The global well-posedness for the compressible viscous fluid flow in - PowerPoint PPT Presentation

. . The global well-posedness for the compressible viscous fluid flow in 3D exterior domains . . . . . Yoshihiro Shibata Math. Department and RISE, Waseda University Mathflows 2015, Porquerolles Sept. 13–18, 2015. Partially supported by Top Global University Project and JSPS Grant-in-aid for Scientific Research (S) # 24224004 Joint work with Yuko Enomoto (Shibaura Institute of Technology). . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 1 / 22

. (NP)  ρ t + div ( ρ u ) = 0 in Ω × (0 , T ) ,    ρ ( u t + u · ∇ u ) − µ ∆ u − ν ∇ div u + ∇ P ( ρ ) = 0 in Ω × (0 , T ) ,   u | ∂ Ω = 0 , ( ρ, u ) | t =0 = ( ρ ∗ + ρ 0 , u 0 ) in Ω .  Ω ⊂ R 3 : exterior domain ( R 3 \ Ω is bounded) ∂ Ω : boundary of Ω, sufficiently smooth ρ ∗ > 0 : mass density of the reference body Ω µ > 0, ν > 0 : viscosity coefficients P ( ρ ) : C ∞ function of ρ > 0 ρ = ρ ( x, t ) : density u = ( u 1 ( x, t ) , u 2 ( x, t ) , u 3 ( x, t )) : velocity . Assumption . . . P ′ ( ρ ) > 0 for any ρ > 0 and P ( ρ ∗ ) = 0 . . . . . If not, we consider P ( ρ ) − P ( ρ ∗ ) instead of P ( ρ ). . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 2 / 22

History (1/3) Local well-posedness • Cauchy Problem J.Nash (1962) N.Itaya (1970) A.I.Volpert-S.I.Hudjaev (1972) • I.B.V.P. V.A.Solonnikov (1976),(1981) Global well-posedness • A.Matsumura & T.Nishida (1980), (1981) Ω : R 3 or exterior domain, Assumption: ∥ ( ρ 0 , u 0 ) ∥ H 3 ≤ ε ρ − ρ ∗ ∈ C 0 ([0 , ∞ ) , H 3 (Ω)) ∩ C 1 ([0 , ∞ ) , H 2 (Ω)), u ∈ C 0 ([0 , ∞ ) , H 3 (Ω)) ∩ C 1 ([0 , ∞ ) , H 1 (Ω)), ρ t , ∇ ρ, u t ∈ L 2 ((0 , ∞ ) , H 2 (Ω)), ∇ u ∈ L 2 ((0 , ∞ ) , H 3 (Ω)) . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 3 / 22

History (2/3) . Global well-posedness • A.Valli (1983), A.Valli & W.Zajaczkowski (1986) non-homogenious boundaly domain, periodic solution • V.A.Solonnikov (1995) Ω : bounded domain, u ∈ W ℓ +2 ,ℓ/ 2+1 ( ℓ > 1 / 2) 2 • M.Kawashita (2002) Ω = R 3 , Cauchy problem, Assumption : ∥ ( ρ 0 , u 0 ) ∥ H 2 ≤ ε ( ρ, u ) ∈ C 0 ([0 , ∞ ) , H 2 ), ∇ u ∈ L 2 ((0 , ∞ ) , H 2 ), ∇ ρ ∈ L 2 ((0 , ∞ ) , H 1 ) • R.Danchin (2000) Critical space for the Cauchy problem • Y.Kagei & T.Kobayashi (2002), (2005) Half space • Y.Kagei & S.Kawashima (2006), Kagei (2012) Layer domain . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 4 / 22

History (3/3) • L p - L q type decay Cauchy problem G.Ponce (1985), Y.Wang & Z.Tan (2011) Exterior domain K.Deckelnick (1992), T.Kobayashi & S. (1999), Y.Enomoto & S (2012) • L p - L q maximal regularity Y.Enomoto & S (2013) Local well-posedness in general unbounded domain Global well-posedness in a bounded domain u ∈ L p ((0 , T ) , W 2 q (Ω)) ∩ W 1 q ((0 , T ) , L q (Ω)) , ρ ∈ W 1 p ((0 , T ) , W 1 q (Ω)) (Lagrangean) . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 5 / 22

