Weak solution for Stochastic Degasperis-Procesi Equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909304714428416 |
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| author | Chemetov, Nikolai V. Cipriano, Fernanda |
| author_facet | Chemetov, Nikolai V. Cipriano, Fernanda |
| contents | This article is concerned with the existence of solution to the stochastic Degasperis-Procesi equation on $\mathbb{R}$ with an infinite dimensional multiplicative noise and integrable initial data.
Writing the equation as a system composed of a stochastic nonlinear conservation law and an elliptic equation, we are able to develop a method based on the conjugation of kinetic theory with stochastic compactness arguments. More precisely, we first apply the stochastic Jakubowski-Skorokhod representation theorem to show the existence of a weak kinetic martingale solution.
Next, we prove the pathwise uniqueness and invoke the Yamada-Watanabe-Engelbert theorem to conclude that the solution is strong in the probabilistic sense.
In this framework, the solution is a stochastic process with sample paths in Lebesgue spaces, which are compatible with peakons and wave breaking physical phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02723 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak solution for Stochastic Degasperis-Procesi Equation Chemetov, Nikolai V. Cipriano, Fernanda Probability Analysis of PDEs 35L65, 60H15, 60H30 This article is concerned with the existence of solution to the stochastic Degasperis-Procesi equation on $\mathbb{R}$ with an infinite dimensional multiplicative noise and integrable initial data. Writing the equation as a system composed of a stochastic nonlinear conservation law and an elliptic equation, we are able to develop a method based on the conjugation of kinetic theory with stochastic compactness arguments. More precisely, we first apply the stochastic Jakubowski-Skorokhod representation theorem to show the existence of a weak kinetic martingale solution. Next, we prove the pathwise uniqueness and invoke the Yamada-Watanabe-Engelbert theorem to conclude that the solution is strong in the probabilistic sense. In this framework, the solution is a stochastic process with sample paths in Lebesgue spaces, which are compatible with peakons and wave breaking physical phenomenon. |
| title | Weak solution for Stochastic Degasperis-Procesi Equation |
| topic | Probability Analysis of PDEs 35L65, 60H15, 60H30 |
| url | https://arxiv.org/abs/2409.02723 |