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On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations
Guangying Lv,
Hongjun Gao,
Jinlong Wei,
Jiang-lun Wu 
Discrete and Continuous Dynamical Systems - B, Volume: 28, Issue: 2, Start page: 1244
Swansea University Author:
Jiang-lun Wu
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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 |
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American Institute of Mathematical Sciences (AIMS)
2023
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa60012 |
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2022-10-18T15:09:24.7325045 v2 60012 2022-05-12 On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations dbd67e30d59b0f32592b15b5705af885 0000-0003-4568-7013 Jiang-lun Wu Jiang-lun Wu true false 2022-05-12 SMA 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. Journal Article Discrete and Continuous Dynamical Systems - B 28 2 1244 American Institute of Mathematical Sciences (AIMS) 1531-3492 1553-524X Nonlocal diffusion, Itô's formula, $L^\infty$ estimates, Hölder estimate. 1 1 2023 2023-01-01 10.3934/dcdsb.2022119 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University Other NSFC of China grants 11771123 2022-10-18T15:09:24.7325045 2022-05-12T13:47:02.7419706 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Guangying Lv 1 Hongjun Gao 2 Jinlong Wei 3 Jiang-lun Wu 0000-0003-4568-7013 4 60012__24120__cb8b313a3701416d8fe15dfbcd904e86.pdf LGWW-DCDS2022.pdf 2022-05-18T10:36:38.4986872 Output 456490 application/pdf Accepted Manuscript true 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 true eng |
| title |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
| spellingShingle |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations Jiang-lun Wu |
| title_short |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
| title_full |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
| title_fullStr |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
| title_full_unstemmed |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
| title_sort |
On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations |
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dbd67e30d59b0f32592b15b5705af885 |
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dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu |
| author |
Jiang-lun Wu |
| author2 |
Guangying Lv Hongjun Gao Jinlong Wei Jiang-lun Wu |
| format |
Journal article |
| container_title |
Discrete and Continuous Dynamical Systems - B |
| container_volume |
28 |
| container_issue |
2 |
| container_start_page |
1244 |
| publishDate |
2023 |
| institution |
Swansea University |
| issn |
1531-3492 1553-524X |
| doi_str_mv |
10.3934/dcdsb.2022119 |
| publisher |
American Institute of Mathematical Sciences (AIMS) |
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Faculty of Science and Engineering |
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Faculty of Science and Engineering |
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School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics |
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| description |
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. |
| published_date |
2023-01-01T04:17:45Z |
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1763754184092418048 |
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11.013148 |