Comments on the local well-posedness . • Using the Lagrange transformation to change the transport equation: ρ t + u · ∇ ρ to ∂ t ρ = ⇒ Quasilinear parabolic system. • We prove the maximal L p in time L q in space maximal regularity for the linearized equation. ρ t + ρ ∗ div u = f, ρ ∗ u t − α ∆ u − β ∇ div u + γ ∗ ∇ ρ = g , u | Γ = 0 , ( ρ, u ) | t =0 = ( ρ 0 , u 0 ) . • To prove the maximal regularity, we construct an R bounded solution operator A λ to the corresponding generalized resolvent problem. λρ + ρ ∗ div u = ˆ f, ρ ∗ λ u − α ∆ u − β ∇ div u + γ ∗ ∇ ρ = ˆ g , u | Γ = 0 . • Apply the Weis operator valued Fourier multipler theorem to the representation formula: u ( · , t ) = L − 1 λ [ A λ L [ f, g ]( λ )]( · , t ) with the help of Laplace transform L and its inverse transform L − 1 in time variable t and its co-variable λ . . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 6 / 22

Global well-posedness in the bounded domain case . • exponential stability of the analytic semigroup associated with the linearized equations. • spectral analysis to the resolvent problem: λρ + ρ ∗ div u = f, ρ ∗ λ u − α ∆ u − β ∇ div u + γ ∗ ∇ ρ = g in Ω , u | Γ = 0 . ⇒ ρ = λ − 1 ( f − ρ ∗ div u ) • λ ̸ = 0 = ρ ∗ λ u − α ∆ u − ( β + γ ∗ ρ ∗ λ − 1 ) ∇ div u = g ′ in Ω , u | Γ = 0 . • New prob. ρ ∗ µ u − α ∆ u − ( β + γ ∗ ρ ∗ λ − 1 ) ∇ div u = g ′ in Ω, u | Γ = 0. • uniqueness implies the unique existence, thus ρ ∗ λ u − α ∆ u − ( β + γ ∗ ρ ∗ λ − 1 ) ∇ div u = 0 in Ω, u | Γ = 0 = ⇒ u = 0 • λ = 0 case: We have to prove the unique existence theorem of the problem: div u = f, µ ∆ u + β ∇ div u − γ ∇ ρ = g in Ω , u | Γ = 0 , ∫ ∫ ∫ Thus, Ω f dx = Ω div u dx = Γ u · n dσ = 0 is necessary to show the exponential decay. • To prove the global well-posedness for the small data, we assume ∫ that Ω ρ 0 dx = 0. . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 7 / 22

Global well-posedness . Ω ⊂ R 3 : exterior domain, ( ρ 0 , u 0 ) ∈ H 2 , ∥ ( ρ 0 , u 0 ) ∥ H 2 ≤ ε ≪ 1 compatibility condition : u 0 | ∂ Ω = 0 = ⇒ the problem (NP) admits a unique solution ( ρ, u ) ρ ∈ C 0 ([0 , ∞ ) , H 2 ) ∩ C 1 ([0 , ∞ ) , H 1 ), u ∈ C 0 ([0 , ∞ ) , H 2 ) ∩ C 1 ([0 , ∞ ) , L 2 ) ∇ ρ, ρ t ∈ L 2 ((0 , ∞ ) , H 1 ), u t ∈ L 2 ((0 , ∞ ) , H 1 ), ∇ u ∈ L 2 ((0 , ∞ ) , H 2 ) Moreover, ∥ ( ρ 0 , u 0 ) ∥ H 2 + ∥ ( ρ 0 , u 0 ) ∥ L 1 = δ ≪ 1 ⇒ ∥ ( ρ ( · , t ) , u ( · , t )) ∥ L 2 ≤ Ct − 3 4 δ , = ∥∇ ( ρ ( · , t ) , u ( · , t )) ∥ H 1 ≤ Ct − 5 4 δ as t → ∞ . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 8 / 22

