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Yingchao Xie

Publications and source records attributed to Yingchao Xie.

29 records · Page 2Linked to original sources
arXiv2019

Strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients

This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to SPDEs.

math.PR↗
arXiv2018

Averaging principle for two dimensional stochastic Navier-Stokes equations

The averaging principle is established for the slow component and the fast component being two dimensional stochastic Navier-Stokes equations and stochastic reaction-diffusion equations, respectively. The classical Khasminskii approach based on time discretization is used for the proof of the slow component strong convergence to the solution of the corresponding averaged equation under some suitable conditions. Meanwhile, some powerful techniques are used to overcome the difficulties caused by the nonlinear term and to release the regularity of the initial value.

math.PR↗
arXiv2018

Stochastic 3D Leray-$α$ Model with Fractional Dissipation

In this paper, we establish the global well-posedness of stochastic 3D Leray-$α$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $θ_1$ in the nonlinear term and a $θ_2$-fractional Laplacian. In the case of $θ_1 \ge 0, θ_2 > 0$ and $θ_1+θ_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special cases such as stochastic hyperviscous Navier-Stokes equation and classical stochastic 3D Leray-$α$ model.

math.AP↗
arXiv2017

Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation

We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,θ}$ for some $θ\in (α/2, 1]$.

math.PR↗
arXiv2017

Well-posedness for SDEs driven by different type of noises

We show the existence and uniqueness of strong solutions for stochastic differential equation driven by partial $α$-stable noise and partial Brownian noise with singular coefficients. The proof is based on the regularity of degenerate mixed type Kolmogorov equation.

math.PR↗
arXiv2017

Pathwise uniqueness for a class of SPDEs driven by cylindrical $α$-stable processes

We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.

math.PR↗
arXiv2016

Stochastic integrals and BDG's inequalities in Orlicz-type spaces

In this paper we extend an inequality of Lenglart, Lépingle and Pratelli \cite[Lemma 1.1]{LLP} to general continuous adapted stochastic processes with values in topology spaces. By this inequality we show Burkholder-Davies-Gundy's inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions.

math.PR↗