Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userVelocity averaging and Holder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients
107
0
0.0
(
0
)
Authors
Yuzhe Zhu
Ask ChatGPT about the research
No Arabic abstract
This article addresses the local boundedness and Holder 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 transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part $partial_t+b(v)cdot abla_x$ is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case $b(v)= v$, that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.
rate research
Read More
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form begin{eqnarray} text{div}big(A(x,u, abla u)big) = B(x,u, abla u)text{ for }xinOmega onumber end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnacks inequality as well as local Holder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.
We consider wave equations with time-independent coefficients that have $C^{1,1}$ regularity in space. We show that, for nontrivial ranges of $p$ and $s$, the standard inhomogeneous initial value problem for the wave equation is well posed in Sobolev spaces $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ over the Hardy spaces $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for Fourier integral operators introduced recently by the authors and Portal, following work of Smith. In spatial dimensions $n = 2$ and $n=3$, this includes the full range $1 < p < infty$. As a corollary, we obtain the optimal fixed-time $L^{p}$ regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.
We consider a class of nonautonomous elliptic operators ${mathscr A}$ with unbounded coefficients defined in $[0,T]timesR^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu={mathscr A}u+f$, $u(0,cdot)=g$.
This paper proves Holder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Holder continuous.
We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
