Related papers: Regularity theory for the spatially homogeneous Bo…

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global…

We consider a simplified Boltzmann equation: spatially homogeneous, two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized around its initial condition. We prove that for a sufficiently singular velocity cross…

This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our…

We study steady solutions to the relativistic Boltzmann equation with hard-sphere interactions in a slab geometry. Under a spatial symmetry assumption in the transverse variables $x_2$ and $x_3$, the problem reduces to a one-dimensional…

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To…

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L^\infty$ in the case of hard potentials. As a consequence,…

We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary…

For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The…

We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some…

Regularity theorems \`a la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general…

We prove a local-in-time existence and uniqueness theorem for a smooth classical solution to the spatially homogeneous Boltzmann equation with cutoff soft potentials. Our proof is based on a series of bilinear estimates for the…

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth…

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence…

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The…

We prove refined space-time regularity for the classical stochastic Allen-Cahn equation with logarithmic potential. This allows to establish a random separation property, i.e. that the trajectories of the solution are strictly separated…

Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory (\cite{Kim11,GKTT1}). In this paper, we prove an H\"older regularity in $C^{0,\frac{1}{2}-}_{x,v}$ for the…

In this paper, we consider a class of spatially homogeneous Boltzmann equation without angular cutoff. We prove that any radial symmetric weak solution of the Cauchy problem become analytic for positive time.

We introduce a practical criterion that justifies the propagation and appearance of $L^{p}$-norms for the solutions to the spatially homogeneous Boltzmann equation with very soft potentials without cutoff. Such criterion also provides a new…

We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator including hard sphere, hard potential and Maxwell molecule models. Our new estimates characterize both regularization and convolution properties…

The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an…