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Regularity of stochastic nonlocal diffusion equations

Guangying Lv, Hongjun Gao , Jinlong Wei , Jiang-lun Wu Orcid Logo

arXiv

Swansea University Author: Jiang-lun Wu Orcid Logo

Abstract

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"older continuity (locally in the whole state space $R^d$) for mild solutions of stochastic no...

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Published in: arXiv
Published: 2018
Online Access: https://arxiv.org/abs/1801.04531
URI: https://cronfa.swan.ac.uk/Record/cronfa39311
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first_indexed 2018-04-04T19:33:14Z
last_indexed 2020-07-03T18:58:40Z
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spelling 2020-07-03T18:30:06.5555114 v2 39311 2018-04-04 Regularity of stochastic nonlocal diffusion equations dbd67e30d59b0f32592b15b5705af885 0000-0003-4568-7013 Jiang-lun Wu Jiang-lun Wu true false 2018-04-04 SMA In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"older continuity (locally in the whole state space $R^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions u belong to the space $C_γ(DT;L^p(Ω))$ with the optimal Ho ̈lder continuity index $γ$ (which is given explicitly), where $D_T := [0,T] × D for T > 0$, and $D ⊂ R^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(Ω; C_{γ^∗} (D_T ))$. What’s more, we give an explicit formula between the two index $γ$ and $γ^∗$. Moreover, we prove H ̈older continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify H\"older continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise. Journal Article arXiv 12 2 2018 2018-02-12 https://arxiv.org/abs/1801.04531 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-07-03T18:30:06.5555114 2018-04-04T15:55:36.0204349 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 39311__17637__a9fa824b3a414c2cb3ce3f3af9e2809d.pdf 39311.pdf 2020-07-03T18:27:22.7436252 Output 242256 application/pdf Author's Original true true eng
title Regularity of stochastic nonlocal diffusion equations
spellingShingle Regularity of stochastic nonlocal diffusion equations
Jiang-lun Wu
title_short Regularity of stochastic nonlocal diffusion equations
title_full Regularity of stochastic nonlocal diffusion equations
title_fullStr Regularity of stochastic nonlocal diffusion equations
title_full_unstemmed Regularity of stochastic nonlocal diffusion equations
title_sort Regularity of stochastic nonlocal diffusion equations
author_id_str_mv dbd67e30d59b0f32592b15b5705af885
author_id_fullname_str_mv dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu
author Jiang-lun Wu
author2 Guangying Lv
Hongjun Gao
Jinlong Wei
Jiang-lun Wu
format Journal article
container_title arXiv
publishDate 2018
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url https://arxiv.org/abs/1801.04531
document_store_str 1
active_str 0
description In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"older continuity (locally in the whole state space $R^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions u belong to the space $C_γ(DT;L^p(Ω))$ with the optimal Ho ̈lder continuity index $γ$ (which is given explicitly), where $D_T := [0,T] × D for T > 0$, and $D ⊂ R^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(Ω; C_{γ^∗} (D_T ))$. What’s more, we give an explicit formula between the two index $γ$ and $γ^∗$. Moreover, we prove H ̈older continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify H\"older continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise.
published_date 2018-02-12T03:49:54Z
_version_ 1763752431832793088
score 10.9979105

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