Convergence towards self-similar asymptotic behavior for the dissipative quasi-geostrophic equations
Volume 74 / 2006
Banach Center Publications 74 (2006), 95-115
MSC: 35Q, 35B40.
DOI: 10.4064/bc74-0-5
Abstract
This work proves the convergence in $L^{1}(\mathbb{R}^{2})$ towards an Oseen vortex-like solution to the dissipative quasi-geostrophic equations for several sets of initial data with suitable decay at infinity. The relative entropy method applies in a direct way for solving this question in the case of signed initial data and the difficulty lies in showing the existence of unique global solutions for the class of initial data for which all properties needed in the entropy approach are met. However, the estimates obtained for the constructed global solutions in $L^1\cap L^2$ show the asymptotic simplification of the solutions even for unsigned initial data emphasizing the character of this equation to behave linearly for large times.
