Related papers: The Gevrey hypoellipticity for kinetic equations

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of…

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining…

This note is devoted to a simple method for proving hypocoercivity of the solutions of a kinetic equation involving a linear time relaxation operator, i.e. the construction of an adapted Lyapunov functional satisfying a Gronwall-type…

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary…

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying…

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a…

In this lectures given at the Morning side center of Mathematics in October 2016, we present in a very simple framework Hilbertian hypocoercive methods in the case of 1d kinetic inhomogeneous equations, and some illustrations concerning…

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing…

We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey…

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue…

The purpose of this paper is to establish the regularity the weak solutions for a nonlinear biharmonic equation.

In this paper we study the effect of randomness in kinetic equations that preserve mass. Our focus is in proving the analyticity of the solution with respect to the randomness, which naturally leads to the convergence of numerical methods.…

In this paper, we study the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann…

We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations whose coefficients lose regularity at one point.

We consider models of one-dimensional chains of non-nearest neighbor and many-body interacting particles subjected to quasi-periodic media. We extend the results in \cite{12Su&delaLlavelongrange} from analytic to Gevrey regularity…

We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to…

We obtain an asymptotic rate of decay for the radius of spatial analyticity of solutions to the nonlinear wave equation with initial data in the analytic Gevrey spaces.

This minicourse contains a description of recent results on the modelling of rarefied gases in weakly non equilibrium regimes, and the numerical methods used to approximate the resulting equations. Therefore this work focuses on BGK type…