Maximal Regularity for the Initial-Boundary Value Problem of some - PowerPoint PPT Presentation

Maximal Regularity for the Initial-Boundary Value Problem of some Evolution Equations Yoshihiro Shibata Waseda University VIII Workshop in Partial Differential Equations August 25-28, Rio 2009 Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 1 / 23

Analytic Semigroup X : Banach space with norm ∥ · ∥ A : D ( A )( ⊂ X ) → X : densely defined closed operator { T ( t ) } t ≥ 0 is an analytic semigroup associated with A ⇐⇒ T ( t ) is a bounded linear operator on X such that T ( t ) T ( s ) = T ( t + s ) for t , s > 0 T ( t ) is strongly continuous in t > 0 lim t → 0 + ∥ T ( t ) x − x ∥ = 0 for any x ∈ X T ( t ) x − x (*) Ax = lim t → 0 + for x ∈ D ( A )( = { x ∈ X | the limit (*) exists } ) t T ( t ) is extended as an analytic function of t to a sector { t ∈ C | | arg t | < θ 0 } with some θ 0 ∈ (0 , π/ 2) . T ( t ) x ∈ D ( A ) , d dt T ( t ) a = AT ( t ) a ( t > 0 ) for any x ∈ X Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 2 / 23

Generation of analytic semigroup Given 0 < γ < π/ 2 we set G γ = { λ ∈ C \{ 0 } | | arg λ | < π 2 + γ } Assumption ∃ b ∈ R > 0 and 0 < γ < π/ 2 such that for any λ ∈ b + G γ the resolvent ( λ − A ) − 1 exists and it holds that ∥ ( λ − A ) − 1 x ∥ ≤ M | λ | − 1 ∥ x ∥ for any x ∈ X . The operator A generates an analytic semigroup { T ( t ) } t ≥ 0 : ∥ d ∥ T ( t ) x ∥ ≤ Me bt ∥ x ∥ , dtT ( t ) x ∥ ≤ Mt − 1 e bt ∥ x ∥ Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 3 / 23

L p maximal regularity Let us consider the abstract Cauchy problem: u ′ − Au = f (**) (0 < t < T ) , u (0) = u 0 Solution class : W 1 p ((0 , T ) , X ) ∩ L p ((0 , T ) , D ( A )) } = M p ((0 , T ) , A ) . Initial data class : Z p ( A )( X , D ( A )) 1 − (1 / p ) , p = the trace class of M p ((0 , T ) , A ) A has an L p maximal regularity ⇐⇒ For any f ∈ L p ((0 , T ) , X ) and u 0 ∈ Z p ( A ) , the problem (**) admits a unique solution u ( t ) ∈ W 1 p ((0 , T ) , X ) ∩ L p ((0 , T ) , D ( A )) Closed graph theorem of S. Banach = ⇒ ∥ u ∥ W 1 p ((0 , T ) , X ) + ∥ Au ∥ L p ((0 , T ) , X ) ≤ C T ( ∥ f ∥ L p ((0 , T ) , X ) + ∥ u 0 ∥ Z p ( A ) ) . A has an L p maximal regularity = ⇒ A generates an analytic semigroup. ⇐ = ? (H. Br´ ezis) : The answer is No in general (Kalton and Lancien). Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 4 / 23

An example of non-linear evolution eq. of parabolic type v t − ∆ v + F ( v ) = 0 ( t > 0) , v (0) = v 0 F ( v ) : semi-linear ⇐⇒ F ( v ) is a non-linear function of v and ∇ v . F ( v ) : quasi-linear ⇐⇒ F ( v ) = G ( v ) ∇ 2 v , G ( v ) is a semi-linear function. Semi-linear = ⇒ Analytic semigroup approach. { T ( t ) } t ≥ 0 : analytic semigroup generated by − ∆ . The Duhamel principle = ⇒ ∫ t v ( t ) = T ( t ) v 0 − T ( t − s ) F ( s ) ds 0 One of the tools to solve this equation is to use the L q - L r estimate: ( 1 ) ∥ T ( t ) f ∥ L r ≤ C q , r Me bt t − k 2 − n q − 1 ∥ f ∥ L q , (1 < q ≤ r < ∞ ) 2 r (1) ( 1 ) ∥∇ T ( t ) f ∥ L r ≤ C q , r Me bt t − 1 2 − n q − 1 ∥ f ∥ L q (1 < q ≤ r ≤ n ) 2 r ( 1 ( 1 q − 1 ) q − 1 ) n 1 − n ∥∇ k T ( t ) f ∥ L q ≤ Me bt t − k r r 2 ∥ f ∥ L q ( k = 0 , 1 , 2 ) PLUS ∥ v ∥ L r ≤ C ∥ v ∥ ∥ v ∥ W 1 L q q ( 1 ) q − 1 ( n )PLUS semigroup property = ⇒ (1) r Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 5 / 23

