Variational formulation of entropy solutions for nonlinear conservation laws

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1 Variational formulation of entropy solutions for nonlinear conservation laws Eitan Tadmor 1 Center for Scientific Computation and Mathematical Modeling CSCAMM) Department of Mathematics, Institute for Physical Science & Technology University of Maryland SIAM Invited Address 1 Joint Mathematical Meeting, Baltimore, MD, January 16, 2014 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 1 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 2

2 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 3 Equations of mass and momentum Hydrodynamics: Euler ) Balance of mass and momentum ρ t + x ρv) = 0 v t + v x v = 1 ρ xp Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 4

3 Equations of mass and momentum Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 5 Equations of mass, momentum... and energy Hydrodynamics: Euler ) Balance of mass and momentum ρ t + x ρv) = 0 v t + v x v = 1 ρ xp Thermodynamics: density ρ, pressure p, temperature T, internal energy e,... First law of thermodynamics: work is converted into gained heat e Single-phase systems: equation of state link any three p = pρ, γ e) 1)ρe Émilie du Châtelet 1749) Helmholtz 1847) 1 Balance of energy e t + v x e = p ρ x v 1 von Mayer 1841) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 6

4 Conservation laws of mass momentum and energy total mass : ρ, t)dx Conservation of mass, momentum, energy ρ t + x ρv) = 0 ρv) t + x v ρv)) + x p = 0 is balanced by flux of mass ρv ndω E t + x ve + p)) = 0 total momentum ρv, t)dx is balanced by momentum flux... and total energy E, t)dx, E := 1 2 ρ v 2 + ρe Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 7 Conservation laws of mass momentum and energy ρ, t)dx, ρv, t)dx and total energy Navier-Stokes equations ): Conservation of mass, momentum, energy ρ t + x ρv) = 0 ρv) t + x v ρv)) + x p = 0 x Σ E t + x ve + p)) = 0 x Σv + κ x T ) E, t)dx, E := 1 2 ρ v 2 + ρe No 100% energy efficiency Viscosity Σ = µ x v + x v ) + λ x v I and heat conduction x T Carnot 1824), Clausius 1865): Second law of thermodynamics Entropy S = Sρ, p) = ρ lnpρ γ ) increases S t + x vs κ x ln T ) T 2 = κ T 2 + 2µ trd2 v)) + λ x v) 2 T Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 8 0

5 Dissipation of entropy Second law of thermodynamics irreversibility It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a cooler body to a hotter body Complete conservation of heat into work is impossible Does lack of conservation vanish with vanishing dissipation κ, µ, λ 0? S t + x vs) = 2µ trd2 v)) + λ x v) 2 0, Dv) = xv + x v T 2 ) x n st Traveling viscous wave: v ɛ x, t) w, ξ ɛ, t with speed s: ss ξ1 + v n S ξ1 w ξ 1 2 independent of ɛ = 2µ + λ 0 T Rate of increase of entropy does not go to zero Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 9 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 10

6 Euler equations as hyperbolic system of conservation laws Conservation laws: u t + x fu) = 0: Conservation: u, t)dx = f n dω, flux f = f 1,..., f d ) Hyperbolicity: u t + j A j u)u xj = 0, A j u) = fj u) u Symmetric hyperbolic systems: ux, t) : R d x R t R m d u t + A j u)u xj = 0 such that j=1 the A j s are symmetrizable: H > 0 : HA j = HA j ) symmterizable ρ Primary example Euler eqs for the conserved variables u := ρv E 0 e f j j 0 u) = A j u) v j I 5 5 +c c = ρv j ρv j v + pδ ij v j E + p) e j 0 γ p ρ Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 11 Basic facts on nonlinear conservation laws Spontaneous formation of shock discontinuities weak solutions: ψ, u t + x fu) dxdt = 0, ψ C0 R t Among the many weak solutions: solution realized as vanishing viscosity limit: u = lim u ɛ u ɛ t + x fu ɛ ) = ɛ x u ɛ d = 1: Bianchini & Bressan 2005) 1 d > 1: P. D. Lax 2007) There is no theory for the initial value problem for compressible flows in two space dimensions once shocks show up, much less in three space dimensions. This is a scientific scandal and a challenge. 2 Vanishing viscosity solutions are entropic solutions 1 For initial data u 0 BV 1; 2 SIAM mini-symposium on entropic solutions...1pm Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 12

