Stochastic Burgers' equation driven by fractional Brownian motion

Abstract

In this paper, we consider the stochastic Burgers' equation driven by a genuine cylindrical fractional Brownian motion with Hurst parameter H > 1 4 . We first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic Burgers' equation. Then we obtain the local and global existence and uniqueness results for the stochastic Burgers' equation.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

“…In [15], the existence and uniqueness results are proved for the solutions of the stochastic generalized Burgers equations driven by multi-parameter fractional noises. Since many Burgers equations cannot be solved analytically, numerical methods for dealing with them are often required, see [4,7,29,33,14,20,34] and references therein. For instance, Adomian decomposition [4], homotopy analysis [29], variational iteration [14], and integral transform [20], finite difference [12,10,35], and finite element methods [28] are some of them which have been studied extensively recently.…”

“…This progress has been motivated by the fact that in many cases, fractional derivatives provide more accurate models of real world problems than integer order derivatives do. For the same reasons, several models of Fractional Burgers Equations (FBEs) have been proposed and investigated recently [15,4,7,29,33,14,20,34,30,26,21,22,6,8]. More precisely, in [15], the existence and uniqueness for the solutions to a class of stochastic generalized Burgers equations driven by multi-parameter fractional noises are proved.…”

“…An application of the timefractional Burgers equation in the framework of nonlinear wave propagation in porous media is discussed in [8]. For more details of special solutions and properties of FBEs, see [34,30,26,21] and references therein. These models are obtained by replacing integer order space/time derivatives and ordinary initial/boundary conditions with their fractional counter parts.…”

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

“…T. Wei investigated in [36] the existence and uniqueness of the solution for a fourth order stochastic heat equation driven by multi-parameter fractional noise, and studied the regularity of the solution. In a recent paper [35], the authors considered a 1-dimensional stochastic Burgers' equation driven by a genuine cylindrical fractional Brownian motion with Hurst parameter H > 1 4 . Through the study of the linear stochastic problem corresponding to the stochastic Burgers' equation, they obtained the global existence and uniqueness results.…”

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

“…So it is interesting to consider the stochastic differential equations with fBm, and the subject of stochastic calculus with respect to fBm has attracted much attentions [2,6,11,16]. In recent years, the existence and uniqueness of solutions for stochastic Burgers equations with fBm have been examined in [11,22]. In addition, Zou and Wang investigated the time-fractional stochastic Burgers equation with standard Brownian motion [27].…”