Invariant measures and global well posedness for SQG equation

Abstract

We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $\mu$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $\mu$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $\mu$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

30 pages

References in corpus (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• SPDE in Hilbert Space with Locally Monotone Coefficients
• Local and global well-posedness of SPDE with generalized coercivity conditions

Cited by in corpus (4)

• Strong illposedness for SQG in critical Sobolev spaces
• Global well-posedness and long-time behavior of the fractional NLS
• Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation
• Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation