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On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations
Discrete and Continuous Dynamical Systems - B, Volume: 28, Issue: 2, Start page: 1244
Swansea University Author:
Jiang-lun Wu
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PDF | Accepted Manuscript
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - B following peer review. The definitive publisher-authenticated version Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022119 is available online at: https://www.aimsciences.org/article/doi/10.3934/dcdsb.2022119
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DOI (Published version): 10.3934/dcdsb.2022119
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 H\"{o}lder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of st...
| Published in: | Discrete and Continuous Dynamical Systems - B |
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| ISSN: | 1531-3492 1553-524X |
| Published: |
American Institute of Mathematical Sciences (AIMS)
2023
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa60012 |
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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 H\"{o}lder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions $u$ belong to the space $C^{\gamma}(D_T;L^p(\Omega))$ with the optimal H\"{o}lder 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 H\"{o}lder continuity for mild solutions on bounded domains. Finally, we present a new criterion to justify H\"{o}lder 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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| Keywords: |
Nonlocal diffusion, Itô's formula, $L^\infty$ estimates, Hölder estimate. |
| College: |
Faculty of Science and Engineering |
| Funders: |
NSFC of China grants 11771123 |
| Issue: |
2 |
| Start Page: |
1244 |