Related papers: Averaging lemmas and hypoellipticity
We introduce a commutator method with multipliers to prove averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. This method requires only elementary techniques in Fourier analysis and shows…
We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results, which display, in some cases, a gain of one full derivative.…
This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach -- based upon the classical energy method -- provides a powerful duality principle in kinetic transport equations which allows for a natural…
This article addresses the local boundedness and H\"older continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and…
We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family…
We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable…
This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport…
We prove new velocity averaging lemmas for multi-dimensional hyperbolic-parabolic partial differential equations. These theorems may be applied to establish several compactness results for both deterministic and stochastic…
This study investigates the regularity of kinetic equations with spatial heterogeneity. Recent progress has shown that velocity averages of weak solutions $h$ in $L^p$ ($p>1$) are strongly $L^1_{\text{loc}}$ compact under the natural…
In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential…
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to…
This paper examines an averaging technique applied to the transport equations as an alternative to vanishing viscosity. Such techniques have been shown to be valid shock-regularizations of the Burgers equation and the Euler equations, but…
In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the…
We introduce a new method which resolves the problem of regularity and compactness of entropy solutions for nonlinear degenerate parabolic equations under non-degeneracy conditions on the sphere. In particular, we address a problem of…
We aim at understanding how the non-commutation phenomena between a linear transport operator and a fractional diffusion allow the transport operator to satisfy hypoelliptic estimates on the whole space. Such hypoelliptic estimates are…
We analyze closed one-dimensional chains of weakly coupled many level systems, by means of the so-called Hilbert space average method (HAM). Subject to some concrete conditions on the Hamiltonian of the system, our theory predicts energy…
The generalized transport equations for a consistent description of kinetic and hydrodynamic processes in dense gases and liquids are considered. The inner structure of the generalized transport kernels for these equations is established.…
We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does…
The paper addresses linear hyperbolic systems in one space dimension with random field coefficients. In many applications, a low degree of regularity of the paths of the coefficients is required, which is not covered by classical stochastic…
We derive two different generalized heat-transport equations: The most general one, of the first order in time and second order in space, encompasses some well known heat equations and describes the hyperbolic regime in the absence of…