Relative entropy method for the growth-fragmentation equation with measure data Tomasz Dębiec Institute of Applied Mathematics and Mechanics, University of Warsaw joint work with Marie Doumic (INRIA), Piotr Gwiazda (IMPAN) and Emil Wiedemann (LU Hannover) Mathflows 2018, Porquerolles, 02.09-07.09.2018 Tomasz Dębiec GRE for measure solutions
Outline of the talk Relative entropy method, Generalized relative entropy (GRE) method Growth-fragmentation equation, asymptotic behaviour Measure-valued solutions Main result, outline of proof Tomasz Dębiec GRE for measure solutions
Classical relative entropy method Consider the system ∂ t U + ∂ α F α = 0 (1) describing the evolution of U : R d × R + → R n , where the fluxes F α : R n → R n are smooth maps. The second law of thermodynamics suggests that the entropy admissibility criterion be met: ∂ t η + ∂ α q α ≤ 0 where the entropy η : R n → R is a smooth convex function and ( q 1 , . . . , q d ) : R d → R n is the associated entropy flux. Tomasz Dębiec GRE for measure solutions
Classical relative entropy method Let U be a BV weak entropy solution and let ¯ U be a classical (Lipschitz) solution. Introduce the following quantities, to compare the distance between the two solutions: relative entropy: η ( U | ¯ U ) = η ( U ) − η ( ¯ U ) − ∇ η ( ¯ U ) · ( U − ¯ U ) relative flux: q α ( U | ¯ U ) = q α ( U ) − q α ( ¯ U ) − ∇ η ( ¯ U ) · ( F α ( U ) − F α ( ¯ U )) A computation leads to: ∂ t η ( U | ¯ U )+div q ( U | ¯ U ) ≤ −∇ 2 η ( ¯ U ) ∂ α ¯ U · [ F α ( U ) − F α ( ¯ U ) −∇ F α ( ¯ U )( U − ¯ U )] . This inequality serves as a starting point to obtain stability results. A typical result reads: Tomasz Dębiec GRE for measure solutions
Classical relative entropy method Let U ( x , t ) be an admissible weak solution and ¯ U ( x , t ) be a Lipschitz solution of (1), defined for t ∈ [ 0 , T ) . Suppose both lie in a convex compact set D in the state space. If (1) is endowed with a strictly convex entropy η , then the following local stability estimate holds: � � U ( x , t ) | 2 dx ≤ ae bt U 0 ( x ) | 2 dx | U ( x , t ) − ¯ | U 0 ( x ) − ¯ | x | < r | x | < r + kt for any r > 0 and t ∈ [ 0 , T ) Tomasz Dębiec GRE for measure solutions
Relative entropy method - applications uniqueness of solutions to scalar conservation laws (Dafermos, DiPerna) weak-strong uniqueness for systems (Dafermos, DiPerna) stability, dimension reduction, asymptotic limits (Bella-Feireisl-Novotný, Giesselmann-Tzavaras, Feireisl-Jin-Novotny, Christoforou-Tzavaras) measure-valued–strong uniqueness (Brenier-De Lellis-Szekelyhidi, Demoulini-Stuart-Tzavaras, Feireisl-Gwiazda-Świerczewska-Gwiazda-Wiedemann) generalized relative entropy (Mischler-Perthame-Ryzhik, Michel-Mischler-Perthame) Tomasz Dębiec GRE for measure solutions
GRE method Extends the notion of relative entropy to equations that are not conservation laws Natural in biological applications (age structured, size structured, maturity structured models) Consequences: a priori bounds, large time convergence to the steady state, attraction to periodic solution, exponential rate of convergence (some cases) Tomasz Dębiec GRE for measure solutions
GRE method Extends the notion of relative entropy to equations that are not conservation laws Natural in biological applications (age structured, size structured, maturity structured models) Consequences: a priori bounds, large time convergence to the steady state, attraction to periodic solution, exponential rate of convergence (some cases) Extended to allow for measure initial data (Gwiazda-Wiedemann 2017, D.-Doumic-Gwiazda-Wiedemann 2018) Tomasz Dębiec GRE for measure solutions
Structured population models Developed for understanding the time evolution of a population. Take into account the distribution of the population along some "structuring" variables. First models date back to early 20th century - age structured models (Lotka & Sharpe 1911, Kermack & McKendrick 1927,1932) Size structure introduced in late 1960’s (Bell & Anderson 1967, Sinko & Streifer 1971) Growth-fragmentation equation is found fitting in various contexts: cell division, polymerization, neurosciences, prion proliferation, telecommunication. Tomasz Dębiec GRE for measure solutions
Growth-fragmentation equation We consider the following growth-fragmentation equation: ∞ � ∂ t n ( t , x ) + ∂ x ( g ( x ) n ( t , x )) + B ( x ) n ( t , x ) = k ( y , x ) B ( y ) n ( t , y ) dy , x g ( 0 ) n ( t , 0 ) = 0 , n ( 0 , x ) = n 0 ( x ) . (2) Here: n ( t , x ) represents the concentration of individuals of size x ≥ 0 at time t > 0 g ( x ) ≥ 0 is their growth rate B ( x ) ≥ 0 is their division rate k ( y , x ) is the quantity of individuals of size x created out of division of individuals of size y . Tomasz Dębiec GRE for measure solutions
