On the L p L q maximal regularity for linear thermoelastic plate - PowerPoint PPT Presentation

On the L p – L q maximal regularity for linear thermoelastic plate equation in a bounded domain Yuka Naito (Sato) Waseda University 25/08/2009 Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 1 / 38

Thermoelastic plate equation ∂ 2 t u + ∆ 2 u + ∆ θ = f 1 in R + × Ω , ∂ t θ − ∆ θ − ∆ ∂ t u = f 2 in R + × Ω , u = ∂ ν u = θ = 0 on R + × ∂ Ω , u ( 0 , x ) = u 0 ( x ) , ∂ t u ( 0 , x ) = u 1 ( x ) , θ ( 0 , x ) = θ 0 ( x ) in Ω . u ( t , x ) : the vertical displacement of the plate, θ ( t , x ) : the absolute temperature Ω ⊂ R n ( n ≥ 2 ) : a bounded domain ∂ Ω : a C 4 hypersurface ν : the unit outer normal vector to ∂ Ω u 0 , u 1 , θ 0 : the initial value. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 2 / 38

The reason of the appearance of ∆ 2 Elasticity displacement : u = X ( t , x ) − x X ( t , x ) : position vector. The case of the plate Considering the displacement of the center of the plate first, I measure the distance from the center of the plate. Necessary assumptions The center of the plate is always center. 1 The vertical segment of the center of the plate keeps vertical. 2 The thickness of x 3 is so small. 3 The curve of the plate is so small. 4 Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 3 / 38

The displacement of the plate in the 3 dimension case. The displacement of the center of the plate: u 1 ( x 1 , x 2 ) , u 2 ( x 1 , x 2 ) , u 3 ( x 1 , x 2 ) . ∂ u 3 U 1 ( x 1 , x 2 , x 3 ) = u 1 ( x 1 , x 2 ) − x 3 ∂ x 1 , ∂ u 3 The displacement of the plate U 2 ( x 1 , x 2 , x 3 ) = u 2 ( x 1 , x 2 ) − x 3 ∂ x 2 , U 3 ( x 1 , x 2 , x 3 ) = u 3 ( x 1 , x 2 ) + x 3 . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 4 / 38

The type of equations I shall consider type of equations. Thermoelastic plate equation ∂ 2 t u + ∆ 2 u + ∆ θ = f 1 Dispersive type ∂ t θ − ∆ θ − ∆ ∂ t u = f 2 Parabolic type Thermoelastic plate equation is the couple of these two type. I consider type of the whole equation. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 5 / 38

The classification of equations by the characteristic root in a Cauchy problem Parabolic type → −| ξ | 2 . Heat eqauton: ∂ t τ − ∆ τ = 0 . − Dispersive type Plate equation: ∂ 2 t u + ∆ 2 u = 0 . − → ± i | ξ | 2 . The couple of these two type { ∂ 2 t u + ∆ 2 u + ∆ θ = 0 , Thermoelastic plate equation ∂ t θ − ∆ θ − ∆ ∂ t u = 0 . β | ξ | 2 ( α : A positive real number , Re β > 0 . ) → − α | ξ | 2 , − β | ξ | 2 , − ¯ − Therefore I know thermoelastic plate equation is parabolic type. But when I consider a initial boundary problem, the classification of equations is not so obvious. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 6 / 38

What is the parabolicity I shall consider a general initial value problem. A initial value problem { ∂ t u ( t ) = Au ( t ) u | t = 0 = u 0 A : a general operator, u 0 : a initial value. The definition of parabolic type ⇒ u ( t ) ∈ D ( A m ) ( ∀ m ≥ 1 ) , t > 0 u 0 ∈ X = D ( A ) : a domain of A . Resolvent estimate = ⇐ ⇒ | λ | ∥ ( λ − A ) − 1 ∥ ≤ C . ∃ λ 0 ∈ R Re λ ≥ λ 0 = In general if it holds resolvent estimate, then it is parabolic. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 7 / 38

The definition of maximal regularity X : Banach space, E : A dense subspace of X , A : E → X ; A closed linear operator. A Cauchy problem. ∂ t u = Au + f ( 0 < t < T ) , u | t = 0 = a ( ∗ ) If ( ∗ ) admits a solution u ∈ L p (( 0 , T ) , E ) ∩ W 1 p (( 0 , T ) , X ) = M p (( 0 , T ) : A ) , then a ∈ [ X , E ] 1 − 1 / p , p : the trace class of M p (( 0 , T ) : A ) of t = 0 , f ∈ L p (( 0 , T ) , X ) . ⇒ if a ∈ [ X , E ] 1 − 1 / p , p and A has the L p -maximal regularity ⇐ f ∈ L p (( 0 , T ) , X ) , then ( ∗ ) admits a unique solution u ∈ M p (( 0 , T ) : A ) A has the L p -maximal regularity. = ⇒ A is parabolic type. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 8 / 38

The known results about thermoelastic plate equation By Kim, Rivera-Racke, Liu-Zeng, Avalos-Lasiecka, Lasiecka-Triggiani, Shibata A bounded domain · · · The exponatial decay with the several boundary conditions By Liu-Renardy, Lasiecka-Triggiani Thermoelastic plate equation is analytic in the Hilbert space setting. In the Hilbert space maximal regularity is generation of analytic semigroup is eqivalent. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 9 / 38

