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48
General relative entropy inequality: an illustration on growth models
 J. Math. Pures Appl
"... We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not concervation laws, the notion of relative entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biol ..."
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Cited by 61 (9 self)
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We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not concervation laws, the notion of relative entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biological applications where birth and death can be described by zeroth order terms. But the concept also has applications to more general growth models as the fragmentation equations. We give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions. Keywords: Relative entropy, fragmentation equations, cell division, selfsimilar solutions, periodic solutions, long time asymptotic. AMS class. No: 35B40, 35B10, 82C21, 92B05, 92D25 1 Introduction: Hyperbolic
Dust and selfsimilarity for the Smoluchowski coagulation equation
, 2004
"... We establish the wellposedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and lo ..."
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Cited by 26 (4 self)
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We establish the wellposedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and loses mass in finite time (gelation phenomena) when λ> 1. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we investigate the qualitative properties of selfsimilar solutions to the Smoluchowski’s coagulation equation when λ < 1. We prove regularity results and sharp uniform small and large size behavior for the selfsimilar profiles.
Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence
, 2006
"... Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributi ..."
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Cited by 20 (2 self)
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Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a selfsimilar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.
Longtime asymptotics for nonlinear growthfragmentation equations
 tel00844123, version 1  12 Jul 2013
"... We are interested in the longtime asymptotic behavior of growthfragmentationequations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Using the General Relative Entropy methodapplie ..."
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Cited by 9 (3 self)
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We are interested in the longtime asymptotic behavior of growthfragmentationequations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Using the General Relative Entropy methodappliedtowellchosenselfsimilarsolutions,weshowthattheequationcan“asymptotically” be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and the existence of periodic solutions is proved with the PoincaréBendixon theorem or by Hopf bifurcation.
Equilibrium solution to the inelastic Boltzmann equation driven by a particle
, 2008
"... ABSTRACT. We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a hostmedium with a fixed distribution. This is achieved by controlling the L pnorms, the moments and the regularity of the solutions for the Cauchy problem toget ..."
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Cited by 6 (6 self)
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ABSTRACT. We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a hostmedium with a fixed distribution. This is achieved by controlling the L pnorms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states. 1.
Highorder WENO scheme for Polymerizationtype equations
, 2013
"... Polymerization of proteins is a biochemical process involved in different diseases. Mathematically, it is generally modeled by aggregationfragmentationtype equations. In this paper we consider a general polymerization model and propose a highorder numerical scheme to investigate the behavior of t ..."
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Cited by 6 (3 self)
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Polymerization of proteins is a biochemical process involved in different diseases. Mathematically, it is generally modeled by aggregationfragmentationtype equations. In this paper we consider a general polymerization model and propose a highorder numerical scheme to investigate the behavior of the solution. An important property of the equation is the mass conservation. The fifthorder WENO scheme is built to preserve the total mass of proteins along time.
OPTIMAL BOUNDS FOR SELFSIMILAR SOLUTIONS TO COAGULATION EQUATIONS WITH MULTIPLICATIVE KERNEL
"... Abstract. We consider massconserving selfsimilar solutions of Smoluchowski’s coagulation equation with multiplicative kernel of homogeneity 2λ ∈ (0,1). We establish rigorously that such solutions exhibit a singular behavior of the form x −(1+2λ) as x → 0. This property had been conjectured, but on ..."
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Cited by 5 (1 self)
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Abstract. We consider massconserving selfsimilar solutions of Smoluchowski’s coagulation equation with multiplicative kernel of homogeneity 2λ ∈ (0,1). We establish rigorously that such solutions exhibit a singular behavior of the form x −(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now. 1.
SPECTRAL ANALYSIS OF SEMIGROUPS AND GROWTHFRAGMENTATION EQUATIONS
, 2013
"... Abstract. The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, so ..."
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Cited by 5 (0 self)
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Abstract. The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl’s Theorems and the KreinRutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the DysonPhillips series in the spirit of those used in [87, 82, 48, 81]. (2) On the other hand, we present the semigroup spectral analysis for three important classes of “growthfragmentation ” equations, namely the cell division equation, the selfsimilar fragmentation equation and the McKendrickVon
Asymptotic behavior of solutions to the fragmentation equation with shattering: an approach via selfsimilar Markov processes
, 2008
"... The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zeromass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the largetime behavior of soluti ..."
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Cited by 4 (1 self)
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The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zeromass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the largetime behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing selfsimilar Markov processes that reach continuously 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We notice that two parameters influence significantly these largetime behaviors: the rate of formation of “nearly1 relative masses ” (this rate is related to the behavior near 0 of the Lévy measure associated to the corresponding selfsimilar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasistationary solutions to the equation. Besides, these quasistationary solutions, or equivalently the quasistationary distributions of the selfsimilar Markov processes, are entirely described.
SELFSIMILARITY FOR BALLISTIC AGGREGATION EQUATION
, 2009
"... We consider ballistic aggregation equation for gases in which each particle is identified either by its mass and impulsion or by its sole impulsion. For the constant aggregation rate we prove existence of selfsimilar solutions as well as convergence to the selfsimilarity for generic solutions. For ..."
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Cited by 3 (2 self)
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We consider ballistic aggregation equation for gases in which each particle is identified either by its mass and impulsion or by its sole impulsion. For the constant aggregation rate we prove existence of selfsimilar solutions as well as convergence to the selfsimilarity for generic solutions. For some classes of mass and/or impulsion dependent rates we are also able to estimate the large time decay of some moments of generic solutions or to build some new classes of selfsimilar solutions. 1