Cross-diffusion systems with entropy structure Ansgar J¨ ungel Vienna University of Technology, Austria asc.tuwien.ac.at/ ∼ juengel + oxygen separator graphite – Introduction and examples 1 Al Li + Cu Analysis 2 Li + Boundedness-by-entropy method 3 A nonstandard example 4 Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 1 / 26
Introduction and examples Multi-species systems Examples: Relative number of publications (in 0.01%): Wildlife populations Tumor growth Gas mixtures Lithium-ion batteries Population herding Nature is composed of multi-species systems + oxygen separator graphite – Al Li + Cu Li + Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 2 / 26
Introduction and examples Modeling of multi-species systems Particle models: Newton’s laws with interactions among species Markov chains: species move to neighboring cells Stochastic differential equations: using Brownian motion Kinetic equations: distribution function depends on age, size, etc. Here: Diffusive equations for population densities Reaction-diffusion systems: u i (0) = u 0 ∂ t u i − div( D i ∇ u i ) = f i ( u ) in Ω , t > 0 , i , no-flux b.c. Flux D i ∇ u i only depends on u i : Fick’s law not always valid! In multicomponent systems, flux may depend on ∇ u 1 , . . . , ∇ u n Cross-diffusion systems: u (0) = u 0 , ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω , t > 0 , no-flux b.c. Meaning: div( A ( u ) ∇ u ) i = � n j =1 div( A ij ( u ) ∇ u j ), A ∈ R n × n , u ∈ R n Cross-diffusion may allow for pattern formation Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 3 / 26
Introduction and examples Example ➊ : Cross-diffusion population dynamics u (0) = u 0 , ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω , t > 0 , no-flux b.c. u = ( u 1 , u 2 ) and u i models population density of i th species Diffusion matrix: ( a ij ≥ 0) � a 10 + a 11 u 1 + a 12 u 2 � a 12 u 1 A ( u ) = a 21 u 2 a 20 + a 21 u 1 + a 22 u 2 Suggested by Shigesada-Kawasaki- Teramoto 1979 to model segregation Derivation from on-lattice model Lotka-Volterra functions: f i ( u ) = ( b i 0 − b i 1 u 1 − b i 2 u 2 ) u i Diffusion matrix is not symmetric, generally not positive definite Figure: Minneapolis-Saint Paul percentage minority population 2010 Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 4 / 26
Introduction and examples Example ➋ : Multicomponent gas mixtures u (0) = u 0 , ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω , t > 0 , no-flux b.c. Volume fractions of gas components u 1 , . . . , u n , u n +1 = 1 − � n i =1 u i Diffusion matrix for n = 2: δ ( u ) = d 1 d 2 (1 − u 1 − u 2 ) + d 0 ( d 1 u 1 + d 2 u 2 ) � d 2 + ( d 0 − d 2 ) u 1 � 1 ( d 0 − d 1 ) u 1 A ( u ) = ( d 0 − d 2 ) u 2 d 1 + ( d 0 − d 1 ) u 2 δ ( u ) Application: Patients with airways obstruction inhale Heliox to speed up diffusion Proposed by Maxwell 1866/Stefan 1871 Duncan-Toor 1962: Fick’s law ( J i ∼ ∇ u i ) not sufficient, include cross-diffusion terms Boudin-Grec-Salvarani 2015: Derivation from Boltzmann equation for simple mixtures Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 5 / 26
Introduction and examples Difficulties and objectives u (0) = u 0 ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω , t > 0 , Main features: Diffusion matrix A ( u ) non-diagonal (cross-diffusion) Matrix A ( u ) may be neither symmetric nor positive definite Variables u i expected to be bounded from below and/or above Objectives: Local-in-time existence and uniqueness of classical solutions Global-in-time existence and uniqueness of weak solutions Positivity and boundedness of solution (if physically expected) Large-time behavior, design of stable numerical schemes Mathematical difficulties: No general theory for diffusion systems Generally no maximum principle, no regularity theory Lack of positive definiteness ⇒ local/global existence nontrivial Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 6 / 26
Introduction and examples Overview 1 Introduction and examples 2 Analysis 3 Boundedness-by-entropy method 4 A nonstandard example Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 7 / 26
