Navier-Stokes equations with time-dependent boundary conditions - PowerPoint PPT Presentation

Navier-Stokes equations with time-dependent boundary conditions Sylvie Monniaux in collaboration with El Maati Ouhabaz (IMB, Bordeaux - France) I2M, Université Aix-Marseille - France Mathflows – September 2015, Porquerolles Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 1 / 14

The equations Given τ > 0, on Ω ⊂ R 3 a bounded C 1 , 1 or convex domain, we consider the following initial condition problem:  ∂ t u − ∆ u + ∇ π − u × curl u = 0 in [ 0 , τ ] × Ω     = [ 0 , τ ] × Ω div u 0 in     ν · u = [ 0 , τ ] × ∂ Ω (RNS) 0 on  ν × curl u = β u [ 0 , τ ] × ∂ Ω  on      u ( 0 ) = Ω . u 0 in  where ν is the unit exterior normal, β : [ 0 , τ ] × ∂ Ω → S 3 ( R ) is bounded, nonnegative a.e. and admits ν as eigenvector a.e. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 2 / 14

About the nonlinearity The identity for u smooth vector field ( u · ∇ ) u = − u × curl u + 1 2 ∇| u | 2 allows to rewrite the usual transport term in the Navier-Stokes equations, so that ∂ t u − ∆ u + ∇ p + ( u · ∇ ) u = 0 becomes ∂ t u − ∆ u + ∇ π − u × curl u = 0 with π = p + 1 2 | u | 2 . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 3 / 14

About the boundary conditions The boundary conditions split into two parts: ν · u = 0 on ∂ Ω � non penetration; ν × curl u = β u � Robin-type condition (with friction coefficient β ). Remark For smooth Ω , the above boundary conditions are equivalent to ν · u = 0 on ∂ Ω , � � S ( u , π ) ν tan + ( β − 2 W ) u = 0 on ∂ Ω � Navier’s slip boundary condition with friction, where W is the Weingarten map on ∂ Ω (depends on the geometry of Ω ), and S ( u , π ) is the Cauchy stress tensor: S ( u , π ) := 1 2 ( ∇ u + ( ∇ u ) ⊤ ) − π I 3 . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 4 / 14

The Leray projection Define u ∈ L 2 (Ω , R 3 ); div u = 0 in Ω and ν · u = 0 on ∂ Ω � � H := . The following orthogonal direct sum holds: ⊥ L 2 (Ω , R 3 ) = H ⊕ ∇ H 1 (Ω) . The projection P : L 2 (Ω , R 3 ) → H is the classical Leray projection. Define now � u ∈ L 2 , div u ∈ L 2 , curl u ∈ L 2 , ν · u = 0 on ∂ Ω � W T := and V := W T ∩ H . Then P : W T − → V . Remark (Amrouche, Bernardi, Dauge, Girault [M2AS, 1998]) In the case of C 1 , 1 or convex bounded domains, W T ֒ → H 1 (Ω , R 3 ) . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 5 / 14

The Leray projection Define u ∈ L 2 (Ω , R 3 ); div u = 0 in Ω and ν · u = 0 on ∂ Ω � � H := . The following orthogonal direct sum holds: ⊥ L 2 (Ω , R 3 ) = H ⊕ ∇ H 1 (Ω) . The projection P : L 2 (Ω , R 3 ) → H is the classical Leray projection. Define now � u ∈ L 2 , div u ∈ L 2 , curl u ∈ L 2 , ν · u = 0 on ∂ Ω � W T := and V := W T ∩ H . Then P : W T − → V . Remark (Amrouche, Bernardi, Dauge, Girault [M2AS, 1998]) In the case of C 1 , 1 or convex bounded domains, W T ֒ → H 1 (Ω , R 3 ) . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 5 / 14

