Relative entropy in diffusive relaxation Relative entropy in diffusive relaxation C. Lattanzio 1 In collaboration with A.E. Tzavaras 2 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications June 25 – 29, 2012, Padova, Italy 1 University of L’Aquila 2 University of Crete
Relative entropy in diffusive relaxation Outline Introduction Diffusive relaxation limits Relative entropy Isentropic gas dynamics with damping Formal analysis Relative entropy estimate Stability and convergence Other applications p –system with damping Keller–Segel type models Viscoelasticity with memory
Relative entropy in diffusive relaxation Introduction Diffusive relaxation limits Main motivation for relaxation limits Hydrodynamic limit for the Boltzmann equation: ν f t + ξ · ∇ x f = 1 ε Q ( f , f ) (1) ν : Mach number and ε : Knudsen number if ν = ε (1) − → Navier–Stokes equations as ε ↓ 0 Simplest discrete velocity model: Carleman’s equations � ∂ t f 1 + 1 ε ∂ x f 1 = 1 ε 2 ( f 2 2 − f 2 1 ) ∂ t f 2 − 1 ε ∂ x f 2 = 1 ε 2 ( f 2 1 − f 2 2 ) ρ = f 1 + f 2 as ε ↓ 0 satisfies ρ t = 1 2 (log( ρ )) xx
Relative entropy in diffusive relaxation Introduction Diffusive relaxation limits Toy model (linear Cattaneo) u ε t + v ε x = 0 x = − 1 t + c 2 u ε v ε ε v ε Time scaling: ∂ t − → ε∂ t t + 1 u ε ε v ε x = 0 t + c 2 x = − 1 v ε ε u ε ε 2 v ε Flux scaling: v ε − → ε v ε � u ε t + v ε x = 0 ε 2 v ε t + c 2 u ε x = − v ε Formal limit: u t − c 2 u xx = 0
Relative entropy in diffusive relaxation Introduction Diffusive relaxation limits References for diffusive relaxation Kurtz, Trans. AMS ‘73; McKean, Israel J. Math. ‘75 Marcati, Milani, Secchi, Manuscr. Math. ‘88; Marcati, Milani, JDE ‘90 Marcati, Natalini, Arch. Rational Mech. Anal. ‘95 Lions, Toscani, Rev. Math. Iberoam. ‘97 Marcati, Rubino, J. Differential Equations ‘00; Donatelli, Marcati, Trans. AMS ‘04 Yong SIAM Appl. Math. ‘04 Coulombel, Goudon, Trans. AMS ‘07; Coulombel, Lin ’11 Donatelli, Di Francesco, Discrete and Cont. Dynamical system. Series B ‘10 L., Yong, Comm. PDE ‘01 L., Natalini, Proc. Roy. Soc. Edinburgh ‘02 Di Francesco, L., Asymptot. Anal. ‘04 Donatelli, L., Commun. Pure Appl. Anal. ‘09
Relative entropy in diffusive relaxation Introduction Relative entropy Relative entropy 3 U (weak, entropy) solution and ¯ U smooth solution of systems of conservation (or balance) laws and ( η, q α ) a convex entropy–entropy flux pair Compute η ( U | ¯ α ∂ α q α ( U | ¯ U ) t + � U ) for η ( U | ¯ U ) = η ( U ) − η (¯ U ) − ∇ U η (¯ U ) · ( U − ¯ U ) q α ( U | ¯ U ) = q α ( U ) − q α (¯ U ) − ∇ U η (¯ U ) · ( F α ( U ) − F α (¯ U )) This shall lead to a stability estimate , used in many different contexts. Recently: ◮ Hyperbolic relaxation: L., Tzavaras ARMA ‘06; Tzavaras Commun. Math. Sci. ‘05; Berthelin, Vasseur SIMA ‘05; Berthelin, Tzavaras, Vasseur J. Stat. Physics ‘09; ◮ Weak–strong uniqueness: Demoulini, Stuart, Tzavaras ARMA ; Feireisl, Novotny ARMA ‘12 3 Dafermos, ARMA ‘79 ; DiPerna, Indiana U. Math. J. ‘79