Purpose of my talk To show L p − L q maximal regularity. We know L p − L q maximal regularity implies the generation of analytic semigroup. But it is not clear the opsite direction in the Banach space case. In fact, Kalton-Lancien proved that if every generator of an analytic on Banach space X has L p maximal regularity, then X must be a Hilbert space. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 10 / 38

Setting function spaces. I rewrite thermoelastic plate equation the following: U t − A q U = F U | t = 0 = U 0 , where I have set  0 1 0   u   0   u 0                  − ∆ 2         A q = 0 − ∆ U = ∂ t u F = f 1 U 0 = u 1   ,    ,    ,                                     ∆ ∆     0 θ f 2 θ 0 X = { F = (˜ f , f 1 , f 2 ) | ˜ f ∈ W 2 q , D (Ω) , f 1 ∈ L q (Ω) , f 2 ∈ L q (Ω) } , E = { U = ( u , v , θ ) | u ∈ W 4 q , D (Ω) , v ∈ W 2 q , D (Ω) , θ ∈ W 2 q , 0 (Ω) } , W 2 q , 0 (Ω) = { u ∈ W 2 q (Ω) | u | ∂ Ω = 0 } , W m q , D (Ω) = { u ∈ W m q (Ω) | u | ∂ Ω = ∂ ν u | ∂ Ω = 0 } . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 11 / 38

The main theorem Ω : a bounded domain. ∂ Ω : C 4 -hypersurface. Let 1 < p , q < ∞ . I assume there exist a γ 0 > 0 such that U 0 , F satsify the following U 0 ∈ [ X , E ] 1 − 1 / p , p , e γ t F ∈ L p ( R + , L q (Ω)) , ( γ ∈ [ 0 , γ 0 ]) , Then there exists a unique solution: U ∈ M p (( 0 , ∞ ) : A ) = L p (( 0 , T ) , E ) ∩ W 1 p (( 0 , T ) , X ) satisfying the following estimates ∥ e γ t U ∥ Lp ( R + , E ) + ∥ e γ t U t ∥ Lp ( R + , X ) ∥ e γ t F ∥ Lp ( R + , Lq (Ω)) + ∥ U 0 ∥ [ X , E ] 1 − 1 / p , p { } ≦ C p , q . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 12 / 38

Outline of proof To prove the theorem, I study two cases. One is the problem with zero right members and initial value. ∂ 2 t u + ∆ 2 u + ∆ θ = 0 in R + × Ω , ∂ t θ − ∆ θ − ∆ ∂ t u = 0 in R + × Ω , u = ∂ ν u = θ = 0 on R + × ∂ Ω u ( 0 , x ) = u 0 ( x ) , ∂ t u ( 0 , x ) = u 1 ( x ) , θ ( 0 , x ) = θ 0 ( x ) in Ω This case can be studied by analytic semigroup theory and interpolaion theory. Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 13 / 38

Outline of proof Another is the problem with non-zero right members and zero initial value. This case is the essential part in my proof. ∂ 2 t u + ∆ 2 u + ∆ θ = f 1 in R + × Ω , ∂ t θ − ∆ θ − ∆ ∂ t u = f 2 in R + × Ω , u = ∂ ν u = θ = 0 on R + × ∂ Ω , u ( 0 , x ) = ∂ t u ( 0 , x ) = θ ( 0 , x ) = 0 in Ω . In a Hilbert space case I know an elegant theory due to Lasiecka-Triggiani. But this theory can not be applied in the Banach space case. And I have to use exact solution formula in a whole and a half space, to get a priori estimate of the solution. Finally by the decomposition of unity, I localize the problem. Then I apply a whole and a half space case . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 14 / 38

The generation of the analytic semigroup and its estimates A half space (N.-Shibata) 1 < q < ∞ . A q generates the analytic semigroup : { T q ( t ) } t ≥ 0 on X . A bounded domain (Denk-Racke-Shibata ’08) Ω : a bounded domain, ∂ Ω : C 4 -hypersurface. Let 1 < q < ∞ . A q generates the analytic semigroup : { T q ( t ) } t ≥ 0 on X and { T q ( t ) } t ≥ 0 satisfys the following estimates: ∥ T q ( t ) G ∥ X ≤ Ce − σ t ∥ G ∥ X . ∃ σ > 0 , ∀ t > 0 , ∀ G ∈ X (Ω) , where C is independent of t and F . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 15 / 38

A model problem ～ a half space ～ ∂ 2 t u + ∆ 2 u + ∆ θ = f 1 in R × R n + , in R × R n ∂ t θ − ∆ θ − ∆ ∂ t u = f 2 + , on R × ∂ R n u = ∂ x n u = θ = 0 + , in R n u = ∂ t u = θ = 0 + . By technical reason instead of this problem I consider the following problem which I shift with repect to time variable. ( ∂ t + 1 ) 2 u + ∆ 2 u + ∆ θ = f 1 in R × R n + , in R × R n ( ∂ t + 1 ) θ − ∆ θ − ( ∂ t + 1 )∆ u = f 2 + , on R × ∂ R n u = ∂ x n u = θ = 0 + , in R n u = ∂ t u = θ = 0 + . Yuka Naito (Sato) (Waseda University) thermoelastic plate equation 25/08/2009 16 / 38