7 Entropy in nonlinear conservation laws u t + x fu) = 0 admits an extension: ηu) t + x qu) = 0 An entropy pair η, q): η u) ) Aj u) = q ju) ), q = q1,..., q d ) NOTATION: ) = ) u, e.g., η u) ) = ηu1,..., η ud ) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 13 Entropy in nonlinear conservation laws u t + x fu) = 0 admits an extension: ηu) t + x qu) = 0 An entropy pair η, q): j A j u)u xj {}}{ η u), u t + x fu) = ηu) t + j η u) ) Aj u) = q ju) ), q = q1,..., q d ) q j u) {}}{ η u)a j u)u u xj = ηu) t + x qu) Lax entropy condition 1971) for all convex η s: ηu) t + x qu) 0 Vanishing viscosity limits: u ɛ t + x fu ɛ ) = ɛ x u ɛ For convex η: η u ɛ ), x u ɛ x ηu ɛ ) x u ɛ, η u ɛ ) x u ɛ x ηu ɛ ) 0 = η u ɛ ), u ɛ t + x fu ɛ ) ɛ u ɛ Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 14

8 Entropy in nonlinear conservation laws u t + x fu) = 0 admits an extension: ηu) t + x qu) = 0 An entropy pair η, q): η u) ) Aj u) = q ju) ), q = q1,..., q d ) η u), u t + x fu) = ηu) t + j η u)a j u)u xj = ηu) t + x qu) Lax entropy condition 1971) for all convex η s: ηu) t + x qu) 0 Vanishing viscosity limits: u ɛ t + x fu ɛ ) = ɛ x u ɛ For convex η: η u ɛ ), x u ɛ x ηu ɛ ) x u ɛ, η u ɛ ) x u ɛ x ηu ɛ ) 0 = η u ɛ ), u ɛ t + x fu ɛ ) ɛ u ɛ = ηu ɛ ) t + x qu ɛ ) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 15 Entropy in nonlinear conservation laws u t + x fu) = 0 admits an extension: ηu) t + x qu) = 0 An entropy pair η, q): η u) ) Aj u) = q ju) ), q = q1,..., q d ) η u), u t + x fu) = ηu) t + j η u)a j u)u xj = ηu) t + x qu) Lax entropy condition 1971) for all convex η s: ηu) t + x qu) 0 Vanishing viscosity limits: u ɛ t + x fu ɛ ) = ɛ x u ɛ For convex η: η u ɛ ), x u ɛ x ηu ɛ ) x u ɛ, η u ɛ ) x u ɛ x ηu ɛ ) 0 = η u ɛ ), u ɛ t + x fu ɛ ) ɛ u ɛ ηu ɛ ) t + x qu ɛ ) ɛ x ηu ɛ ) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 16

9 Entropy in nonlinear conservation laws u t + x fu) = 0 admits an extension: ηu) t + x qu) = 0 An entropy pair η, q): η u) ) Aj u) = q ju) ), q = q1,..., q d ) η u), u t + x fu) = ηu) t + j η u)a j u)u xj = ηu) t + x qu) Lax entropy condition 1971) for all convex η s: ηu) t + x qu) 0 Vanishing viscosity limits: u ɛ t + x fu ɛ ) = ɛ x u ɛ For convex η: η u ɛ ), x u ɛ x ηu ɛ ) x u ɛ, η u ɛ ) x u ɛ x ηu ɛ ) 0 = η u ɛ ), u ɛ t + x fu ɛ ) ɛ u ɛ ηu ɛ ) t + x qu ɛ ) ɛ x ηu ɛ ) Vanishing viscosity limits u = lim u ɛ are entropic: ηu) t + x fu) 0 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 17 Entropic solutions for nonlinear conservation laws Krushkov 1970), Lax 1971): entropy pair η u) ) Aj u) = q ju) ) u is an entropic solution of u t + x fu) = 0 for all convex η s: ηu) t + x qu) 0 Linear η s: ηu) = ±u k q j u) = ±f j k : u t + x fu) = 0 Nontrivial extension for strictly convex η s decrease of entropy Primary example: from Naiver-Stokes to Euler equations specific entropy s = sρ, p) = lnpρ γ ) T 2 s t + v x s = 0 κ x ln T = κ T 2 + 2µ trd2 v)) + λ x v) 2 T Convex entropy functions 1,2 : η, q) = ρhs), ρvhs)), ḣ γḧ > 0 Physical entropy is concave: Su) = ρs increases hs) = s) S = ρs : S t + x vs) 0-1 Harten 1984) 2 T. J. R. Hughes 1986) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 18 0