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Formal analysis The model Isentropic gas dynamics in three space dimensions with a damping term: ρ t + 1 ε div x m = 0 (2) m t + 1 m ⊗ m + 1 ε ∇ x p ( ρ ) = − 1 ε 2 m , ε div x ρ t ∈ R , x ∈ R 3 , density ρ ≥ 0 and momentum flux m ∈ R 3 . The pressure p ( ρ ) satisfies p ′ ( ρ ) > 0 which makes the system hyperbolic. γ –law: p ( ρ ) = k ρ γ with γ ≥ 1 and k > 0. In the diffusive relaxation limit ε ↓ 0, solutions of (2) formally converge to those of the porous medium equation ρ t − △ x p (¯ ¯ ρ ) = 0
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Formal analysis Hilbert’s expansion/1 We now use the standard Hilbert’s expansion ρ = ρ 0 + ερ 1 + ε 2 ρ 2 + . . . , m = m 0 + ε m 1 + ε 2 m 2 + . . . , into (2) and into | m | 2 η ( ρ, m ) t + 1 ε div x q ( ρ, m ) = − 1 ε 2 ∇ m η ( ρ, m ) · m = − 1 ≤ 0 ε 2 ρ | m | 2 for the mechanical energy η ( ρ, m ) = 1 + h ( ρ ) 2 ρ 2 m | m | 2 and the associated flux q ( ρ, m ) = 1 + mh ′ ( ρ ) ρ 2 h ′′ ( ρ ) = p ′ ( ρ ) ρ ; ρ h ′ ( ρ ) = p ( ρ ) + h ( ρ )
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Formal analysis Hilbert’s expansion/2 From the equations we get: O ( ε − 1 ) div x m 0 = 0 O (1) ∂ t ρ 0 + div x m 1 = 0 O ( ε ) ∂ t ρ 1 + div x m 2 = 0 O ( ε − 2 ) m 0 = 0 O ( ε − 1 ) − m 1 = ∇ x p ( ρ 0 ) O (1) − m 2 = ∇ x ( p ′ ( ρ 0 ) ρ 1 ) � m 1 ⊗ m 1 p ′ ( ρ 0 ) ρ 2 + 1 � � � 2 p ′′ ( ρ 0 ) ρ 2 O ( ε ) − m 3 = ∂ t m 1 + div x + ∇ x 1 ρ 0 In particular, we recover the equilibrium relation m 0 = 0 for the state variables, the Darcy’s law m 1 = −∇ x p ( ρ 0 ), and that ρ 0 satisfies porous medium equation
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Formal analysis Hilbert’s expansion/3 From the entropy we get: = −| m 1 | 2 � m 1 h ′ ( ρ 0 ) � O (1) h ( ρ 0 ) t + div x ρ 0 � h ′ ( ρ 0 ) ρ 1 � � m 2 h ′ ( ρ 0 ) + m 1 h ′′ ( ρ 0 ) ρ 1 � O ( ε ) ∂ t + div x = | m 1 | 2 ρ 1 − 2 m 1 · m 2 ρ 2 ρ 0 0 Thus we recover the entropy dissipation relation associated to the porous medium equation for ρ 0 = −|∇ x p ( ρ ) | 2 � h ′ ( ρ ) ∇ x p ( ρ ) � h ( ρ ) t − div x ρ