Linearized equation .  ρ t + ρ ∗ div u = 0 in Ω × (0 , ∞ ) ,    (LP) u t − µ ∗ ∆ u − ν ∗ ∇ div u + γ ∗ ∇ ρ = 0 in Ω × (0 , ∞ ) ,   u | ∂ Ω = 0 , ( ρ, u ) | t =0 = ( ρ 0 , u 0 ) in Ω  where Ω ⊂ R N ( N ≥ 3) is an exterior domain. . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 9 / 22

L p - L q decay for linearized equation . Let 1 < q < ∞ and let N ≥ 2. Then problem (LP) generates C 0 semigroup { T ( t ) } t ≥ 0 on H q (Ω) = { ( ρ 0 , u 0 ) ∈ W 1 q (Ω) × L q (Ω) } which is analytic. Let 1 ≤ q ≤ 2 ≤ p ≤ ∞ , let N ≥ 2, and let [ ρ 0 , u 0 ] p,q = ∥ ρ 0 ∥ W 1 p + ∥ u 0 ∥ L p + ∥ ( ρ 0 , u 0 ) ∥ L q . Then, ( ) ∥ ( ρ, u )( · , t ) ∥ L p ≤ C p,q t − N q − 1 1 2 p [ ρ, u 0 ] p,q ,  ( ) t − N 1 q − 1 − 1 2 [ ρ, u 0 ] p,q , 2 p p ≤ N  ∥∇ ( ρ, u )( · , t ) ∥ L p ≤ C p,q t − N 2 q [ ρ, u 0 ] p,q , p ≥ N   ( ) t − N 1 q − 1 − 1 [ ρ, u 0 ] p,q , p ≤ N/ 2 , 2 p  ∥∇ 2 u ( · , t ) ∥ L p ≤ C p,q t − N 2 q [ ρ, u 0 ] p,q . p ≥ N/ 2  . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 10 / 22

Idea of proof . Global well-posedness : A.Matsumura & T.Nishida method • first energy is obtained directly in Ω • To obtain higher order energy estimate, we consider the half-space problem: x 3 > 0. • energy method for tangential derivative and time derivative • To estimate normal derivative D 3 , we use the formula D 3 ( ρ t + u · ∇ ρ ) + δ ∗ D 3 ρ = ( · · · there are no D 2 3 terms) . • To estimate D 2 3 ρ , we use the formula D 2 3 ( ρ t + u · ∇ ρ ) + δ ∗ D 2 3 ρ = · · · Multiplying this formula by D 2 3 ρ , we have 1 d 3 ρ ( · , t ) ∥ 2 + δ ∗ ∥ D 2 3 ρ ( · , t ) ∥ 2 − (div u D 2 dt ∥ D 2 3 ρ, D 2 3 ρ ) = · · · 2 . . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 11 / 22

The worst term of the nonlinear estimate: ∫ t ∫ t ∥∇ 2 ρ ∇ u ∥ 2 ∥∇ 2 ρ ∥ 2 L 2 ∥∇ u ∥ 2 L 2 ds ≤ L ∞ ds 0 0 ∫ t ∥∇ 2 ρ ∥ 2 L 2 ∥∇ u ∥ 2 ≤ C H 2 ds 0 ∫ t ∥∇ 2 ρ ( · , s ) ∥ 2 ∥∇ u ∥ 2 ≤ C sup H 2 ds L 2 0 <s<t 0 To estimate nonlinear terms, we use ∥ v ∥ L 6 ≤ ∥∇ v ∥ L 2 , ∥ v ∥ L 2 (Ω ∩ B R ) ≤ C R ∥∇ v ∥ L 2 , ∥ v ∥ L ∞ ≤ C ∥ v ∥ L 6 ≤ C ∥ v ∥ H 1 In this way, we can enclosed our estimations in ( ρ, u ) ∈ L ∞ ((0 , ∞ ) , H 2 (Ω)) , ( ρ t , u t ) ∈ L ∞ ((0 , ∞ ) , L 2 (Ω)) ∇ u ∈ L 2 ((0 , ∞ ) , H 2 (Ω)) , u t , ∇ ρ, ρ t ∈ L 2 ((0 , ∞ ) , H 1 (Ω)) . . . . . . Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow Sept. 13–18, 2015 12 / 22