Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded Domains
Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded Domains
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DOI：
10.1007/s00021-015-0234-5
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发表时间：
2016-03
影响因子：
1.3
通讯作者：
Latifa Debbi
中科院分区：
文献类型：
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作者：
Latifa Debbi
在这项工作中，我们引入并研究了有界域上和环面上的多维分数次随机N-S方程(简称DD-FSNSE)的适定性。对于亚临界区域，我们建立了存在最大局部温解且满足所要求的空间和时间正则性的阈值。我们证明了在Beale-Kato-Majda类型的条件下，这些解是整体唯一的。对于环面上的2D-FSNSE，如果初始数据具有H1-正则性，并且扩散项满足对应于H1-空间的增长和Lipschitz条件，则这些条件被自动满足。分别研究了环面上的2D-FSNSE的情况。特别地，我们建立了次临界状态下弱(强概率)解的整体存在性、唯一性、空间和时间正则性的阈值。对于一般的情形，我们证明了在分式Sobolev空间上，在Serrin类型的条件下，我们证明了一个鞅解的存在性和唯一性。
In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier–Stokes equations on bounded domains and on the torus (briefly dD-FSNSE). For the subcritical regime, we establish thresholds for which a maximal local mild solution exists and satisfies required space and time regularities. We prove that under conditions of Beale–Kato–Majda type, these solutions are global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data hasH1-regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding toH1-spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime. For the general regime, we prove the existence of a martingale solution and we establish the uniqueness under a condition of Serrin’s type on the fractional Sobolev spaces.