Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

By M. J. Cáceres, José A. Cañizo, St'ephane Mischler. 2010

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Abstract We study the asymptotic behavior of linear evolution equations of the type ∂ t g = D g + L g − λ g , where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator D g + L g . In the case D g = − ∂ x g , this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case D g = − ∂ x ( x g ) , it is known that λ = 1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂ t f = L f . By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L 2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.

Subfields: Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f55ec8f56b290d0e563593437ec09b8ac604f953

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Citations in the corpus

Markovian growth-fragmentation processes (2017, background): “Roughly speaking, growth-fragmentation processes can further be viewed as the stochastic counterparts of growth-fragmentation equations, which have received some attention in the recent years; see [6, 7, 8], and references therein.”
Spectral analysis of semigroups and growth-fragmentation equations (2013, methodology): “31) an entropy - dissipation of entropy method has been implemented in [18, 8] where the inequality DJ ≥ cJ is proved for j(s) = (s− 1)(2).”
Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case (2013, methodology): “Our proof follows a strategy of “enlarging the functional space of semigroup spectral gap” initiated in [32] for studying long time convergence to the equilibrium for the homogeneous Boltzmann equation, and then developed in [30, 25, 12, 11, 29] (see also [31]) in the framework of kinetic equations ”
A non‐conservative Harris ergodic theorem (2019, background): “The question of obtaining an exponential rate of convergence once the eigentriplet is known has been treated in several works after [64], by means of functional inequalities [2, 36], semigroup methods [11], or a combination of both [18, 17].”
Factorization for non-symmetric operators and exponential H-theorem (2010, background): “We also refer to [25, 24] where polynomial weights are considered for a fragmentation equation.”