10 Key questions Entropic systems of conservation laws { u t + x fu) = 0, ηu) t + x qu) 0, convex entropy pairs η, q) The notion of entropy imposing the compatibility requirement η, q) = ηu), qu)) is an entropy pair: q j u) ) = η u) ) Aj u), A j u) = f j u) u Questions: What does ) really mean? How rich is the class of systems endowed with such entropy pairs? Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 19 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 20

11 Entropy and symmetry One-dimensional systems: u t + fu) x = 0, f = f 1,... f m ) η, q) = ηu), qu)) is an entropy pair: q u) ) = η u) ) Au), Akj u) = f ku) u j q u j = k η u k f k u j Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 21 Entropy and symmetry One-dimensional systems: u t + fu) x = 0, f = f 1,... f m ) η, q) = ηu), qu)) is an entropy pair: q u) ) = η u) ) Au), Akj u) = f ku) u j 2 q u i u j = ) η f k u i u k u j k k k 2 η u i u k f k u j + T ij =T ji {}}{ η 2 f k u k u i u j k An entropy Hessian η symmetrizes A: q = η A + T, T ij = T ji Conversely: if η A is symmetric: q such that η f k = q u k u j u j k Friedrichs & Lax 1971): Entropy symmetrizer Hu):=η u) u t + x fu) = 0 Hu t + j HA j u xj = 0, HA j = symmetric Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 22

12 Entropy and symmetry ) u t + x fu) = 0 Sever-Mock 1980) If ηu) is a convex entropy of ): Then ) is symmetric w.r.t. the entropy variables w := η u) u t + x fu) = 0 symmetric uw) t + x fuw)) = 0 uw) t + x fuw)) = u w w t + f j u u w w x j = 0 j Recall the left symmetrizer H = η u) = w u : HA j are symmetric u Symmetrization on the right : w = H 1 > 0 and A j H 1 symmetric Godunov 1961) If ) is symmetrizable under change of variables: Then the system u t + x fu) = 0 has an entropy Example ET 1986). Euler eqs: w = pρ) γ γ+1 E, ρv, ρ) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 23 So how rich are entropic systems? Scalar equations m = 1): all convex η s are entropy functions 2 2 systems m = 2): η u 1, u 2 ) fu 1, u 2 ) u 1, u 2 ) is symmetric1 : η F η : η u1 u 1 f 1 u 2 + η u1 u 2 f 2 u 2 = η u1 u 2 f 1 u 1 + η u2 u 2 f 2 u 1 There are m m entropic rich systems ; Temple class 2 But general m 3-systems are non-entropic: mm 1)/2 symmetry constraints of η u) f is over-determined u u 2 Example: u t + u 3 u x =0 has no conservative extension u 1 Many physically relevant systems are endowed with entropy functions: Euler eqs, MHD eqs, elasticity eqs, shallow-water eqs,... 1 Lax, DiPerna, G.-Q. Chen,... 2 Serre, Temple,... Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 24

