Ryan Aschoff | Searcharxiv

arXiv2026

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

We investigate weak solutions to the two-dimensional quasi-geostrophic equations without dissipation. We establish global existence of weak solutions for temperature bounded and lacking spatial decay and velocity in the space $L^2_{ul}(\mathbb{R}^2)$. Our methods rely on a spectral Serfati identity, which we use to establish uniform $L^2_{ul}$ bounds on a sequence of velocities satisfying the dissipative equations. These bounds, combined with a maximum principle on the scalar temperature, allow us to pass to the zero-dissipation limit, giving global-in-time weak solutions.

math.AP↗

arXiv2025

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. with fractional diffusion of order $2\alpha$ for $\alpha>\frac{1}{2}$.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. One obstacle is that for $L^{\infty}$ data, the constitutive law may not be applicable, as Riesz transforms are unbounded. However, for $L^{\infty}$ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solution locally in time, and that the constitutive law continues to hold at positive times. In the case that $\alpha\in(\frac{1}{2},1]$ and that the initial data has some smoothness (specifically, if the data is in $C^{2}$), we demonstrate a maximum principle and show that this unique solution is actually classical and global in time. Then, a density argument allows us to show that mild solutions with only $L^{\infty}$ data are also global in time, and also possess this maximum principle. Finally, we introduce a related problem in which we replace the usual constitutive law for the surface quasi-geostrophic equation with a generalization of Sertfati type, and prove the same results for this relaxed model.

math.AP↗