Showing papers by "Xicheng Zhang published in 2012"
TL;DR: In this article, a stochastic Lagrangian particle path approach for the Navier-Stokes equations is proposed and a self-contained proof for the existence of a local unique solution for the fractal Navier Stokes equation with initial data in \(n W 1,p) is provided, and in the case of two dimensions or large viscosity, the existence is also obtained.
Abstract: In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Levy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in \({{\mathbb W}^{1,p}}\) is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Levy processes with time dependent and discontinuous drifts are proved.
36 citations
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TL;DR: In this article, the ergodicity of stochastic Burgers equations driven by cylindrical Brownian motions was shown by showing the strong Feller property and accessibility to zero.
Abstract: In this work, by showing the strong Feller property and accessibility to zero, we prove the ergodicity of stochastic Burgers equations driven by $\alpha/2$-subordinated cylindrical Brownian motions with $\alpha\in(1,2)$. To prove the strong Feller property, we truncate the nonlinearity and use the derivative formula for SDEs driven by $\alpha$-stable noises established in Zhang (arXiv:1204.2630v2).
7 citations
TL;DR: In this article, the authors proved the existence of local smooth solutions in Sobolev spaces for a class of second order quasi-linear parabolic partial differential equations (possibly degenerate) with smooth coefficients.
1 citations
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TL;DR: In this article, the authors considered the problem of estimating the gradient and the Holder continuity of an elliptic differential operator on a complete connected Riemannian manifold with two-sided Gaussian bounds.
Abstract: Let $L$ be an elliptic differential operator on a complete connected Riemannian manifold $M$ such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let $L^{(a)}$ be the $a$-stable subordination of $L$ for $a\in (1,2).$ We found some classes $\mathbb K_a^{\gg,\bb} (\bb,\gg\in [0,a))$ of time-space functions containing the Kato class, such that for any measurable $b: [0,\infty)\times M\to TM$ and $c: [0,\infty)\times M\to M$ with $|b|, c\in \mathbb K_a^{1,1},$ the operator $$L_{b,c}^{(a)}(t,x):= L^{(a)}(x)+ +c(t,x),\ \ (t,x)\in [0,\infty)\times M$$ has a unique heat kernel $p_{b,c}^{(a)}(t,x;s,y), 0\le s 1$, where $\rr$ is the Riemannian distance. The estimate of
$
abla_yp^{(a)}_{b,c}$ and the Holder continuity of $
n_x p_{b,c}^{(a)}$ are also considered. The resulting estimates of the gradient and its Holder continuity are new even in the standard case where $L=\DD$ on $\R^d$ and $b,c$ are time-independent.