Quasi-linear = ⇒ we need L p - L q maximal regularity approach. ∫ t ∥∇ 2 v ( t ) ∥ L q ≤ ∥∇ 2 T ( t ) v 0 ∥ L q + 0 ∥∇ 2 T ( t − s ) F ( s ) ∥ L q ds ∥∇ 2 T ( t − s ) F ( s ) ∥ L q ≤ C ( t − s ) − 1 ∥ G ( s ) ∇ v ( s ) ∥ L q This approach fails because the singurality: ( t − s ) − 1 appears. L p - L q maximal regularity theorem is applied to the linearized eq. v t − ∆ v = − G ( w ) ∇ 2 w = ⇒ ∥ v t ∥ L p ((0 , T ) , L q ) + ∥ v ∥ L p ((0 , T ) , W 2 q ) q ) 1 − (1 / p ) , p + ∥ G ( w ) ∇ 2 w ∥ L p ((0 , T ) , L q ) ≤ C T ( ∥ v 0 ∥ ( L q , W 2 toghether with the embedding theorem: W 1 p ((0 , T ) , L q )) ∩ L p ((0 , T ) , W 2 q ) ⊂ BC ([0 , T ] , ( L q , W 2 q ) 1 − (1 / p ) , p ) implies at least a local in time unique existence theorem Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 6 / 23

Navier-Stokes equation v t + ( v · ∇ ) v − µ ∆ v + ∇ p = f , div v = 0 v = ( v 1 , . . . , v n ) velocity vector, p scalor pressure j = 1 v j D j θ , v t + ( v · ∇ ) v = Dv v · ∇ θ = ∑ n Dt : material derivative µ = 1 / R , R = ρ LV / ˆ µ : Reynolds number. ρ : mass, L : length, V : velocity, ˆ µ : viscosity coefficient Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 7 / 23

Fujita-Kato principle The scaling: v λ ( x , y ) = λ 2 v ( λ x , λ 2 t ) , p λ ( x , t ) = λ 2 p ( λ x , λ 2 t ) does not change the NS eq. ∥ v λ ∥ L p ((0 , ∞ ) , L q ( R n )) = ∥ v ∥ L p ((0 , ∞ ) , L q ( R n )) = ⇒ 2 p + n q = 1 (Serrin Condition) Theorem (Kato) v t + ( v · ∇ ) v − µ ∆ v + ∇ p = 0 , div v = 0 in R n × (0 , ∞ ) , v | t = 0 = a (2) Given initial data a ∈ L n ( R n ) with div a = 0 , there exists a time T > 0 such that the Navier-Stokes equation (2) admits a unique strong solution u ∈ C 0 ([0 , T ) , L n ( R n )) ∩ C 0 ((0 , T ) , L q ( R n ) ∩ W 1 n ( R n )) with some q ∈ ( n , ∞ ) . Moreover, if ∥ a ∥ L n ( R n ) is small enough, then T = ∞ . Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 8 / 23

Observation for the global in time existence theorem If the local existence time T depends only on ∥ a ∥ L n = ⇒ the scaling argument implies the prolongation of a local in time solution to any time interval = ⇒ global in time unique existence of the NS equation for any initial data in L n !! But, unfortunately T depends on some properties of ∥∇ u ( · , t ) ∥ L n and ∥ u ( · , t ) ∥ L q , so that to get global in time unique existence theorem, so far we have to assume some smallness on the initial data. Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 9 / 23

Navier-Stokes equation in the time dependent domain Ω ( t ) v t + v · ∇ v − µ ∆ v + ∇ p = 0 , div v = 0 ( x ∈ Ω ( t ) , t > 0) , v | t = 0 = v 0 with some boundary condition. Free boundary problem A viscous incompressible fluid flow past rotating bodies. Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 10 / 23

Reduction to the Quasi-linear equation Suitable change of variables and unknown function: Free boundary problem ⇐⇒ Lagrangian coordinate Rotating obstacle ⇐⇒ Galdi transform, Inoue-Wakimoto transform implies the quasi-linear equation: v t − µ ∆ v + ∇ p + F ( t , v , ∇ v , ∇ 2 v , ∇ p ) = 0 ( x ∈ Ω , t > 0) , div ( v + G ( t , v )) = 0 ( x ∈ Ω , t > 0) , v t = 0 = v 0 with suitable boundary condition on a fixed domain Ω . F and G are some nonlinear functions such that F | t = 0 = G | t = 0 = 0 . Serrin cond. = ⇒ L p ((0 , T ) , L q ( Ω )) type maximal regularity is necessary. Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 11 / 23

Some results about L p maximal regularity 1. Stokes Equation u t − µ ∆ u + ∇ p = f , div u = 0 ( x ∈ Ω , t > 0) u | t = 0 = u 0 with the following boundary condition on the boundary Γ : Non-slip: u = 0 (Solonnikov, Giga-Sohr) Slip: D ( u ) ν − < ν, D ( u ) ν > ν = g , u · ν = 0 (Saal, Shimada) Free bc: µ D ( u ) ν − p ν = g (Shibata-Shimizu, Solonnikov). ν stands for the unit outer normal to Γ and < · , · > is the standard inner-product in R n , D ( u ) = ∇ u + T D ( u ) (the symmetric part of ∇ u ). Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 12 / 23