Clément Mouhot

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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(More)
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(More)
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give more general estimates on the tail and the regularity of generic solutions. In particular we(More)
For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated(More)
We consider a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing (in the framework of so-called constant normal restitution coefficients α ∈ [0, 1] for the inelas-ticity). In the physical regime of a small inelasticity (that is α ∈ [α * , 1) for some constructive α * ∈ [0, 1)) we prove uniqueness(More)
We prove an inequality on the Kantorovich-Rubinstein distance – which can be seen as a particular case of a Wasserstein metric– between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the(More)
We prove the appearance of an explicit lower bound on the solution to the full Boltzmann equation in the torus for a broad family of collision kernels including in particular long-range interaction models, under the assumption of some uniform bounds on some hydrodynamic quantities. This lower bound is independent of time and space. When the collision kernel(More)
We prove a fractional version of Poincaré inequalities in the context of R n endowed with a fairly general measure. Namely we prove a control of an L 2 norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that(More)
We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semi-groups of positive, bounded operators (P t) t≥0 thereon with lim t→0+ P t f (x) = f (x) are in fact strongly continuous. This result applies to prove optimal rates of convergence of splitting schemes for stochastic(More)