Work

Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation

Year: 2026

Type: article

Abstract: For any initial datum $$\theta _0\in L^{\frac{4}{3}}_x$$ θ 0 ∈ L x 4 3 it is proven that the existence of a global-in-time weak solution $$\theta \in L^\infty _t L^{\frac{4}{3}}_x$$ θ ∈ L t ∞ L x 4 3 to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $$\dot{H}^{-\frac{1}{2}}_x$$ H ˙ x - 1 2 norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the $$L^{\frac{4}{3}}_x$$ L x 4 3 norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.

Source: Archive for Rational Mechanics and Analysis

DOI: 10.1007/s00205-026-02231-2

Authors: Luigi De Rosa, Mickaël Latocca, Jaemin Park

Institutions: Gran Sasso Science Institute, Laboratoire de Mathématiques et Modélisation d'Évry, Université d'Évry Val-d'Essonne, Yonsei University

Language: English

Topic: Navier-Stokes equation solutions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090