Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General Case

Abstract

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.