Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

Yan Wang,Guanggan Chen

doi:10.3934/cpaa.2019140

Abstract

This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

Highlights

Burgers equation plays an important role in describing the interaction of dissipative and non-linear inertial terms in the motion of the turbulent fluid [1]
U(x, 0) = u0(x), where D = (−1, 1) ⊆ R1, Dc = R1\D, and W (t) is a degenerate noise specified in detail
There are few works of the Burgers equation with the fractional Laplacian operator on bounded domains driven by the degenerate noise

Summary

Introduction

Burgers equation plays an important role in describing the interaction of dissipative and non-linear inertial terms in the motion of the turbulent fluid [1]. The fractional Laplacian operator (−∆)s is defined as Stochastic Burgers equation, degenerate noise, fractional Laplacian operator, invariant measure, asymptotically strong Feller.

Results

Conclusion