Relative entropy methods in the mathematical theory of complete fluid systems Eduard Feireisl Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague and Erwin Schroedinger International Institute for Mathematical Physics, Vienna joint work with A. Novotn´ y (Toulon) 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications June 25-29, 2012 - University of Padua, Italy Eduard Feireisl Relative entropies and complete fluids
Mathematical model State variables Mass density Absolute temperature Velocity field ̺ = ̺ ( t , x ) ϑ = ϑ ( t , x ) u = u ( t , x ) Thermodynamic functions Pressure Internal energy Entropy p = p ( ̺, ϑ ) e = e ( ̺, ϑ ) s = s ( ̺, ϑ ) Transport Viscous stress Heat flux S = S ( ϑ, ∇ x u ) q = q ( ϑ, ∇ x ϑ ) Eduard Feireisl Relative entropies and complete fluids
Field equations Equation of continuity ∂ t ̺ + div x ( ̺ u ) = 0 Momentum balance Claude Louis Marie Henri Navier ∂ t ( ̺ u ) + div x ( ̺ u ⊗ u ) + ∇ x p ( ̺, ϑ ) = div x S + ̺ ∇ x F [1785-1836] Entropy production � q � ∂ t ( ̺ s ( ̺, ϑ )) + div x ( ̺ s ( ̺, ϑ ) u ) + div x = σ ϑ George σ = ( ≥ ) 1 � S : ∇ x u − q · ∇ x ϑ � Gabriel ϑ ϑ Stokes [1819-1903]
Constitutive relations Fourier’s law q = − κ ( ϑ ) ∇ x ϑ Fran¸ cois Marie Charles Fourier [1772-1837] Newton’s rheological law � x u − 2 � ∇ x u + ∇ t S = µ ( ϑ ) 3 div x u + η ( ϑ ) div x u I Isaac Newton [1643-1727] Eduard Feireisl Relative entropies and complete fluids
Gibbs’ relation Gibbs’ relation: � 1 � ϑ Ds ( ̺, ϑ ) = De ( ̺, ϑ ) + p ( ̺, ϑ ) D ̺ Willard Gibbs [1839-1903] Thermodynamics stability: ∂ p ( ̺, ϑ ) > 0 , ∂ e ( ̺, ϑ ) > 0 ∂̺ ∂ϑ Eduard Feireisl Relative entropies and complete fluids
Boundary conditions Impermeability u · n | ∂ Ω = 0 No-slip No-stick u tan | ∂ Ω = 0 [ S n ] × n | ∂ Ω = 0 Thermal insulation q · n | ∂ Ω = 0 Eduard Feireisl Relative entropies and complete fluids
A bit of history of global existence for large data Olga Aleksandrovna Jean Leray [1906-1998] Ladyzhenskaya Global existence of weak [1922-2004] Global solutions for the existence of classical incompressible solutions for the Navier-Stokes system (3D) incompressible 2D Navier-Stokes system Pierre-Louis Lions [*1956] Global existence of weak solutions for the compressible barotropic Navier-Stokes system (2,3D) and many, many others... Eduard Feireisl Relative entropies and complete fluids
Relative entropy (energy) Dynamical system d d t u ( t ) = A ( t , u ( t )) , u ( t ) ∈ X , u (0) = u 0 Relative entropy U : t �→ U ( t ) ∈ Y a “trajectory” in the phase space Y ⊂ X � � � E u ( t ) � U ( t ) , E : X × Y → R � Eduard Feireisl Relative entropies and complete fluids
