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 R 2 ◮ Finite energy ◮ No surface tension ◮ µ 1 = µ 2 We consider: 1. Open curves vanishing at infinity α →∞ ( z ( α, t ) − ( α, 0 )) = 0 , lim 2. Periodic curves in the space variable z ( α + 2 k π, t ) = z ( α, t ) + 2 k π ( 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 R 2 � ρ t + u · ∇ ρ = 0 Two-dimensional mass balance µ κ u = −∇ p − ( 0 , g ρ ) equation in porous media (2D IPM) div u = 0

Incompressible porous media equation in R 2 � ρ t + u · ∇ ρ = 0 Two-dimensional mass balance µ κ u = −∇ p − ( 0 , g ρ ) equation in porous media (2D IPM) div u = 0 Remark: let µ = κ = g = 1 � | y | 4 , y 2 1 − y 2 1 R 2 ( − 2 y 1 y 2 | y | 4 ) ρ ( x − y ) dy − 1 ◮ u ( x ) = 2 π PV 2 ( 0 , ρ ( x )) , 2 ◮ || ρ || L p ( t ) = || ρ || L p ( 0 ) p ∈ [ 1 , ∞ ] = ⇒ || u || L p ( t ) ≤ C p ∈ ( 1 , ∞ ) ◮ ( ∂ t + u · ∇ ) ∇ ⊥ ρ = ( ∇ u ) ∇ ⊥ ρ.

Muskat: Contour equation We consider � ρ 1 x ∈ Ω 1 ( t ) ρ ( x , t ) = ρ 2 x ∈ Ω 2 ( t ) with ∂ Ω j ( t ) = { z ( α, t ) = ( z 1 ( α, t ) , z 2 ( α, t )) : α ∈ R } .

Muskat: Contour equation We consider � ρ 1 x ∈ Ω 1 ( t ) ρ ( x , t ) = ρ 2 x ∈ Ω 2 ( t ) with ∂ Ω j ( t ) = { z ( α, t ) = ( z 1 ( α, t ) , z 2 ( α, t )) : α ∈ R } . Darcy’s law: u = −∇ p − ( 0 , ρ ) ⇒ ∇ ⊥ · u = − ∂ x 1 ρ. ∇ ⊥ · u ( x , t ) = − ( ρ 2 − ρ 1 ) ∂ α z 2 ( α, t ) δ ( x − z ( α, t )) .

Muskat: Contour equation We consider � ρ 1 x ∈ Ω 1 ( t ) ρ ( x , t ) = ρ 2 x ∈ Ω 2 ( t ) with ∂ Ω j ( t ) = { z ( α, t ) = ( z 1 ( α, t ) , z 2 ( α, t )) : α ∈ R } . Darcy’s law: u = −∇ p − ( 0 , ρ ) ⇒ ∇ ⊥ · u = − ∂ x 1 ρ. ∇ ⊥ · u ( x , t ) = − ( ρ 2 − ρ 1 ) ∂ α z 2 ( α, t ) δ ( x − z ( α, t )) . Biot-Savart: � u ( x , t ) = − ρ 2 − ρ 1 ( x − z ( β, t )) ⊥ PV | x − z ( β, t ) | 2 ∂ α z 2 ( β, t ) d β, 2 π R for x � = z ( α, t ) .

Muskat: Contour equation We consider � ρ 1 x ∈ Ω 1 ( t ) ρ ( x , t ) = ρ 2 x ∈ Ω 2 ( t ) with ∂ Ω j ( t ) = { z ( α, t ) = ( z 1 ( α, t ) , z 2 ( α, t )) : α ∈ R } . Darcy’s law: u = −∇ p − ( 0 , ρ ) ⇒ ∇ ⊥ · u = − ∂ x 1 ρ. ∇ ⊥ · u ( x , t ) = − ( ρ 2 − ρ 1 ) ∂ α z 2 ( α, t ) δ ( x − z ( α, t )) . Biot-Savart: � u ( x , t ) = − ρ 2 − ρ 1 ( x − z ( β, t )) ⊥ PV | x − z ( β, t ) | 2 ∂ α z 2 ( β, t ) d β, 2 π R for x � = z ( α, t ) . � u � L 2 ( t ) < ∞ .

Muskat: Contour equation Taking limits u j ( z ( α, t ) , t ) = − ( ρ 2 − ρ 1 ) BR ( z , ∂ α z 2 )( α, t ) ∂ α z 2 ( α, t ) ∓ 2 | ∂ α z ( α, t ) | 2 ∂ α z ( α, t ) , where � ( z ( α, t ) − z ( β, t )) ⊥ BR ( z , f )( α, t ) = 1 2 π PV | z ( α, t ) − z ( β, t ) | 2 f ( β, t ) d β. R

Muskat: Contour equation It yields z t ( α, t ) = − ( ρ 2 − ρ 1 ) BR ( z , ∂ α z 2 )( α, t ) + c ( α, t ) ∂ α z ( α, t ) , for c parametrization freedom.

Muskat: Contour equation It yields z t ( α, t ) = − ( ρ 2 − ρ 1 ) BR ( z , ∂ α z 2 )( α, t ) + c ( α, t ) ∂ α z ( α, t ) , for c parametrization freedom. � z t ( α ) = ρ 2 − ρ 1 z 1 ( α ) − z 1 ( β ) PV | z ( α ) − z ( β ) | 2 ( ∂ α z ( α ) − ∂ β z ( β )) d β. 2 π R ◮ 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 )) . � α t = ρ 2 − ρ 1 ( α − β )( ∂ α α − ∂ β β ) ( α − β ) 2 + ( f ( α ) − f ( β )) 2 d β 2 π R ( 0 = 0 ) � f t ( α ) = ρ 2 − ρ 1 ( α − β )( ∂ α f ( α ) − ∂ β f ( β )) ( α − β ) 2 + ( f ( α ) − f ( β )) 2 d β 2 π R with initial data z 1 ( α, 0 ) = α z 2 ( α, 0 ) = f ( α, 0 ) = f 0 ( α ) .

