Factorisation for non-symmetric operators and exponential H-theorems
Seminar Room 1, Newton Institute
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].