Basic properties Positivity(distance) E { u | U } is a “distance” between u , and U , meaning E ( u | U ) ≥ 0 and E { u | U } = 0 only if u = U Lyapunov function � � u ( t ) | ˜ is a Lyapunov function provided ˜ E U is an equilibrium U � � � � ˜ t �→ E u ( t ) U is non-increasing � Gronwall inequality � τ � � � � � � � � � E u ( τ ) � U ( τ ) ≤ E u ( s ) � U ( s ) + c ( T ) E u ( t ) � U ( t ) d t � � � s if U is a solution of the same system (in a “better” space) Y
Applications Stability of equilibria Any solution ranging in X stabilizes to an equilibrium belonging to Y (to be proved!) Weak-strong uniqueness Solutions ranging in the “better” space Y are unique among solutions in X . Singular limits Stability and convergence of a family of solutions u ε corresponding to A ε to a solution U = u of the limit problem with generator A . Eduard Feireisl Relative entropies and complete fluids
Navier-Stokes-Fourier system revisited Total energy balance (conservation) d � � 1 � 2 ̺ | u | 2 + ̺ e ( ̺, ϑ ) − ̺ F d x = 0 d t Ω Total entropy production d � � ̺ s ( ̺, ϑ ) d x = σ d x ≥ 0 d t Ω Ω Total dissipation balance d � � 1 � � 2 ̺ | u | 2 + ̺ e ( ̺, ϑ ) − Θ ̺ s ( ̺, ϑ ) − ̺ F d x + σ d x = 0 d t Ω Ω Eduard Feireisl Relative entropies and complete fluids
Equilibrium (static) solutions Equilibrium solutions minimize the entropy production u ≡ 0 , ϑ ≡ Θ > 0 a positive constant Static problem ∇ x p (˜ ̺, Θ) = ˜ ̺ ∇ x F Total mass and energy are constants of motion � � ̺ d x = M 0 , ˜ ̺ e (˜ ˜ ̺, Θ) − ˜ ̺ F d x = E 0 Ω Ω Eduard Feireisl Relative entropies and complete fluids
Total dissipation balance revisited � 1 � d � 2 ̺ | u | 2 + H Θ ( ̺, ϑ ) − ∂ H Θ (˜ ̺, Θ) ( ̺ − ˜ ̺ ) − H Θ (˜ ̺, Θ) d x d t ∂̺ Ω � + σ d x = 0 Ω Ballistic free energy � � H Θ ( ̺, ϑ ) = ̺ e ( ̺, ϑ ) − Θ s ( ̺, ϑ ) Eduard Feireisl Relative entropies and complete fluids
Coercivity of the ballistic free energy ̺,̺ H Θ ( ̺, Θ) = 1 ∂ 2 ̺∂ ̺ p ( ̺, Θ) ∂ ϑ H Θ ( ̺, ϑ ) = ̺ ( ϑ − Θ) ∂ ϑ s ( ̺, ϑ ) Coercivity ̺ �→ H Θ ( ̺, Θ) is convex ϑ �→ H Θ ( ̺, ϑ ) attains its global minimum (zero) at ϑ = Θ Eduard Feireisl Relative entropies and complete fluids
Relative entropy � � � E ̺, ϑ, u � r , Θ , U � � � 1 2 ̺ | u − U | 2 + H ( ̺, ϑ ) − ∂ H Θ ( r , Θ) � = ( ̺ − r ) − H Θ ( r , Θ) d x ∂̺ Ω Eduard Feireisl Relative entropies and complete fluids
Dissipative solutions Relative entropy inequality �� τ � � � E ̺, ϑ, u � r , Θ , U � t =0 � τ � Θ � S ( ϑ, ∇ x u ) : ∇ x u − q ( ϑ, ∇ x ϑ ) · ∇ x ϑ � + d x d t ϑ ϑ 0 Ω � τ ≤ R ( ̺, ϑ, u , r , Θ , U ) d t 0 for any r > 0, Θ > 0, U satisfying relevant boundary conditions Eduard Feireisl Relative entropies and complete fluids
Remainder R ( ̺, ϑ, u , r , Θ , U ) � � � � � = ̺ ∂ t U + u · ∇ x U · ( U − u ) + S ( ϑ, ∇ x u ) : ∇ x U d x Ω � div U + ̺ �� � � + p ( r , Θ) − p ( ̺, ϑ ) r ( U − u ) · ∇ x p ( r , Θ) d x Ω � � � � � � − s ( ̺, ϑ ) − s ( r , Θ) ∂ t Θ + ̺ s ( ̺, ϑ ) − s ( r , Θ) u · ∇ x Θ ̺ Ω + q ( ϑ, ∇ x ϑ ) � · ∇ x Θ d x ϑ � r − ̺ � � + ∂ t p ( r , Θ) + U · ∇ x p ( r , Θ) d x r Ω Eduard Feireisl Relative entropies and complete fluids