Analysis Local existence analysis u (0) = u 0 ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω ⊂ R d , t > 0 , Theorem (Amann 1990) Let a ij , f i smooth, A ( u ) normally elliptic, u 0 ∈ W 1 , p (Ω; R n ) with p > d. Then ∃ unique local solution u 0 < T ∗ ≤ ∞ u ∈ C 0 ([0 , T ∗ ); W 1 , p (Ω)) , u ∈ C ∞ (Ω × [0 , T ∗ ); R n ) , A ( u ) normally elliptic = all eigenvalues have positive real parts Linear algebra: If H ( u ) symmetric positive definite such that H ( u ) A ( u ) positive definite then A ( u ) normally elliptic Application: Let h ( u ) convex and set H ( u ) := h ′′ ( u ). Then, if f = 0, � � � d h ′ ( u ) · ∂ t udx = − ∇ u : h ′′ ( u ) A ( u ) ∇ u h ( u ) dx = dx dt � �� � Ω Ω Ω ≥ 0 if h ′′ ( u ) A ( u ) pos. def. � Aim: find a Lyapunov functional (entropy) Ω h ( u ) dx Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 8 / 26
Analysis State of the art ∂ t u − div( A ( u ) ∇ u ) = f ( u ) in Ω ⊂ R d , t > 0 Global existence if . . . Growth conditions on nonlinearities (Ladyˇ zenskaya ... 1988) Control on W 1 , p (Ω) norm with p > d (Amann 1989) Positivity, mass control, diagonal A ( u ) (Pierre-Schmitt 1997) Unexpected behavior: Finite-time blow-up of H¨ older solutions (Star´ a-John 1995) Weak solutions may exist after L ∞ blow-up (Pierre 2003) Cross-diffusion may lead to pattern formation (instability) or may avoid finite-time blow-up (Hittmeir-A.J. 2011) Special structure needed for global existence theory: gradient-flow or entropy structure Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 9 / 26
Analysis Entropy and gradient flows Entropy: Measure of molecular disorder or energy dispersal Introduced by Clausius (1865) in thermodynamics Boltzmann, Gibbs, Maxwell: statistical interpretation Shannon (1948): concept of information entropy Entropy in mathematics: ∼ convex Lyapunov functional Hyperbolic conservation laws (Lax), kinetic theory (Lions) Relations to stochastic processes (Bakry, Emery) and optimal transportation (Carrillo, Otto, Villani) Gradient flow: ∂ t u = − grad H | u on differential manifold Example: R d with Euclidean structure ⇒ ∂ t u = − H ′ ( u ) H ( u ) is Lyapunov functional since ∂ t H ( u ) = −| H ′ ( u ) | 2 Gradient flow of entropy w.r.t. Wasserstein distance (Otto), entropy � u log udx : ∂ t u = div( u ∇ H ′ ( u )) = ∆ u H ( u ) = Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 10 / 26
Analysis Gradient flows: Cross-diffusion systems Main assumption ∂ t u − div( A ( u ) ∇ u ) = f ( u ) possesses formal gradient-flow structure � � ∂ t u − div B ∇ grad H ( u ) = f ( u ) , � where B is positive semi-definite, H ( u ) = Ω h ( u ) dx entropy Equivalent formulation: grad H ( u ) ≃ h ′ ( u ) =: w (entropy variable) B = A ( u ) h ′′ ( u ) − 1 ∂ t u − div( B ∇ w ) = f ( u ) , Consequences: 1 H is Lyapunov functional if f = 0: � � dH ∂ t u · h ′ ( u ) dt = dx = − ∇ w : B ∇ wdx ≤ 0 Ω � �� � Ω = w 2 L ∞ bounds for u : Let h ′ : D → R n ( D ⊂ R n ) be invertible ⇒ u = ( h ′ ) − 1 ( w ) ∈ D (no maximum principle needed!) Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 11 / 26
Analysis Example: Maxwell-Stefan systems for n = 2 Volume fractions of gas components u 1 , u 2 , u 3 = 1 − u 1 − u 2 u (0) = u 0 , ∂ t u − div( A ( u ) ∇ u ) = 0 in Ω , t > 0 , no-flux b.c. � d 2 + ( d 0 − d 2 ) u 1 � 1 ( d 0 − d 1 ) u 1 A ( u ) = δ ( u ) ( d 0 − d 2 ) u 2 d 1 + ( d 0 − d 1 ) u 2 � Entropy: H ( u ) = Ω h ( u ) dx , where h ( u ) = u 1 (log u 1 − 1) + u 2 (log u 2 − 1) + (1 − u 1 − u 2 )(log(1 − u 1 − u 2 ) − 1) Entropy variables: w = h ′ ( u ) ∈ R 2 or u = ( h ′ ) − 1 ( w ) e w i w i = ∂ h = log u i , u i = 1 + e w 1 + e w 2 ∈ (0 , 1) ∂ u i u 3 Entropy production: � 2 � � |∇ u i | 2 |∇ u 3 | 2 dH � dt ( u ) = − + d 0 u 1 u 2 dx ≤ 0 d i u i u 3 Ω i =1 Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 12 / 26
Analysis Overview 1 Introduction and examples 2 Analysis 3 Boundedness-by-entropy method 4 A nonstandard example Ansgar J¨ ungel (TU Wien) Nonstandard entropies asc.tuwien.ac.at/ ∼ juengel 13 / 26
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