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Relative entropy estimate Reformulation of the limiting equation We rewrite ¯ ρ t − △ x p (¯ ρ ) = 0 as follows ρ t + 1 ¯ ε∂ x i ¯ m i = 0 (3) m t + 1 m ) = − 1 ¯ ε∂ x i f i (¯ ρ, ¯ ε 2 ¯ m + e (¯ ρ, ¯ m ) for (¯ ρ, ¯ m = − ε ∇ x p (¯ ρ )) and the error term � ¯ m ) = 1 m ⊗ ¯ m � ¯ e := e (¯ ρ, ¯ − ε∂ t ∇ x p (¯ ρ ) ε div x ρ ¯ � ∇ x p (¯ ρ ) ⊗ ∇ x p (¯ ρ ) � = ε div x − ε ∇ x ( p ′ (¯ ρ ) △ x p (¯ ρ )) ρ ¯ = O ( ε )
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Relative entropy estimate Relative entropy/1 The relative entropy is of the form η ( ρ, m | ¯ ρ, ¯ m ) := η ( ρ, m ) − η (¯ ρ, ¯ m ) − η ρ (¯ ρ, ¯ m )( ρ − ¯ ρ ) − ∇ m η (¯ ρ, ¯ m ) · ( m − ¯ m ) 2 � � = 1 m ρ − ¯ m � � + h ( ρ | ¯ ρ ) 2 ρ � � ρ ¯ � � while the corresponding relative entropy-flux reads q i ( ρ, m | ¯ ρ, ¯ m ) := q i ( ρ, m ) − q i (¯ ρ, ¯ m ) − η ρ (¯ ρ, ¯ m )( m i − ¯ m i ) − ∇ m η (¯ ρ, ¯ m ) · ( f i ( ρ, m ) − f i (¯ ρ, ¯ m )) = 1 ρ − ¯ ρ − ¯ + ¯ � m m 2 � m i m i m i � � � + ρ ( h ′ ( ρ ) − h ′ (¯ 2 m i ρ )) ρ h ( ρ | ¯ ρ ) � � ¯ ¯ ¯ ρ � ρ
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Relative entropy estimate Relative entropy/2 Proposition Let ( ρ, m ) be a weak entropy solution of (2) and let (¯ ρ, ¯ m ) be a smooth solution of (3) . Then, m ) t + 1 m ) ≤ − 1 η ( ρ, m | ¯ ρ, ¯ ε div x q ( ρ, m | ¯ ρ, ¯ ε 2 R ( ρ, m | ¯ ρ, ¯ m ) − Q − E , where � ¯ Q = 1 � � 0 � ρ x i ε ∇ 2 ( ρ, m ) η (¯ ρ, ¯ m ) · m x i ¯ f i ( ρ, m | ¯ ρ, ¯ m ) � = − ∂ x i x j h ′ (¯ ρ ) f ij ( ρ, m | ¯ ρ, ¯ m ) i , j 2 � � m ρ − ¯ m m ) · ρ � m ρ − ¯ m � � � R ( ρ, m | ¯ ρ, ¯ m ) = ρ E = e (¯ ρ, ¯ � � ρ ¯ ρ ¯ ρ ¯ � �
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Stability and convergence Control of the quadratic term Q Lemma If p ( ρ ) satisfies p ′′ ( ρ ) ≤ A p ′ ( ρ ) ∀ ρ > 0 for some A > 0 , then h ( ρ ) ρ verifies p ( ρ | ¯ ρ ) ≤ ch ( ρ | ¯ ρ ) ∀ ρ , ¯ ρ > 0 for a given constant c > 0 . Moreover, there exists a C > 0 such that for any fixed i | f i ( ρ, m | ¯ ρ, ¯ m ) | ≤ C η ( ρ, m | ¯ ρ, ¯ m ) � m i � � m j ρ − ¯ � ρ − ¯ m i m j f ij ( ρ, m | ¯ ρ, ¯ m ) = ρ + p ( ρ | ¯ ρ ) δ ij ¯ ¯ ρ ρ Remark: For a γ –law gases: 1 γ > 1, h ( ρ ) = γ − 1 p ( ρ ); γ = 1, p ( ρ | ¯ ρ ) = 0
Relative entropy in diffusive relaxation Isentropic gas dynamics with damping Stability and convergence Possible framework We assume ( H 1 ) ¯ ρ is a smooth, positive solution of the multidimensional porous medium equation ( ρ, m ) be a weak solution of (2) such that ρ ≥ 0, satisfying ρ ∈ L 1 ( R 3 ), the entropy inequality, ρ − ¯ � � � R 3 η ( ρ, m | ¯ ρ, ¯ m ) dx < + ∞ � � t =0 and q ( ρ, m | ¯ ρ, ¯ m ) → 0 , as | x | → + ∞ the pressure p ( ρ ) satisfies p ′′ ( ρ ) ≤ A p ′ ( ρ ) ∀ ρ > 0 ; for ρ instance, p ( ρ ) = ρ γ , γ ≥ 1
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