由乘法时空白噪声驱动的随机时分数布尔格斯方程的$L_p$$$可解性和霍尔德正则性

IF 1.4 3区 数学 Q2 MATHEMATICS, APPLIED
Beom-Seok Han
{"title":"由乘法时空白噪声驱动的随机时分数布尔格斯方程的$L_p$$$可解性和霍尔德正则性","authors":"Beom-Seok Han","doi":"10.1007/s40072-024-00329-w","DOIUrl":null,"url":null,"abstract":"<p>This paper establishes <span>\\(L_p\\)</span>-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise: </p><span>$$\\begin{aligned} \\partial _t^\\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\\bar{b}}^i u u_{x^i} + \\partial _t^\\beta \\int _0^t \\sigma (u)dW_t,\\quad t&gt;0;\\quad u(0,\\cdot ) = u_0, \\end{aligned}$$</span><p>where <span>\\(\\alpha \\in (0,1)\\)</span>, <span>\\(\\beta &lt; 3\\alpha /4+1/2\\)</span>, and <span>\\(d&lt; 4--2(2\\beta -1)_+/\\alpha \\)</span>. The operators <span>\\(\\partial _t^\\alpha \\)</span> and <span>\\(\\partial _t^\\beta \\)</span> are the Caputo fractional derivatives of order <span>\\(\\alpha \\)</span> and <span>\\(\\beta \\)</span>, respectively. The process <span>\\(W_t\\)</span> is an <span>\\(L_2(\\mathbb {R}^d)\\)</span>-valued cylindrical Wiener process, and the coefficients <span>\\(a^{ij}, b^i, c, {\\bar{b}}^{i}\\)</span> and <span>\\(\\sigma (u)\\)</span> are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant <span>\\(T&lt;\\infty \\)</span>, small <span>\\(\\varepsilon &gt;0\\)</span>, and almost sure <span>\\(\\omega \\in \\varOmega \\)</span>, </p><span>$$\\begin{aligned} \\sup _{x\\in \\mathbb {R}^d}|u(\\omega ,\\cdot ,x)|_{C^{\\left[ \\frac{\\alpha }{2}\\left( \\left( 2-(2\\beta -1)_+/\\alpha -d/2 \\right) \\wedge 1 \\right) +\\frac{(2\\beta -1)_{-}}{2} \\right] \\wedge 1-\\varepsilon }([0,T])}&lt;\\infty \\end{aligned}$$</span><p>and </p><span>$$\\begin{aligned} \\sup _{t\\le T}|u(\\omega ,t,\\cdot )|_{C^{\\left( 2-(2\\beta -1)_+/\\alpha -d/2 \\right) \\wedge 1 - \\varepsilon }(\\mathbb {R}^d)} &lt; \\infty . \\end{aligned}$$</span><p>The Hölder regularity of the solution in time changes behavior at <span>\\(\\beta = 1/2\\)</span>. Furthermore, if <span>\\(\\beta \\ge 1/2\\)</span>, then the Hölder regularity of the solution in time is <span>\\(\\alpha /2\\)</span> times that in space.</p>","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":"57 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$$L_p$$ -solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise\",\"authors\":\"Beom-Seok Han\",\"doi\":\"10.1007/s40072-024-00329-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper establishes <span>\\\\(L_p\\\\)</span>-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise: </p><span>$$\\\\begin{aligned} \\\\partial _t^\\\\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\\\\bar{b}}^i u u_{x^i} + \\\\partial _t^\\\\beta \\\\int _0^t \\\\sigma (u)dW_t,\\\\quad t&gt;0;\\\\quad u(0,\\\\cdot ) = u_0, \\\\end{aligned}$$</span><p>where <span>\\\\(\\\\alpha \\\\in (0,1)\\\\)</span>, <span>\\\\(\\\\beta &lt; 3\\\\alpha /4+1/2\\\\)</span>, and <span>\\\\(d&lt; 4--2(2\\\\beta -1)_+/\\\\alpha \\\\)</span>. The operators <span>\\\\(\\\\partial _t^\\\\alpha \\\\)</span> and <span>\\\\(\\\\partial _t^\\\\beta \\\\)</span> are the Caputo fractional derivatives of order <span>\\\\(\\\\alpha \\\\)</span> and <span>\\\\(\\\\beta \\\\)</span>, respectively. The process <span>\\\\(W_t\\\\)</span> is an <span>\\\\(L_2(\\\\mathbb {R}^d)\\\\)</span>-valued cylindrical Wiener process, and the coefficients <span>\\\\(a^{ij}, b^i, c, {\\\\bar{b}}^{i}\\\\)</span> and <span>\\\\(\\\\sigma (u)\\\\)</span> are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant <span>\\\\(T&lt;\\\\infty \\\\)</span>, small <span>\\\\(\\\\varepsilon &gt;0\\\\)</span>, and almost sure <span>\\\\(\\\\omega \\\\in \\\\varOmega \\\\)</span>, </p><span>$$\\\\begin{aligned} \\\\sup _{x\\\\in \\\\mathbb {R}^d}|u(\\\\omega ,\\\\cdot ,x)|_{C^{\\\\left[ \\\\frac{\\\\alpha }{2}\\\\left( \\\\left( 2-(2\\\\beta -1)_+/\\\\alpha -d/2 \\\\right) \\\\wedge 1 \\\\right) +\\\\frac{(2\\\\beta -1)_{-}}{2} \\\\right] \\\\wedge 1-\\\\varepsilon }([0,T])}&lt;\\\\infty \\\\end{aligned}$$</span><p>and </p><span>$$\\\\begin{aligned} \\\\sup _{t\\\\le T}|u(\\\\omega ,t,\\\\cdot )|_{C^{\\\\left( 2-(2\\\\beta -1)_+/\\\\alpha -d/2 \\\\right) \\\\wedge 1 - \\\\varepsilon }(\\\\mathbb {R}^d)} &lt; \\\\infty . \\\\end{aligned}$$</span><p>The Hölder regularity of the solution in time changes behavior at <span>\\\\(\\\\beta = 1/2\\\\)</span>. Furthermore, if <span>\\\\(\\\\beta \\\\ge 1/2\\\\)</span>, then the Hölder regularity of the solution in time is <span>\\\\(\\\\alpha /2\\\\)</span> times that in space.</p>\",\"PeriodicalId\":48569,\"journal\":{\"name\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"volume\":\"57 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-04-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s40072-024-00329-w\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics and Partial Differential Equations-Analysis and Computations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40072-024-00329-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

