Non-decaying solutions to the 2D dissipative quasi-geostrophic equations

We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. fractional diffusion of order 2α for α>1/2.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. For L∞ data, the constitutive law may notbe applicable, as Riesz transforms are unbounded. However, forL∞ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solutionlocally in time, and that the constitutive law continues to hold at positivetimes. In the case that α∈(1/2,1], we show that mild solutions withL∞ data are global in time, and also possess a maximum principle. For smoother initial data, we demonstrate that the solutions hold classically. Finally, we introduce a related problem in which we replace the usual constitutive law for SQG with a generalization of Sertfati type, and prove the same results for this relaxed model.