The result Theorem (Monniaux-Ouhabaz, 2015) ∃ ε > 0 s.t. ∀ u 0 ∈ V 1 / 2 := [ H , V ] 1 / 2 with � u 0 � V 1 / 2 ≤ ε , there is a unique u ∈ H 1 ( 0 , τ ; V ′ 1 / 2 ) with t �→ A β ( t ) u ( t ) ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) and π ∈ L 2 ( 0 , τ ; H 1 / 2 ) such that ( u , π ) satisfies ( RNS ) in the sense of distributions. Recall  ∂ t u − ∆ u + ∇ π − u × curl u = [ 0 , τ ] × Ω 0 in   ( RNS ) div u = 0 in [ 0 , τ ] × Ω  ν · u = 0 , ν × curl u = β u [ 0 , τ ] × ∂ Ω .  on Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 6 / 14

The bilinear form On the Hilbert space H , we consider the bilinear form a β : V × V − → R a β ( u , v ) := � curl u , curl v � Ω + � β u , v � ∂ Ω . Thanks to the properties of β , a β is symmetric, closed and coercive. We denote by A β the associated (self adjoint) operator � � D ( A β ) = u ∈ V ; ∃ v ∈ H : a β ( u , φ ) = � v , φ � Ω , A β u = v . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 7 / 14

The operator Theorem (Monniaux-Ouhabaz, 2015) The operator A β is given by u ∈ L 2 (Ω , R 3 ); div u = 0 in Ω , curl u ∈ L 2 (Ω , R 3 ) , � D ( A β ) = curl curl u ∈ L 2 (Ω , R 3 ) , ν · u = 0 and ν × curl u = β u on ∂ Ω � , u ∈ D ( A β ) , p ∈ H 1 (Ω) . = P ( curl curl u ) = − ∆ u + ∇ p , A β u In addition, − A β generates an analytic semigroup of contractions on H and D ( A 1 / 2 β ) = V. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 8 / 14

Non autonomous maximal regularity Under the above assumptions on β : β : [ 0 , τ ] × ∂ Ω → S 3 ( R ) bounded measurable, piecewise Hölder continuous in t of order α > 1 / 4, admits a.e. ν as eigenvector for a.e. t ∈ ( 0 , τ ) and β ( t , x ) ξ · ξ ≥ 0 for almost all ( t , x ) ∈ [ 0 , τ ] × ∂ Ω and all ξ ∈ R 3 , we have that ∀ f ∈ L 2 ( 0 , τ ; V ′ ) ∃ ! u ∈ H 1 ( 0 , τ, V ′ ) ∩ L 2 ( 0 , τ ; V ) ∗ ∀ u 0 ∈ H or (Lions, 1957) ∀ f ∈ L 2 ( 0 , τ ; H ) ∃ ! u ∈ H 1 ( 0 , τ, H ) with A β u ∈ L 2 ( 0 , τ ; H ) ∗ ∀ u 0 ∈ V (Arendt-Monniaux, 2015) such that ∂ t u + A β u = f and u ( 0 ) = u 0 , with the accompanying estimates. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 9 / 14

� � � � � � Non autonomous maximal regularity, cont’d Interpolation spaces Define V 1 / 2 = [ H , V ] 1 / 2 and V ′ 1 / 2 = [ V ′ , H ] 1 / 2 : A β A 1 / 2 A 1 / 2 β β � V � V 1 / 2 � H � V ′ � V ′ D ( A β ) � � � � � � � � � � 1 / 2 A 1 / 2 A 1 / 2 β β A β Under the previous assumptions on β , interpolating between the two results, ∀ f ∈ L 2 ( 0 , τ ; V ′ ∀ u 0 ∈ V 1 / 2 , 1 / 2 ) ∃ ! u ∈ H 1 ( 0 , τ ; V ′ 1 / 2 ) with A β u ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) such that ∂ t u + A β u = f and u ( 0 ) = u 0 , with the accompanying estimates. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 10 / 14

