Stochastic homogenization of nonlinear evolution equations with space-time nonlocality

2023/10/29 by Junlong Chen, Yanbin Tang, Chen, Junlong +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Fractional Differential Equations Solutions

paper · pdf · doi:10.48550/arxiv.2310.19146

openalex publication_date 2023/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel J(x,t;y,s) is re-scaled in a suitable way and the oscillation coefficient ν(x,t;y,s) possesses periodic and stationary structure, we show that the solutions u^ε(x,t) to the perturbed equations converge to u₀(x,t), the solution of corresponding local nonlinear parabolic equation as scale parameter ε→ 0⁺. Then for the nonlocal linear index p=2 we give the convergence rate such that ||u^ε -u₀||__{L²(ℝ^d×(0,T))}≤ Cε. Furthermore, we obtain that the normalized difference (1)/(ε)[u^ε(x,t)-u₀(x,t)]-χ((x)/(ε), \fractε²) ∇_xu₀(x,t) converges to a solution of an SPDE with additive noise and constant coefficients. Finally, we give some numerical formats for solving non-local space-time homogenization.

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