (β, α)
• 1 − ρ2

Given (ε(t), z(s, t)) we have (ρ, u, m)

Constructing a subsolution (ρ, u, m) from z0 ∈ Hk

If there is a solution (ε(t), z(s, t)) to the following system ∂tz(s, t) =M[z, ε](s, t) z(s, 0) = z0(s) ∂tε(t) = c ε(0) = 0, where c > 0 is a constant and the velocity M[z, ε](s, t) is given by

M[z, ε](s, t) = − 1 4ε2π ε

−ε

∞

−∞

ε

−ε

(∂sz(s) − ∂sz(s′))(z1(s) − z1(s′)) |z(s) − z(s′) + (λ − λ′)(0, 1)|2 dλ′ds′dλ. then the solution is in the Mixing Hull.

Contour dynamics

◮ We have to solve the equation ∂tz(s, t) =M[z, ε](s, t) z(s, 0) = z0(s) (2) ∂tε(t) = c ε(0) = 0, (3) ◮ We can use that the the interface can be parametrize as the graph a function f(x, t), (z1(s, t), z2(s, t)) = (x, f(x, t)) ◮ Quasi-linearization. We take ∂5

xf(x, t) = F(x, t). We can write

∂tF(x, t) =

• K∂xf(x,t)(x − y)∂xF(y, t)dy + a(x, t)∂xF(y, t)dy + G(x, t)

(4) where G(x, t) and a(x, t) are low order functions.

The kernel KA

Taking ε = t (c = 1) KA(y, t) = 1 4πt2

• −2Ay arctan (A) + (2t + Ay) arctan

2t + Ay y

• + (Ay − 2t) arctan

Ay − 2t y

• + y log
• y2(1 + A2)
• − y

2 log

• y2 + (2t + Ay)2

− y 2 log

• y2 + (Ay − 2t)2

ˆ KA(ξ, t) = −isign(ξ) 2π|ξ|t

• 1 +

1 4π|ξ|t

• e−4π|ξ|tσ (cos(4π|ξ|tσA)

−A sin(4π|ξ|tσA)) − 1)} .

A very simplified model

Toy model ∂tf = D−1 ∗ Λf

• D−1 =

1 1 + t|ξ| f(ξ, 0) = f0(x) In the Fourier side ∂tˆ f(ξ) = |ξ| 1 + |ξ|t ˆ f(ξ, t) = ⇒ ˆ f(ξ, t) = (1 + t|ξ|)ˆ f0(ξ) Energy estimates d||D−1f||L2 dt = 0, ||D−1f||L2 =||f||L2 ||f||Hs =||f0||Hs+1

Existence of solutions z(s, t) ∈ Hk

Let F(s, t) = ∂5)

s z(s, t) then

∂tF = L ∗ ΛF + a(s, t)∂sF + l.o.t. where L ∗ f(ξ) ∼

1 1+t|ξ|ˆ

f(ξ). Goal: By choosing the correct energy we can prove ||F(t)||Hs ≤ C(T)||F0||Hs+1.

Mixing in the stable regime

We can obtain mixing in the stable regime.

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