# Factorization for non-symmetric operators and exponential H-theorem

* Corresponding author
Abstract : We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

Cited literature [100 references]

https://hal.archives-ouvertes.fr/hal-00495786
Contributor : Clément Mouhot <>
Submitted on : Tuesday, November 19, 2013 - 7:36:29 PM
Last modification on : Thursday, April 15, 2021 - 6:04:01 PM
Long-term archiving on: : Thursday, February 20, 2014 - 9:46:36 AM

### Files

SpectreFPBoltz48.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00495786, version 3
• ARXIV : 1006.5523

### Citation

Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot. Factorization for non-symmetric operators and exponential H-theorem. 2013. ⟨hal-00495786v3⟩

Record views