Invariant Measures and Global Well Posedness for the SQG Equation
Abstract
We construct an invariant measure μ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of μ are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of μ 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 measure a μ 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.
- Publication:
-
Archive for Rational Mechanics and Analysis
- Pub Date:
- July 2021
- DOI:
- 10.1007/s00205-021-01650-7
- arXiv:
- arXiv:2002.09555
- Bibcode:
- 2021ArRMA.241..187F
- Keywords:
-
- Mathematics - Analysis of PDEs;
- Mathematics - Probability;
- 35Q35;
- 76D03;
- 35R60;
- 60H15;
- 37C40
- E-Print:
- 30 pages