"Main" asymptotic behaviour To state the entropy structure need existence of first eigenelements ( λ, N , ϕ ) . Eigenvalue problem and adjoint eigenvalue problem: � ∞ ∂ ∂ x ( g ( x ) N ( x )) + ( B ( x ) + λ ) N ( x ) = k ( x , y ) B ( y ) N ( y ) dy x � x − g ( x ) ∂ ∂ x ( ϕ ( x )) + ( B ( x ) + λ ) ϕ ( x ) = B ( x ) k ( y , x ) ϕ ( y ) dy 0 Existence and uniqueness: Doumic-Gabriel, 2009. Then by Generalized Relative Entropy � � n ( t , x ) e − λ t − � n ( 0 ) , ϕ � N ( x ) � � � ϕ ( x ) dx → 0 as t → ∞ R + Tomasz Dębiec GRE for measure solutions
"Main" asymptotic behaviour To state the entropy structure need existence of first eigenelements ( λ, N , ϕ ) . Eigenvalue problem and adjoint eigenvalue problem: � ∞ ∂ ∂ x ( g ( x ) N ( x )) + ( B ( x ) + λ ) N ( x ) = k ( x , y ) B ( y ) N ( y ) dy x � x − g ( x ) ∂ ∂ x ( ϕ ( x )) + ( B ( x ) + λ ) ϕ ( x ) = B ( x ) k ( y , x ) ϕ ( y ) dy 0 Existence and uniqueness: Doumic-Gabriel, 2009. Then by Generalized Relative Entropy � � n ( t , x ) e − λ t − � n ( 0 ) , ϕ � N ( x ) � � � ϕ ( x ) dx → 0 as t → ∞ R + And for measure-valued solutions? Tomasz Dębiec GRE for measure solutions
Generalized Young measures By a generalised Young measure on Ω = R + × R + we mean a parameterised family ( ν t , x , m ) where for ( t , x ) ∈ Ω , ν t , x is a family of probability measures on R and m is a nonnegative Radon measure on Ω . For f : R → R + even continuous with at most linear growth define, whenever it exists, the recession value of f as f ( s ) f ( − s ) f ∞ = lim = lim . s →∞ s s →∞ s (Alibert-Bouchitté): Let ( u n ) be a bounded sequence in L 1 loc (Ω; µ, R ) There exists a subsequence ( u n k ) , a nonnegative Radon measure m on Ω and a parametrized family of probabilities ( ν ζ ) such that for any admissible function f f ( u n k ( ζ )) µ ∗ ⇀ � ν ζ , f � µ + f ∞ m Tomasz Dębiec GRE for measure solutions
Compactness property The above fact be generalised to say that every bounded sequence of generalised Young measures possesses a weak ∗ convergent subsequence: (Kristensen, Rindler, 2012) Let ( ν n , m n ) be a sequence of generalised Young measures on Ω such that The map ζ �→ � ν n ζ , | · |� is uniformly bounded in L 1 , The sequence ( m n (¯ Ω)) is uniformly bounded. Then there is a generalised Young measure ( ν, m ) on Ω such that ( ν n , m n ) converges weakly ∗ to ( ν, m ) . Tomasz Dębiec GRE for measure solutions
Measure-valued solutions A measure-valued solution is a generalised Young measure ( ν, m ) , where the oscillation measure is a family of parameterised probabilities over the state domain R + such that equation (2) is satisfied by its barycenters � ν t , x , ξ � , i.e. the following equation ∂ t ( � ν t , x , ξ � + m ) + ∂ x ( g ( x )( � ν t , x , ξ � + m )) + B ( x )( � ν t , x , ξ � + m ) � ∞ � ∞ = k ( x , y ) B ( y ) � ν t , x , ξ � dy + k ( x , y ) B ( y ) dm ( y ) x x holds in the sense of distributions on R ∗ + × R ∗ + . Natural in reference to biology: 1 single cell − → Dirac Well-posedness for measure-valued solutions: e.g. (Cañizo, Carrillo, Cuadrado, 2013), (Carrillo, Colombo, Gwiazda, Ulikowska, 2012) Tomasz Dębiec GRE for measure solutions
Main result, asymptotic convergence Theorem Let n 0 ∈ M ( R + ) and let n solve the growth-fragmentation equation. Then under suitable assumptions on the coefficients � ∞ ϕ ( x ) d | n ( t , x ) e − λ t − m 0 N ( x ) L 1 | = 0 lim t →∞ 0 � ∞ 0 ϕ ( x ) d n 0 ( x ) and L 1 denotes the Lebesgue where m 0 := measure. T.Dębiec, M.Doumic, P.Gwiazda, E.Wiedemann, Relative entropy method for measure solutions of the growth-fragmentation equation to appear in SIAM J. Math. Anal. , 2018. Tomasz Dębiec GRE for measure solutions
Outline of proof Regularize n 0 and consider the corresponding solutions n ε . Usual generalised relative entropy for a regularizing sequence: � ∞ � n ε ( t , x ) e − λ t � H ε ( t ) := ϕ ( x ) N ( x ) H d x N ( x ) 0 For H convex; the entropy dissipation is � � ∞ � ∞ D H ε ( t ) = 0 ϕ ( x ) N ( y ) B ( y ) k ( x , y ) 0 � n ε ( t , y ) e − λ t � � n ε ( t , x ) e − λ t � − H H N ( y ) N ( x ) �� − H ′ � n ε ( t , x ) e − λ t � � n ε ( t , y ) e − λ t − n ε ( t , x ) e − λ t dx dy . N ( x ) N ( y ) N ( x ) Tomasz Dębiec GRE for measure solutions
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