Relative entropy method for the growth-fragmentation equation with - - PowerPoint PPT Presentation

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 ∂tU + ∂αFα = 0 (1) describing the evolution of U : Rd × R+ → Rn, where the fluxes Fα : Rn → Rn are smooth maps. The second law of thermodynamics suggests that the entropy admissibility criterion be met: ∂tη + ∂αqα ≤ 0 where the entropy η : Rn → R is a smooth convex function and (q1, . . . , qd) : Rd → Rn 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: qα(U| ¯ U) = qα(U) − qα( ¯ U) − ∇η( ¯ U) · (Fα(U) − Fα( ¯ U)) A computation leads to: ∂tη(U| ¯ U)+div q(U| ¯ U) ≤ −∇2η( ¯ U)∂α ¯ U·[Fα(U)−Fα( ¯ U)−∇Fα( ¯ 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:

• |x|<r

|U(x, t) − ¯ U(x, t)|2 dx ≤ aebt

• |x|<r+kt

|U0(x) − ¯ U0(x)|2 dx 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

• f 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

• f 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: ∂tn(t, x) + ∂x(g(x)n(t, x)) + B(x)n(t, x) =

∞

• x

k(y, x)B(y)n(t, y)dy, g(0)n(t, 0) = 0, n(0, x) = n0(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) = ∞

x

k(x, y)B(y)N(y)dy −g(x) ∂ ∂x (ϕ(x)) + (B(x) + λ)ϕ(x) = B(x) x k(y, x)ϕ(y)dy Existence and uniqueness: Doumic-Gabriel, 2009. Then by Generalized Relative Entropy

• R+
• n(t, x)e−λt − n(0), ϕN(x)
• ϕ(x)dx → 0 as t → ∞

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) = ∞

x

k(x, y)B(y)N(y)dy −g(x) ∂ ∂x (ϕ(x)) + (B(x) + λ)ϕ(x) = B(x) x k(y, x)ϕ(y)dy Existence and uniqueness: Doumic-Gabriel, 2009. Then by Generalized Relative Entropy

• R+
• n(t, x)e−λt − n(0), ϕN(x)
• ϕ(x)dx → 0 as t → ∞

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 (νt,x, m) where for (t, x) ∈ Ω, νt,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 ∞ = lim

s→∞

f (s) s = lim

s→∞

f (−s) s . (Alibert-Bouchitté): Let (un) be a bounded sequence in L1

loc(Ω; µ, R) There exists a subsequence (unk), a nonnegative

Radon measure m on Ω and a parametrized family of probabilities (νζ) such that for any admissible function f f (unk(ζ))µ ∗ ⇀ νζ, f µ + f ∞m

Tomasz Dębiec GRE for measure solutions

Compactness property

The above fact be generalised to say that every bounded sequence

• f generalised Young measures possesses a weak∗ convergent

subsequence: (Kristensen, Rindler, 2012) Let (νn, mn) be a sequence of generalised Young measures on Ω such that

The map ζ → νn

ζ, | · | is uniformly bounded in L1,

The sequence (mn(¯ Ω)) is uniformly bounded.

Then there is a generalised Young measure (ν, m) on Ω such that (νn, mn) 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 νt,x, ξ, i.e. the following equation ∂t (νt,x, ξ + m) + ∂x (g(x)(νt,x, ξ + m)) + B(x)(νt,x, ξ + m) = ∞

x

k(x, y)B(y)νt,x, ξdy + ∞

x

k(x, y)B(y)dm(y) 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 n0 ∈ M(R+) and let n solve the growth-fragmentation

• equation. Then under suitable assumptions on the coefficients

lim

t→∞

∞ ϕ(x)d|n(t, x)e−λt − m0N(x)L1| = 0 where m0 := ∞

0 ϕ(x)dn0(x) and L1 denotes the Lebesgue

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 n0 and consider the corresponding solutions nε. Usual generalised relative entropy for a regularizing sequence: Hε(t) := ∞ ϕ(x)N(x)H nε(t, x)e−λt N(x)

• dx

For H convex; the entropy dissipation is DH

ε (t) =

∞ ∞

0 ϕ(x)N(y)B(y)k(x, y)

• H
• nε(t,y)e−λt

N(y)

• − H
• nε(t,x)e−λt

N(x)

• −H′

nε(t,x)e−λt N(x) nε(t,y)e−λt N(y)

− nε(t,x)e−λt

N(x)

• dx dy.

Tomasz Dębiec GRE for measure solutions

We then have (Michel-Mischler-Perthame) d dt ∞ ϕ(x)N(x)H nε(t, x)e−λt N(x)

• dx
• = −DH

ε (t) ≤ 0

and ∞ DH

ε (t)dt ≤ Hε(0).

The sequence uε(t, x) := nε(t,x)e−λt

N(x)

is uniformly bounded in L∞(R+; L1

ϕ,loc(R+)) −

→ generates a generalised Young measure (ν, m) on R+ × R+.

Tomasz Dębiec GRE for measure solutions

Deduce that lim

ε→0

∞

0 χ(t)Hε(t)dt and lim ε→0

∞

0 DH ε (t)dt exist

for any χ ∈ Cc([0, ∞)). These define measure-valued relative entropy H(t) and entropy dissipation DH(t) for a.e. t.

H(t) =

∞

• ϕ(x)N(x)νt,x(α), H(α)dx +

∞ H∞dmt(x) DH(t) = ∞ ∞ Φ(x, y)νt,y(λ) ⊗ νt,x(α), H(λ) − H(α) − H′(α)(λ − α)dxdy + ∞ ∞ Φ(x, y)νt,x(α), H∞ − H′(α)dmt(y)dx with Φ(x, y) = ϕ(x)N(y)B(y)k(y, x)

Tomasz Dębiec GRE for measure solutions

Deduce that d

dt H(t) ≤ 0 in the sense of distributions, and

∞

0 DH(t)dt ≤ H(0).

Extract a sequence of times tn with lim

n→∞ DH(tn) = 0.

By the compactness property (νtn,x, mtn,x)

∗

⇀ ( ¯ νx, ¯ m) in the sense of measures. Show that the corresponding entropy dissipation DH

∞ is zero.

(Indeed, DH

∞ = lim n→∞ DH(tn))

Deduce that ¯ n is a Dirac measure concentrated at m0 = ∞

0 ϕ(x)dn0(x) and ¯

m = 0 Taking H = | · −m0| we get lim

n→∞

∞ ϕ(x)d|n(tn, x)e−λtn − m0N(x)L1| = 0 From monotonicity of H this convergence holds of the whole time limit

Tomasz Dębiec GRE for measure solutions

Thank you for your attention!

Tomasz Dębiec GRE for measure solutions