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.

arXiv comment30 pages

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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