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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2101.06934 |
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| _version_ | 1866909075093061632 |
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| author | Galimberti, Luca Karlsen, Kenneth H. |
| author_facet | Galimberti, Luca Karlsen, Kenneth H. |
| contents | We analyze continuity equations with Stratonovich stochasticity, $\partial ρ+ div_h \left[ ρ\circ\left(u(t,x)+\sum_{i=1}^N a_i(x) \dot W_i(t) \right) \right]=0$, defined on a smooth closed Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms $\dot W_1(t),\ldots,\dot W_N(t)$ driven by smooth spatially dependent vector fields $a_1(x),\ldots,a_N(x)$ on $M$. The velocity $u$ belongs to $L^1_t W^{1,2}_x$ with $div_h u$ bounded in $L^p_{t,x}$ for $p>d+2$, where $d$ is the dimension of $M$ (we do not assume $div_h u \in L^\infty_{t,x}$). We show that by carefully choosing the noise vector fields $a_i$ (and the number $N$ of them), the initial-value problem is well-posed in the class of weak $L^2$ solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this "regularization by noise" result reveals a link between the nonlinear structure of the underlying domain $M$ and the noise, a link that is somewhat hidden in the Euclidian case ($a_i$ constant) \cite{Beck:2019,Flandoli-Gubinelli-Priola,Neves:2015aa}. The proof is based on an a priori estimate in $L^2$, which is obtained by a duality method, and a weak compactness argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_06934 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Well-posedness of stochastic continuity equations on Riemannian manifolds Galimberti, Luca Karlsen, Kenneth H. Analysis of PDEs We analyze continuity equations with Stratonovich stochasticity, $\partial ρ+ div_h \left[ ρ\circ\left(u(t,x)+\sum_{i=1}^N a_i(x) \dot W_i(t) \right) \right]=0$, defined on a smooth closed Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms $\dot W_1(t),\ldots,\dot W_N(t)$ driven by smooth spatially dependent vector fields $a_1(x),\ldots,a_N(x)$ on $M$. The velocity $u$ belongs to $L^1_t W^{1,2}_x$ with $div_h u$ bounded in $L^p_{t,x}$ for $p>d+2$, where $d$ is the dimension of $M$ (we do not assume $div_h u \in L^\infty_{t,x}$). We show that by carefully choosing the noise vector fields $a_i$ (and the number $N$ of them), the initial-value problem is well-posed in the class of weak $L^2$ solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this "regularization by noise" result reveals a link between the nonlinear structure of the underlying domain $M$ and the noise, a link that is somewhat hidden in the Euclidian case ($a_i$ constant) \cite{Beck:2019,Flandoli-Gubinelli-Priola,Neves:2015aa}. The proof is based on an a priori estimate in $L^2$, which is obtained by a duality method, and a weak compactness argument. |
| title | Well-posedness of stochastic continuity equations on Riemannian manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2101.06934 |