Applications Existence Dissipative (weak) solutions exist (under certain constitutive restrictions) globally in time for any choice of the initial data. Unconditional stability of the equilibrium solutions Any (weak) solution of the Navier-Stokes-Fourier system stabilizes to an equilibrium (static) solution for t → ∞ . Weak-strong uniqueness Weak and strong solutions emanating from the same initial data coincide as long as the latter exists. Strong solutions are unique in the class of weak solutions. Eduard Feireisl Relative entropies and complete fluids
Singular limit in the incompressible, inviscid regime Solutions of the Navier-Stokes-Fourier system converge in the limit of low Mach and high Reynolds and P´ eclet number to the Euler-Boussinesq system. Jean Claude Osborne Ernst Mach Eug` ene Reynolds [1838-1916] P´ eclet [1842-1912] [1793-1857] Eduard Feireisl Relative entropies and complete fluids
Scaled Navier-Stokes-Fourier system Equation of continuity ∂ t ̺ + div x ( ̺ u ) = 0 Balance of momentum 1 ε 2 ∇ x p ( ̺, ϑ ) = ε a div x S ∂ t ( ̺ u ) + div x ( ̺ u ⊗ u ) + Entropy production � q ( ϑ, ∇ x ϑ ) � ∂ t ( ̺ s ( ̺, ϑ )) + div x ( ̺ s ( ̺, ϑ ) u ) + ε b div x ϑ = 1 � ε 2+ a S : ∇ x u − ε b q ( ϑ, ∇ x ϑ ) · ∇ x ϑ � ϑ ϑ Eduard Feireisl Relative entropies and complete fluids
Target system incompressibility div x v = 0 Euler system ∂ t v + v · ∇ x v + ∇ x Π = 0 temperature transport ∂ t T + v · ∇ x T = 0 Basic assumption The incompressible Euler system possesses a strong solution v on a time interval (0 , T max ) for the initial data v 0 = H [ u 0 ] . Eduard Feireisl Relative entropies and complete fluids
Prepared data ̺ (0 , · ) = ̺ + ε̺ (1) 0 ,ε , ̺ (1) 0 ,ε → ̺ (1) in L 2 (Ω) and weakly-(*) in L ∞ (Ω) 0 ϑ (0 , · ) = ϑ + εϑ (1) 0 ,ε , ϑ (1) 0 ,ε → ϑ (1) in L 2 (Ω) and weakly-(*) in L ∞ (Ω) 0 u (0 , · ) = u 0 ,ε → u 0 in L 2 (Ω; R 3 ) , v 0 = H [ u 0 ] ∈ W k , 2 (Ω; R 3 ) , k > 5 2 Eduard Feireisl Relative entropies and complete fluids
Boundary conditions Navier’s complete slip condition u · n | ∂ Ω = 0 , [ S n ] × n | ∂ Ω = 0 Eduard Feireisl Relative entropies and complete fluids
Convergence b > 0 , 0 < a < 10 3 ess sup � ̺ ε ( t , · ) − ̺ � L 2 + L 5 / 3 (Ω) ≤ ε c t ∈ (0 , T ) √ ̺ ε u ε → � loc ((0 , T ]; L 2 loc (Ω; R 3 )) ̺ v in L ∞ and weakly-(*) in L ∞ (0 , T ; L 2 (Ω; R 3 )) ϑ ε − ϑ loc ((0 , T ]; L q loc (Ω; R 3 )) , 1 ≤ q < 2 , → T in L ∞ ε and weakly-(*) in L ∞ (0 , T ; L 2 (Ω)) Eduard Feireisl Relative entropies and complete fluids
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