Searcharxiv⌕ Search

arXiv subjects

Guangying Lv

Publications and source records attributed to Guangying Lv.

27 records · Page 2Linked to original sources
arXiv2017

Strong solution for stochastic transport equations with irregular drift: existence and non-existence

We prove some existence, uniqueness and non-existence results of stochastic strong solutions for a class of stochastic transport equations with a $q$-integrable (in time), bounded and $α$-Hölder continuous (in space) drift coefficient. More precisely, we show that for a Sobolev differentiable initial condition, there exists a unique stochastic strong solution when $α>2/q$, while for $α+1<2/q$ with spatial dimension higher than one, we can choose proper initial data and drift coefficients so that there is no stochastic strong solutions.

math.AP↗
arXiv2017

Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises

In this paper, we consider the extended stochastic Navier-Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity of the generalized Ornstein-Uhlenbeck process. Then we discuss the existence, uniqueness, and Hölder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order $α$ and Hurst parameter $H$. The results obtained in this study improve some results in existing literature.

math.NA↗
arXiv2017

Kinetic solutions for nonlocal scalar conservation laws

This work is devoted to examine the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator. Our proof for uniqueness is based upon the analysis on a microscopic contraction functional and the existence is enabled by a parabolic approximation. As an illustration, we obtain the existence and uniqueness of kinetic solutions for the generalized fractional Burgers-Fisher type equations. Moreover, we demonstrate the kinetic solutions' Lipschitz continuity in time, and continuous dependence on nonlinearities and Lévy measures.

math.AP↗
arXiv2017

BMO estimates for stochastic singular integral operators and its application to PDEs with Lévy noise

In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=Δ^{\fracα{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $Δ^{\fracα{2}}=-(-Δ)^{\fracα{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump Lévy processes with Lévy measure of $ν^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\fracα{q_0}$-order derivative of $u$ is controlled by the norm of $g$.

math.PR↗
arXiv2017

Morrey-Campanato estimates for the moments of stochastic integral operators and its application to SPDEs

In this paper, we are concerned with the estimates for the moments of stochastic convolution integrals. We first deal with the stochastic singular integral operators and we aim to derive the Morrey-Campanato estimates for the $p$-moments (for $p\ge1$). Then, by utilising the embedding theory between the Campanato space and Hölder space, we establish the norm of $C^{θ,θ/2}(\bar D)$, where $θ\ge0, \bar D=\bar G\times[0,T]$ for arbitrarily fixed $T\in(0,\infty)$ and $G\subset\mathbb{R}^d$. As an application, we consider the following stochastic (fractional) heat equations with additive noises \bess du_t(x)=Δ^αu_t(x)dt+g(t,x)dη_t,\ \ \ u_0=0,\ 0\leq t\leq T, x\in G, \eess where $Δ^α=-(-Δ)^α$ with $0<α\leq1$ (the fractional Laplacian), $g:[0,T]\times G\timesΩ\to\mathbb{R}$ is a joint measurable coefficient, and $η_t, t\in[0,T]$, is either the Brownian motion or a Lévy process on a given filtered probability space $(Ω,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. The Schauder estimate for the $p$-moments of the solution of the above equation is obtained. The novelty of the present paper is that we obtain the Schauder estimate for parabolic stochastic partial differential equations with Lévy noise.

math.PR↗
arXiv2015

A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift

Despite there are numerous theoretical studies of stochastic differential equations with a symmetric $α$-stable Lévy noise, very few regularity results exist in the case of $0<α\leq1$. In this paper, we study the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that there exists a unique solution, which is $C^\infty$ in space for $t>0$ when $α\in (0, 2]$.

math.DS↗
arXiv2014

Martingale and Weak Solutions for a Stochastic Nonlocal Burgers Equation on Bounded Intervals

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on bounded intervals. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

math.PR↗
arXiv2014

Impacts of Noise on a Class of Partial Differential Equations

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic parabolic equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen-Cahn equation does not have finite time singularities and the unique solution exists globally.

math.AP↗