Mathematical tools for the study of hydrodynamic limits Laure - PowerPoint PPT Presentation

Mathematical tools for hydrodynamic limits Mathematical tools for the study of hydrodynamic limits Laure Saint-Raymond D´ epartement de Math´ ematiques et Applications Ecole Normale Sup´ erieure de Paris, France 17 juin 2008

Mathematical tools for hydrodynamic limits Notations Notations • Nondimensional form of the Boltzmann equation 1 Ma ∂ t f + v · ∇ x f = Kn Q ( f , f ) • Fluctuations around a global equilibrium M f = M (1 + Ma g ) controlled by the relative entropy �� � � f log f dvdx ≤ C Ma 2 H ( f | M ) = M − f + M • Perturbative form of the Boltzmann equation Ma ∂ t g + v · ∇ x g = − 1 Kn L g + Ma Kn Q ( g , g )

Mathematical tools for hydrodynamic limits Physical a priori estimates The entropy inequality Physical a priori estimates ◮ The entropy inequality Starting from • the local conservation of mass, momentum and energy • the local entropy inequality and integrating by parts using • Maxwell’s boundary condition with accomodation coefficient α we get formally the entropy inequality � t � t 1 � D ( f )( s , x ) dsdx + α � H ( f | M )( t ) + E ( f | M )( s , x ) d σ x ds KnMa Ma 0 Ω 0 ∂ Ω ≤ H ( f in | M ) ≤ C Ma 2 (which will be actually satisfied even for very weak solutions of the Boltzmann equation)

Mathematical tools for hydrodynamic limits Physical a priori estimates The entropy inequality The three controlled quantities are • the relative entropy �� H ( f | M ) = Mh ( Ma g ) dvdx with h ( z ) = (1 + z ) log(1 + z ) − z • the entropy dissipation � D ( f ) = − Q ( f , f ) log fdv � f ′ f ′ � = 1 � ∗ − 1 bdvdv ∗ d ω with r ( z ) = z log(1 + z ) ff ∗ r 4 ff ∗ • the Darroz` es-Guiraud information 1 E ( f | M ) = √ � h ( Ma g ) − h ( � Ma g � ∂ Ω ) � ∂ Ω 2 π √ def � with � G � ∂ Ω = 2 π ( v · n ( x )) + dv GM

Mathematical tools for hydrodynamic limits Physical a priori estimates The relative entropy ◮ The relative entropy The relative entropy bound �� Mh ( Ma g ) dvdx ≤ C Ma 2 controls the size of the fluctuation . • By Young’s inequality (1 + | v | 2 ) g = O (1) L ∞ loc ( dx : L 1 ( Mdv ))) . t ( L 1 • Heuristically 1 2 z 2 h ( z ) ∼ z → 0 so that we expect g to be almost in L ∞ t ( L 2 ( dxMdv )).

Mathematical tools for hydrodynamic limits Physical a priori estimates The relative entropy • We therefore define the renormalized fluctuation 2 � ˆ g = Ma ( 1 + Ma g − 1) . The functional inequality √ 1 1 + z − 1) 2 , 2 h ( z ) ≥ ( ∀ z > − 1 implies that g = O (1) L ∞ ˆ t ( L 2 ( dxMdv )) . That refined a priori estimate will be used together with the identity g + 1 g 2 . g = ˆ 4 Ma ˆ

Mathematical tools for hydrodynamic limits Physical a priori estimates The entropy dissipation ◮ The entropy dissipation The bound on the entropy dissipation � t � f ′ f ′ 1 � � � ∗ bdvdv ∗ d ω dxds ≤ C Ma 3 Kn ff ∗ r − 1 4 ff ∗ 0 controls some renormalized collision integral. The functional inequality y ≥ 4( √ x − √ y ) 2 , ( x − y ) log x x , y > 0 coupled with the Cauchy-Schwarz inequality, implies indeed √ √ 1 1 ˆ = √ M Q ( Mf , Mf ) q Ma 3 Kn = O (1) L 2 loc ( dt , L 2 ( M ν − 1 dvdx ) Remark : In order to control the relaxation process, we will further need estimates on the nonlinearity based on the continuity properties of Q and bounds on g .

Mathematical tools for hydrodynamic limits Physical a priori estimates The Darroz` es-Guiraud information ◮ The Darroz` es-Guiraud information The bound on the boundary term � t � h ( Ma g ) − h ( � Ma g � ∂ Ω ) � ∂ Ω d σ x ds ≤ C Ma 3 � α 0 ∂ Ω controls the variation of the trace in v . By Taylor’s formula (with cancellation of the first order), one indeed has � α �� � � η ˆ = 2 1 + Ma g − � 1 + Ma g � ∂ Ω Ma 3 = O (1) L 2 loc ( dt , L 2 ( M ( v · n ( x )) + d σ x dv )) Remark : In order to control the trace g | ∂ Ω , we will further need estimates coming from the inside, on g and on v · ∇ x g .

