Related papers: Strichartz estimates for the kinetic transport equ…

We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for…

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…

We generalize known results on transport equations associated to a Lipschitz field $\mathbf{F}$ on some subspace of $\mathbb{R}^N$ endowed with some general space measure $\mu$. We provide a new definition of both the transport operator and…

We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required…

We give new computable necessary conditions for a class of optimal transportation problems to have smooth solutions.

In the previous studies, from the Fokker-Planck equation the general spatial transport equation, which contains an infinite number of spatial derivative terms $T_n=\kappa_{nz}\partial^n{F}/ \partial{z^n}$ with $n=1, 2, 3, \cdots$, was…

An efficient new method is presented to calculate the quantum transports using periodic boundary conditions. This new method is based on a method we developed previously, but with an essential change in solving the Schrodinger's equation.…

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic…

A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.

We develop quantum electrodynamics into a kinetic-theory-like evolution equation for electrons, positrons and photons. To keep the "collision rules" simple, we make use of longitudinal and temporal photons in addition to the usual…

Interband effects such as coherence/tunneling have recently been shown to give an important contribution to the charge and heat transport properties under certain conditions. These can be captured by adding corrective terms to the…

We prove global, scale invariant Strichartz estimates for the linear magnetic Schr\"odinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global…

We prove quantitative regularity estimates for the solutions to non-linear continuity equations and their discretized numerical approximations on Cartesian grids when advected by a rough force field. This allow us to recover the known…

A spatially-periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearised limit of the macroscopic conservation equations within…

Strichartz estimates are derived from $\ell^2$-decoupling for phase functions satisfying a curvature condition. Bilinear refinements without loss in the high frequency are discussed. Estimates are established from uniform curvature…

For stochastic reaction-diffusion equations with L\'evy noises and non-Lipschitz reaction terms, we prove that $W_1H$ transportation cost inequalities hold for their invariant probability measures and for their process-level laws on the…

Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.

We prove global Strichartz estimates without loss for the wave equation outside two strictly convex obstacles, following the roadmap introduced in [Lafontaine, 2017] for the Schr\"odinger equation. Moreover, we show a first step toward the…

We study stochastic differential equations(SDEs) with a small perturbation parameter. Under the dissipative condition on the drift coefficient and the local Lipschitz condition on the drift and diffusion coefficients we prove the existence…

A derivation is given of the Boltzmann-Vlasov equation beginning from multiple scattering considerations. The motivation for the discussion, which is purely pedagogical in nature, is the current interest in understanding the origins of…