The linearized equation t ( α, t ) = − ρ 2 − ρ 1 Λ( f L )( α, t ) , Λ = ( − ∆) 1 / 2 . f L 2 Fourier transform: � � − ρ 2 − ρ 1 � f L ( ξ, t ) = � f 0 ( ξ, t ) exp | ξ | t . 2 ◮ ρ 2 > ρ 1 stable case, ◮ ρ 2 < ρ 1 unstable case.

Local existence theory For a general interface ∂ Ω j ( t ) = { z ( α, t ) = ( z 1 ( α, t ) , z 2 ( α, t )) , α ∈ R } after taking k derivatives ( k ≥ 3 ) it can be shown that α z ( α, t ) = − ( ρ 2 − ρ 1 ) ∂ α z 1 ( α, t ) ∂ t ∂ k Λ ∂ k α z ( α, t ) + l.o.t. | ∂ α z ( α, t ) | 2 � �� � σ ( α, t ) ≡ R − 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 ρ 2 > ρ 1 For k = 3: � d dt || f || 2 σ ( α ) ∂ 3 α f ( α )Λ ∂ 3 H 3 = − α f ( α ) d α + Controlled Quantities Then, since σ > 0, yields � � � � 2 d α α f ( α ) ≤ − 1 σ ( α ) ∂ 3 α f ( α )Λ ∂ 3 ∂ 3 − σ ( α )Λ α f ( α ) 2 � � � 2 ≤ − 1 ∂ 3 Λ σ ( α ) α f ( α ) 2 Finally we obtain d dt || f || 2 H 3 ≤ C || f || m H 3

Local existence results in the stable regime ◮ D.C. and F. Gancedo (2007). Local existence in H 3 (and ill-posedness for ρ 2 < ρ 1 ). ◮ A. Cheng, R. Granero and S. Shkoller (2016). Local existence in H 2 . ◮ P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol (2017). Local existence in W 2 , p for p>1. 3 2 + ǫ . ◮ B-V. Matioc (arxiv). Local existence in H

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α.

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α. ◮ Maximum principle for the L 2 − norm � � 2 � � f ( α ) − f ( β ) � T � � || f ( · , t ) || 2 d α d β = || f 0 || 2 L 2 ( R ) + log 1 + L 2 ( R ) α − β 0 R R

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α. ◮ Maximum principle for the L 2 − norm � � 2 � � f ( α ) − f ( β ) � T � � || f ( · , t ) || 2 d α d β = || f 0 || 2 L 2 ( R ) + log 1 + L 2 ( R ) α − β 0 R R Compare with the linear case � T � � f ( α ) − f ( β ) � 2 dt = || f 0 || 2 || f ( · , t ) || L 2 ( R ) + d α d β L 2 ( R ) α − β 0 R � �� � 1 � 2 f ( · , t ) || 2 = R f ( x )Λ f ( x ) dx = || Λ L 2 ( R )

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α. ◮ Maximum principle for the L 2 − norm � � 2 � � f ( α ) − f ( β ) � T � � || f ( · , t ) || 2 d α d β = || f 0 || 2 L 2 ( R ) + log 1 + L 2 ( R ) α − β 0 R R Compare with the linear case � T � � f ( α ) − f ( β ) � 2 dt = || f 0 || 2 || f ( · , t ) || L 2 ( R ) + d α d β L 2 ( R ) α − β 0 R � �� � 1 � 2 f ( · , t ) || 2 = R f ( x )Λ f ( x ) dx = || Λ L 2 ( R ) But � � 2 � � f ( α ) − f ( β ) � � 1 log 1 + d α d β ≤ C || f ( · , t ) || L 1 α − β 2 R R

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α. ◮ Maximum principle for the L 2 − norm � � 2 � � f ( α ) − f ( β ) � T � � || f ( · , t ) || 2 d α d β = || f 0 || 2 L 2 ( R ) + log 1 + L 2 ( R ) α − β 0 R R ◮ Maximum principle: � f � L ∞ ( t ) ≤ � f � L ∞ ( 0 ) . Periodic case: � � � f − 1 f 0 d α � L ∞ ( t ) ≤ � f 0 − 1 f 0 d α � L ∞ e − Ct . 2 π 2 π T T Flat at infinity: � f � L ∞ ( t ) ≤ � f 0 � L ∞ 1 + Ct .

Conserved quantities in the stable regime: ( z 1 , z 2 ) = ( α, f ( α, t )) � � ◮ f ( α, t ) d α = f 0 ( α ) d α. ◮ Maximum principle for the L 2 − norm � � 2 � � f ( α ) − f ( β ) � T � � || f ( · , t ) || 2 d α d β = || f 0 || 2 L 2 ( R ) + log 1 + L 2 ( R ) α − β 0 R R ◮ Maximum principle: � f � L ∞ ( t ) ≤ � f � L ∞ ( 0 ) . Periodic case: � � � f − 1 f 0 d α � L ∞ ( t ) ≤ � f 0 − 1 f 0 d α � L ∞ e − Ct . 2 π 2 π T T Flat at infinity: � f � L ∞ ( t ) ≤ � f 0 � L ∞ 1 + Ct . ◮ Maximum principle: If � f α � L ∞ ( 0 ) < 1 then � f α � L ∞ ( t ) ≤ � f α � L ∞ ( 0 ) .

Global existence for || ∂ α f 0 || L ∞ ( R ) < 1