13 Entropy and stability Kruzkov 1970) L 1 -stability theory for scalar eqs m = 1) ηu; c) = u c : η u 2 x, t); u 1 x, t)) dx η u 20 x); u 10 x)) dx R d R d Tartar 1979), DiPerna 1983): m = 2 with η F 2 2 systems: compensated compactness ηu ɛ ) t + qu ɛ ) x H 1 x,t {u ɛ } L 2 -compact Dafermos 1979), DiPerna 1979) relative entropies ηu; c) = ηu) ηc) η c) u c) u c 2 u 1 Lip : u 2, t) u 1, t) L 2 u 20 u 10 L 2 Entropy stable difference schemes: ET 1987) d dt u νt)+ 1 ) ) f x ν+ 1 f 2 ν 1 = x D + x Q 2 ν 1 D x w ν 2 : Q Q Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 25 Key questions - revisited Entropic systems of conservation laws { u t + x fu) = 0, ηu) t + x qu) 0, convex entropies η and compatible flux q) The notion of entropy imposing the symmetry requirement η = ηu) is an entropy: η u)a j u) = η u)a j u) ), Aj u) = fj u) u Questions: Why imposing the symmetry requirement )? Where do entropy symmetrizers come from? Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 26

14 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 27 Physically relevant systems kinetic description Hydrodynamic description: ρ, v ρ t + x ρv) = 0 v t + v x v = 1 ρ xp Human scale: ε 0, t = εt, x = εx) v Kinetic description: f x, ξ, t) f t + ξ x f = 1 ɛ Qf, f ) N including meso-scale phenomena) Particle description: {x i, ξ i } 1 i N { ẋi = ξ i ξ i = F i, F i Fx i, ξ i )x, ξ) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 28

15 Entropy at the kinetic level: the H-theorem Boltzmann equation 1872): f t + ξ x f = Qf, f ) Conservation of mass... momentum energy... ρ ρv E = R 3 1 ξ 1/2 ξ 2 f x, ξ, t)dξ Entropy S = k log W 1 N log W = f i log f i : W = # of microscopic states compatible with the observed macroscopic states Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 29 Entropy at the kinetic level: the H-theorem Boltzmann equation 1872): f t + ξ x f = Qf, f ) ) f log f dξ ξ x f log f )dξ = Qf, f ) log f dξ 0 + t Conservation of mass... momentum energy... ρ ρv E = R 3 1 ξ 1/2 ξ 2 f x, ξ, t)dξ Entropy S = k log W 1 N log W = f i log f i : ) S = k B H: H := f log f dξ Hf ) t + x ξf log f dξ R 3 R 3 Maxwell 1872): f M ρ,v,t x, ξ, t) = S = M log M dξ = ρ lnρt d /2 ) : ρ v ξ 2 2πT ) d e 2T /2 S t + x vs) 0 0 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 30

16 Maximum Entropy Principle MEP) Clausius 1865), Gibbs 1875,...,1902): Entropy is maximized for a given expectation of energy E.T. Jaynes 1957) Out of the many possibilities choose the one w/little structure as possible MaxEnt) Shannon 1948): entropy as a measure of uncertainty: The only such measure which is increasing and additive: Sp 1,..., p N ) = i p i log p i maximize entropy subject to observables with unknown probabilities {}}{ max { }}{{ }}{ p i log p i : pi E µ ξ i ) pi = 1 {p i } pξ) = e Eµ ξ)/t µ = e Eξ)/k BT w/partition ZT ) = e µ ZT ) ZT ) ξ Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 31 Eµ Tµ Key questions summary Entropic systems of conservation laws 1. Conservation law: u t + x fu) = 0, 2. Entropy condition: ηu) t + x qu) 0, Minimum Entropy Principle for a convex entropy η) t2 3. MEP : #u, t 1 ) u, t) min ηux, τ))dxdτ τ=t 1 Maximum Entropy principle concave η Imposing the symmetry requirement: symmetry requirement {}}{ η A j = η A j ) Now, in many branches of physics...symmetries play a fundamental role, but all these symmetries as it seems to me are assumed and not derived. I now wonder whether or not...symmetries can also be derived. K. O. Friedrichs, 1979 John von Neumann Lecture Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 32

