Fazel Hadadifard,
University of California, Riverside

Abstract: The long-term dynamics of the equations arising in fluid mechanics is a ubiquitous and well-studied subject, and several methods have been developed. In this talk, we introduce the scaled variable method of Gallay-Wayne. We expand the method to cover a wider range of equations/models.  The method is then applied to the quasi-geostrophic equation and the Boussinesq system, both subject to fractional dissipation. We also present the stability of the plane wave equations in higher dimensions. The method produces sharp time rates, the leading order terms as well as sharp asymptotics.  Our work, joint with Prof. A. Stefanov, generalizes the classical works on the Navier-Stokes system. Since the Green's functions in the fractional dissipation context are not sufficiently decaying at infinity,  the center-stable manifold construction of Gallay-Wayne appears to be out of reach. Instead, we rely on appropriate a priori estimates for the solutions (both in weighted and unweighted settings) to derive the asymptotic profiles.