Mathematical tools for hydrodynamic limits Additional integrability in v coming from the relaxation Control of the relaxation Additional integrability in v coming from the relaxation ◮ Control of the relaxation The fundamental identity From the bilinearity of Q and the definition of ˆ g , we have obviously √ √ = Ma 2 1 L ˆ g 2 Q (ˆ g , ˆ g ) − M Q ( Mf , Mf ) Ma √ = Ma 2 Q (ˆ g , ˆ g ) − 2 MaKn ˆ q For simplicity, we assume that ν is bounded from up and below. Else we would have to use some truncated ˜ b , ˜ L and ˜ Q Control of the quadratic term By the continuity of Q : L 2 ( Mdv ) × L 2 ( M ν dv ) → L 2 ( M ν − 1 dv ) and the L 2 bound on ˆ g , we get Ma 2 Q (ˆ g , ˆ g ) = O ( Ma ) L ∞ t ( L 1 x ( L 2 ( Mdv ))

Mathematical tools for hydrodynamic limits Additional integrability in v coming from the relaxation Control of the relaxation Control coming from the entropy dissipation By the entropy dissipation bound, √ √ 2 MaKn ˆ q = O ( MaKn ) L 2 loc ( dt , L 2 ( dxMdv )) The relaxation estimate From the coercivity inequality for L � g L M g ( v ) M ( v ) dv ≥ C � g − Π g � 2 L 2 ( M ν dv ) . we then deduce √ g − Πˆ ˆ g = O ( Ma ) L ∞ x ( L 2 ( Mdv )) + O ( MaKn ) L 2 t ( L 1 loc ( dt , L 2 ( dxMdv ))

Mathematical tools for hydrodynamic limits Additional integrability in v coming from the relaxation Control of large velocities ◮ Control of large velocities By Young’s inequality Ma 2 | Ma g | (1 + | v | p ) 2 δ 2 (1 + | v | p ) 2 | ˆ g | 2 ≤ δ 2 δ 2 � (1 + | v | p ) 2 � �� h ( Ma g ) + h ∗ ≤ Ma 2 δ 2 Therefore, for any δ > 0, p < 1, q < + ∞ � C δ, q � (1 + | v | p ) | ˆ g | = O ( δ ) L ∞ t ( L 2 ( Mdvdx )) + O Ma L ∞ t , x ( L q ( Mdv )) Remark : for p = 1 one can actually obtain a bound.

Mathematical tools for hydrodynamic limits Additional integrability in v coming from the relaxation Moments and equiintegrability in v ◮ Moments and equiintegrability in v From the decomposition g = (ˆ ˆ g − Πˆ g ) + Πˆ g we deduce that for r < 2, q < + ∞ , p < 1 (1 + | v | p ) 2 | ˆ g | 2 = (1 + | v | 2 p )ˆ g + (1 + | v | 2 p )(ˆ g Πˆ g − Πˆ g )ˆ g x ( L r ( Mdv )) + (1 + | v | p ) | ˆ = O (1) L ∞ g − Πˆ g | O ( δ ) L ∞ t ( L 1 t ( L 2 ( Mdvdx )) � C δ, q � +(1 + | v | p ) | ˆ g − Πˆ g | O Ma L ∞ t , x ( L q ( Mdv )) By the relaxation estimate, choosing δ sufficiently small, we get g | 2 = O (1) L 1 (1 + | v | p ) 2 | ˆ loc ( dtdx , L 1 ( Mdv )) uniformly integrable in v .

Mathematical tools for hydrodynamic limits Additional integrability in x coming from the free transport Additional integrability in x coming from the free transport In viscous regime, we further use properties of the free-transport equation Ma ∂ t g + v · ∇ x g = S (1) • The free transport is the prototype of hyperbolic operators � t g ( t , x , v ) = g in ( x − Ma tv , v ) + S ( x − Ma sv , v , t − s ) ds 0 No regularizing effect on g . Propagation of singularities at finite speed. • Ellipticity of the symbol outside from a small subset of R 3 v a ( τ, ξ, v ) = i ( Ma τ + v · ξ ) � Regularity in x of the averages g ϕ ( v ) dv (moments).

Mathematical tools for hydrodynamic limits Additional integrability in x coming from the free transport Averaging properties ◮ Averaging properties v 2 ξ |St τ +v. ξ | < α |St τ +v. ξ | > α Ellipticity of the symbol v 1 |St τ +v. ξ | > α Ellipticity of Small contribution the symbol to the average

Mathematical tools for hydrodynamic limits Additional integrability in x coming from the free transport Averaging properties Theorem [ L 2 averaging lemma] (Golse, Lions, Perthame, Sentis) : Let g ∈ L 2 t , x , v be the solution of the transport equation (1). Then, for all ϕ ∈ L ∞ ( R 3 v ) � � � ≤ C ϕ � g � 1 / 2 t , x , v � S � 1 / 2 � � g ϕ ( v ) dv t , x , v . � � L 2 L 2 L 2 ( R t , H 1 / 2 � � ) x Sketch of the proof • Take Fourier transform • Split the integral into two contributions • Estimate each contribution with the Cauchy-Schwarz inequality • Optimize with respect to α Can be extended to L p spaces with 1 < p < ∞ .

Mathematical tools for hydrodynamic limits Additional integrability in x coming from the free transport Averaging properties Remark 1 : Because of concentration phenomena, velocity averaging fails in L 1 and L ∞ (as proved by the following counterexample). Consider ( S n ) bounded in L 1 t , x , v such that S n → St χ ′ ( t ) δ x − Ma − 1 v 0 t ⊗ δ v − v 0 Let ( f n ) be the corresponding solutions to (1). Then, � R 3 f n ϕ ( v ) dv ⇀ ρ in M t , x , support( ρ ) ⊂ R × R + v 0 . Remark 2 : It is actually sufficient to control the concentration effects in v (non concentration in x will follow automatically).

Mathematical tools for hydrodynamic limits Additional integrability in x coming from the free transport Mixing properties ◮ Mixing properties v (t-s)v E(s) E(t) x A set of “small measure in x ” becomes a set of “small measure in v ”