We introduce and study the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations defined either on the torus \( {\mathbb {T}}^d\) or \( {\mathbb {R}}^d\) or \( O\subsetneq {\mathbb {R}}^d\) bounded, \( d\ge 1\). This class includes, among others the stochastic; dD-quasi-geostrophic equation, fractional Burgers equation, fractional nonlocal transport equation and the 2D-fractional vorticity Navier–Stokes equation. We consider a locally Lipschitz diffusion term and a cylindrical Q-Wiener process with finite trace covariance Q. In particular, for \( O={\mathbb {T}}^d\) or \( O\subsetneq {\mathbb {R}}^d\) bounded, we prove the existence and the uniqueness of a global mild solution for the free divergence mode in the subcritical regime (\(\alpha >\alpha _0(d)\ge 1\)), martingale solutions in the general regime (\(\alpha \in (0, 2)\), supercritical, critical and subcritical) and free divergence mode and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. For \( O={\mathbb {R}}^d\), we studied the subcritical regime and we proved the existence of a global martingale solution for the free divergence mode and a local mild solution for the general mode.

中文翻译：

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推广多维拟地转和二维Navier–Stokes方程的分数阶随机活动标量方程：一般情况

我们介绍并研究适当的适定性：在圆环\（{\\ mathbb {T}} ^ d \）或\上定义的一类d维分数阶随机主动标量方程的整体存在性，解的唯一性和正则性。 （{\\ mathbb {R}} ^ d \）或\（O \ subsetneq {\ mathbb {R}} ^ d \）有界，\（d \ ge 1 \）。该课程除其他外，还包括随机课程；dD-准地转方程，分数Burgers方程，分数非局部输运方程和二维分数涡度Navier-Stokes方程。我们考虑局部Lipschitz扩散项和具有有限轨迹协方差Q的圆柱Q-维纳过程。特别是对于\（O = {\ mathbb {T}} ^ d \）或\（O \ subsetneq {\ mathbb {R}} ^ d \）有界，我们证明了亚临界状态下自由散模的全局温和解的存在性和唯一性（\（\ alpha> \ alpha _0（ d）\ ge 1 \）），一般体制下的mar解决方案（\（\ alpha \ in（0，2）\），超临界，临界和亚临界）和自由发散模式以及一般模式和亚临界政权。还为这些解决方案建立了各种规则。对于\（O = {\ mathbb {R}} ^ d \），我们研究了亚临界状态，并证明了存在自由for散模式的全局mar解和一般模式的局部温和解的存在。