Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Wensheng Wang

doi:10.3390/sym13071251

Abstract

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

Highlights

The fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs are used to the model of pattern formation phenomena accompanying the appearance of turbulence
X ∈ Rd, where ∆ is the d-dimensional Laplacian operator, R+ = (0, ∞), (ε, β) ∈ R+ × R is a pair of parameters, and the noise term ∂d+1 W/∂t∂x is the space-time white noise corresponding to the real-valued Brownian sheet W on R+ × Rd, d = 1, 2, 3
Our spatial results are crucially dependent on the following spatial spectral density for L-KS SPDEs and their gradient

Summary

Introduction

The fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs are used to the model of pattern formation phenomena accompanying the appearance of turbulence (see [1,2,3,4,5,6]). In a series of articles [1,5,6,7], Allouba investigated the existence/uniqueness, sharp dimension-dependent L p , and Hölder regularity of the linear and nonlinear noise version of Equation (1) These results naturally lead to the following list of motivating questions:. Allouba and Xiao [4] investigated the exact, spatio-temporal, dimension-dependent, uniform, and local moduli of continuity for the fourth order L-KS SPDEs and their gradient. These results gave the answers to spatial continuity and exact moduli of continuity of the solutions to Equation (1), and gave partial answers to above questions.

Rigorous Kernel Stochastic Integral Equations Formulations

Spatial Spectral Density for L-KS SPDEs and Their Gradient

Extremes for L-KS SPDEs and Their Gradient

Spatial Zero-One Laws for L-KS SPDEs and Their Gradient

Spatial Moduli of Non-Differentiability for L-KS SPDEs and Their Gradient

Conclusions