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
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.
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