� � � � � � Non autonomous maximal regularity, cont’d Interpolation spaces Define V 1 / 2 = [ H , V ] 1 / 2 and V ′ 1 / 2 = [ V ′ , H ] 1 / 2 : A β A 1 / 2 A 1 / 2 β β � V � V 1 / 2 � H � V ′ � V ′ D ( A β ) � � � � � � � � � � 1 / 2 A 1 / 2 A 1 / 2 β β A β Under the previous assumptions on β , interpolating between the two results, ∀ f ∈ L 2 ( 0 , τ ; V ′ ∀ u 0 ∈ V 1 / 2 , 1 / 2 ) ∃ ! u ∈ H 1 ( 0 , τ ; V ′ 1 / 2 ) with A β u ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) such that ∂ t u + A β u = f and u ( 0 ) = u 0 , with the accompanying estimates. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 10 / 14

The space of maximal regularity u ∈ H 1 ( 0 , τ ; V ′ 1 / 2 ); A β u ∈ L 2 ( 0 , τ ; V ′ � � E := 1 / 2 ) and u ( 0 ) ∈ V 1 / 2 Lemma → L 4 ( 0 , τ ; V ) . E ֒ Idea of the proof. 1 / 2 ) implies A 1 / 2 A β u ∈ L 2 ( 0 , τ ; V ′ β u ∈ L 2 ( 0 , τ ; V 1 / 2 ) ; u ∈ H 1 ( 0 , τ ; V ′ 1 / 2 ) and A β u ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) imply, by interpolation, that A 1 / 2 β u ∈ H 1 / 2 ( 0 , τ ; V ′ 1 / 2 ) ; we conclude by interpolation that A 1 / 2 β u ∈ H 1 / 4 ( 0 , τ ; H ) ֒ → L 4 ( 0 , τ ; H ) . Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 11 / 14

Fixed point theorem The idea is to rewrite the time-dependent Robin-Navier-Stokes system as a fixed point problem u = a + B ( u , u ) , u ∈ E , with a the solution of ∂ t a + A β a = 0 , a ( 0 ) = u 0 ∈ V 1 / 2 , and w = B ( u , u ) the solution of ∂ t w + A β w = P ( u × curl u ) ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) , w ( 0 ) = 0 . Lemma (Picard’s contraction principle) 1 Assume that � a � E ≤ 4 � B � E × E → E . Then v �→ a + B ( v , v ) has a unique fixed point in E. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 12 / 14

Fixed point theorem The idea is to rewrite the time-dependent Robin-Navier-Stokes system as a fixed point problem u = a + B ( u , u ) , u ∈ E , with a the solution of ∂ t a + A β a = 0 , a ( 0 ) = u 0 ∈ V 1 / 2 , and w = B ( u , u ) the solution of ∂ t w + A β w = P ( u × curl u ) ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) , w ( 0 ) = 0 . Lemma (Picard’s contraction principle) 1 Assume that � a � E ≤ 4 � B � E × E → E . Then v �→ a + B ( v , v ) has a unique fixed point in E. Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 12 / 14

Proof of: P ( u × curl u ) ∈ L 2 ( 0 , τ ; V ′ 1 / 2 ) for u ∈ E If u ∈ E , then curl u ∈ L 4 ( 0 , τ, L 2 (Ω , R 3 )) ; u ∈ L 4 ( 0 , τ ; V ) and V ֒ → H 1 (Ω , R 3 ) ֒ → L 6 (Ω , R 3 ) , so that u ∈ L 4 ( 0 , τ, L 6 (Ω , R 3 )) . This proves that u × curl u ∈ L 2 ( 0 , τ, L 3 / 2 (Ω , R 3 )) since 1 6 + 1 2 = 2 3 and 4 + 1 1 4 = 1 2 , and therefore P ( u × curl u ) ∈ L 2 ( 0 , τ, V ′ 1 / 2 ) → L 3 (and then P L 3 / 2 ֒ → V ′ since V 1 / 2 ֒ 1 / 2 ). Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 13 / 14

Thank you for your attention! Sylvie Monniaux (I2M) Robin-Navier-Stokes Mathflows 14 / 14