Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials

By Clément Mouhot. 2006

154 citations recorded by the source corpus.

For the spatially homogeneous Boltzmann equation with hard potentials and Grad's cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gap-like estimates valid near the equilibrium, and then connecting the latter to the quantitative nonlinear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger than the usual linearization setting.

Subfields: Applied Mathematics, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/a43282db7479de5aed8394ca3bf1e68bc563eddb

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In Applied Mathematics

In Ocean Engineering

Citations in the corpus

Spectral analysis of semigroups and growth-fragmentation equations (2013, methodology): “The first author also would like to thank C. Mouhot for the many discussions of the spectral analysis of semigroups issue.”
Factorization for non-symmetric operators and exponential H-theorem (2010, methodology): “We shall extend the estimate of [22] to the full inhomogeneous case in the torus.”
Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations (2010, background): “9 with some recent result obtained in [5] (see also [11] for the first results in that direction) we may enlarge the space in which the spectral gap holds and prove part 2 of Theorem 1.”
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 ”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, methodology): “It continues the program of research [19, 10] where quantitative methods for enlarging the functional space of spectral gap estimates were developed with application to kinetic equations; specifically in [10] spectral gap estimates were obtained in Lebesgue spaces for Boltzmann and Fokker-Planck equ”
Kac’s program in kinetic theory (2011, background): “Let us start from an auxiliary result from [60].”