We have just preprinted together with Maria Pia Gualdani and Stephane Mischler a new paper entitled “Factorization for non-symmetric operators and exponential H-theorem”:
This is part of a long-term project (started with the paper hal-00076709) on how to obtain relaxation estimates in larger spaces than the usual “symmetrization” space for PDEs arising in statistical physics. A general abstract spectral theory is developped in order to enlarge the functional space of decay estimates for semigroups. Then applications are given for Fokker-Planck and linear(ized) Boltzmann equations (homogeneous and inhomogeneous). The main outcome of the paper is the proof of exponential convergence in the H-theorem for the full Boltzmann equation under a priori smoothness conditions. The detailed abstract can be found below.
Comments are welcome.
We present a factorization method for estimating resolvents of non-symmetric operators in Banach or Hilbert spaces in terms of estimates in another (typically smaller) “reference” space. This applies to a class of operators writing as a “regularizing” part (in a broad sense) plus a dissipative part. Then in the Hilbert case we combine this factorization approach with an abstract Plancherel identity on the resolvent into a method for enlarging the functional space of decay estimates on semigroups. In the Banach case, we prove the same result however with some loss on the norm. We then apply these functional analysis approach to several PDEs: the Fokker-Planck and kinetic Fokker-Planck equations, the linear scattering Boltzmann equation in the torus, and, most importantly the linearized Boltzmann equation in the torus (at the price of extra specific work in the latter case). In addition to the abstract method in itself, the main outcome of the paper is indeed the first proof of exponential decay towards global equilibrium (e.g. in terms of the relative entropy) for the full Boltzmann equation for hard spheres, conditionnally to some smoothness and (polynomial) moment estimates. This improves on the result in [Desvillettes-Villani, Invent. Math., 2005] where the rate was “almost exponential”, that is polynomial with exponent as high as wanted, and solves a long-standing conjecture about the rate of decay in the H-theorem for the nonlinear Boltzmann equation, see for instance [Cercignani, Arch. Mech, 1982] and [Rezakhanlou-Villani, Lecture Notes Springer, 2001].