On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

Guangying Lv; Hongjun Gao; Jinlong Wei; Jiang-Lun Wu

doi:10.3934/dcdsb.2022119

Article Contents

2023, Volume 28, Issue 2: 1244-1266. Doi: 10.3934/dcdsb.2022119

This issue Previous Article Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion Next Article Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation

On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

1.
College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2.
School of Mathematics and Computational Sciences, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China
3.
School of Mathematics, School of Mathematical Science, Southeast University, Nanjing 211189, China
4.
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
5.
Department of Mathematics, Computational Foundry, Swansea University, Swansea SA1 8EN, UK

^* Corresponding author: Guangying Lv
^* Corresponding author: Guangying Lv

Received: January 2022

Revised: April 2022

Early access: June 2022

Published: February 2023

This work is partially supported by China NSF Grant Nos. 12171247, 12171084, 11771123, 11501577 and the Startup Foundation for Introducing Talent of NUIST.

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Abstract

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Campanato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space $ \mathbb{R}^d $) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions belong to the space $ C^{\gamma}(D_T;L^p(\Omega)) $ with the optimal Hölder continuity index $ \gamma $ (which is given explicitly), where $ D_T: = [0, T]\times D $ for $ T>0 $, and $ D\subset\mathbb{R}^d $ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $ L^p(\Omega;C^{\gamma^*}(D_T)) $. What's more, we give an explicit formula between the two indexes $ \gamma $ and $ \gamma^* $. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criterion to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method is suitable to the case of space-time white noise.

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Mathematics Subject Classification: Primary: 60H15, 60H40; Secondary: 35K20.

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