摘要

本文建立了由乘法时空白噪声驱动的随机时分数布尔格斯方程的可解性: $$\begin{aligned}\u = a^{ij}u_{x^ix^j}+ b^{i}u_{x^i}+ cu + {\bar{b}}^i u u_{x^i}+ partial _t^\beta int _0^t \sigma (u)dW_t,\quad t>0;\quad u(0,\cdot ) = u_0, \end{aligned}$其中(\alpha \in (0,1)\),\(\beta < 3\alpha /4+1/2\), and\(d< 4--2(2\beta -1)_+/\alpha \)。算子\(\partial _t^\alpha\) 和\(\partial _t^\beta\) 分别是阶\(\alpha \)和\(\beta \)的卡普托分数导数。过程 \(W_t\) 是一个 \(L_2(\mathbb {R}^d)\) 值圆柱维纳过程,系数 \(a^{ij}, b^i, c, {\bar{b}}^{i}\) 和 \(\sigma (u)\) 是随机的。除了解的唯一性和存在性之外,解的赫尔德正则性也被建立起来。例如,对于任意常数(T<\infty \)、小(varepsilon >0\)和几乎确定的(\omega \in \varOmega \),$$\begin{aligned}。|u(\omega ,\cdot ,x)|_{C^{left[ \frac\{alpha }{2}\left( \left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 \right) +\frac{(2\beta -1)_{-}}{2}\(右)\1-\varepsilon }([0,T])}<(infty) (end{aligned}}$$和 $$(begin{aligned})$$。\end{aligned}$$The Hölder regularity of the solution in time change behavior at \(\beta = 1/2\).此外,如果 \(\beta \ge 1/2\), 那么解在时间上的霍尔德正则性是空间上的\(\alpha /2\)倍。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
$$L_p$$ -solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

This paper establishes \(L_p\)-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise:

$$\begin{aligned} \partial _t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\bar{b}}^i u u_{x^i} + \partial _t^\beta \int _0^t \sigma (u)dW_t,\quad t>0;\quad u(0,\cdot ) = u_0, \end{aligned}$$

where \(\alpha \in (0,1)\), \(\beta < 3\alpha /4+1/2\), and \(d< 4--2(2\beta -1)_+/\alpha \). The operators \(\partial _t^\alpha \) and \(\partial _t^\beta \) are the Caputo fractional derivatives of order \(\alpha \) and \(\beta \), respectively. The process \(W_t\) is an \(L_2(\mathbb {R}^d)\)-valued cylindrical Wiener process, and the coefficients \(a^{ij}, b^i, c, {\bar{b}}^{i}\) and \(\sigma (u)\) are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant \(T<\infty \), small \(\varepsilon >0\), and almost sure \(\omega \in \varOmega \),

$$\begin{aligned} \sup _{x\in \mathbb {R}^d}|u(\omega ,\cdot ,x)|_{C^{\left[ \frac{\alpha }{2}\left( \left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 \right) +\frac{(2\beta -1)_{-}}{2} \right] \wedge 1-\varepsilon }([0,T])}<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{t\le T}|u(\omega ,t,\cdot )|_{C^{\left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 - \varepsilon }(\mathbb {R}^d)} < \infty . \end{aligned}$$

The Hölder regularity of the solution in time changes behavior at \(\beta = 1/2\). Furthermore, if \(\beta \ge 1/2\), then the Hölder regularity of the solution in time is \(\alpha /2\) times that in space.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.70
自引率
13.30%
发文量
54
期刊介绍: Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信