17 Outline 1 Entropy and dissipation Equations of mass, momentum, energy Conservation laws of mass momentum and energy Dissipation of entropy 2 Hyperbolic conservation laws Entropic solutions 3 Entropy and symmetry Entropy and stability 4 Entropy at the kinetic level H-Theorem Maximum entropy principle 5 A new variational formulation Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 33 A new variational formulation of ) u t + x fu) = 0 η C 1 0,convex, u BV R d x R + t A critical point of J η, u) := R m ) is a variational solution of ): η u), u t + x fu) φ dxdt t2 t 1 Original m m)-system is embedded in m + 1 dimensional phase space BV framework for non-conservative products 1,2, φ u BV, au) L : au), u x φ := Borel measure µ such that µb) = au), u x dx, u CB) µx 0 ) φ := B 1 s=0 aus)), us) ds, us) = uφs)) : [0, 1] [u 0, u 0 +] Integration by parts Gauss-Green holds Au)u x φ independent of path φ iff Au) = f u : Au)u x φ = f x 1 Volpert 1967) 2 DelMaso, LeFloch, Murat 1995) Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 34

18 A new variational formulation of ) u t + x fu) = 0 η C 1 0,convex, u BV R d x R + t A critical point of J η, u) := R m ) is a variational solution of ): η u), u t + x fu) φ dxdt t2 t 1 Original m m)-system is embedded in m + 1 dimensional phase space variation in η-variable: δ η J η, u) = δηu)), u t + x fu) φ = 0 1. A variational formulation of the conservation law ): for all admissible δη ): δηu)), u t + x fu) φ = 0 THEOREM. A variational solution is a weak solution of ) Renormalized solution a la DiPerna-Lions 1989)) Different paths φ encode different vanishing viscosity limits Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 35 A new variational formulation of ) u t + x fu) = 0 η C 1 0,convex, u BV R d x R + t A critical point of J η, u) := R m ) is a variational solution of ): η u), u t + x fu) φ dxdt t2 t 1 Original m m)-system is embedded in m + 1 dimensional phase space variation in u: δ u J η, u) = η u), δu) t + xj A j u)δu) = 0 φ η u) ) t j A j u) η u) ) x j, δu φ = 0 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 36

19 A new variational formulation of ) u t + x fu) = 0 η C 1 0,convex, u BV R d x R + t A critical point of J η, u) := R m ) is a variational solution of ): η u), u t + x fu) φ dxdt t2 t 1 Original m m)-system is embedded in m + 1 dimensional phase space variation in u: δ u J η, u) = η u), δu) t + xj A j u)δu) = 0 φ 2. η j A j u xj {}}{ η η u)u t Derivation of an entropy η: η u xj η symmetrize A j : η A j = A j η j A j { }} { A j u) η u) ), δu x j φ = 0 j η u)a j u) = q ju) Existence of critical points) existence of entropy functions)! Non entropic systems: no extremum is attained. Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 37 A new variational formulation of ) u t + x fu) = 0 η C 1 0,convex, u BV R d x R + t A critical point of J η, u) := R m ) is a variational solution of ): η u), u t + x fu) φ dxdt t2 t 1 Original m m)-system is embedded in m + 1 dimensional phase space since a critical point η is an entropy: η, x f φ = x q Derivation of a minimum entropy principle: arg min J arg min ηu) t + x qu) dxdt 3. [t 1,t 2 ] t2 =arg min ηu, t)) dx qu) n dω t 2 + t 1 t 1 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 38

20 A variational solution of ) u t + x fu) = 0 DEFINITION. η, u) Con, XBV ) is a variational solution of ): A critical point of J η, u) := t2 t 1 η u), u t + x fu) φ dxdt A variational solution : 1. Satisfies the conservation law ) is a renormalized sense ; 2. Admits an entropy function or approximate one); 3. Satisfies a minimum entropy principle. Justify formal steps Topological methods Palais-Smale condition: Is the class of approximate entropy solutions compact ηu ɛ ) t + x qu ɛ )? µ 0 Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 39 THANK YOU Eitan Tadmor Variational formulation of entropy solutions for nonlinear conservation laws 40