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

2025/09/01 by De Rosa, Luigi, Latocca, Mickaël, Park, Jaemin · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.01268

Abstract

For any initial datum θ₀∈ L^(4)/(3)_x it is proved the existence of a global-in-time weak solution θ∈ L^∞_t L^\frac43_x to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the H^-(1)/(2)_x norm, is constant in time. The solution is obtained as a vanishing viscosity limit. Outside the classical strong compactness setting, the main idea is to propagate in time the non-concentration of the L^(4)/(3)_x norm of the initial data, from which strong compactness in the Hamiltonian norm is deduced. General no anomalous dissipation results under minimal Onsager supercritical assumptions are also obtained.

Citations

• Dissipation concentration in two-dimensional fluids
• Wellposedness of inviscid SQG in the half-plane
• Absence of anomalous dissipation for vortex sheets
• Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
• An Onsager-type theorem for SQG
• A proof of Onsager's Conjecture for the SQG equation
• Sharp conditions for energy balance in two-dimensional incompressible ideal flow with external force
• No anomalous dissipation in two-dimensional incompressible fluids
• Non-uniqueness up to the Onsager threshold for the forced SQG equation
• Non-unique stationary solutions of forced SQG
• Small scale creation for solutions of the SQG equation
• Regularity results for a class of generalized surface quasi-geostrophic equations
• Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation
• Global weak solutions for SQG in bounded domains
• Nonuniqueness of weak solutions to the SQG equation
• Global smooth solutions for the inviscid SQG equation
• A pointwise inequality for fractional laplacians
• Holder Continuous Solutions of Active Scalar Equations
• A variation on a theme of Caffarelli and Vasseur
• Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

Cited by

• Weak Solutions for Inviscid SQG with Lorentz Data

Related

• Global and local existence for the dissipative critical SQG equation\n with small oscillations
• Quasi-stationary solutions of the surface quasi-geostrophic equation
• The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation
• On the critical dissipative quasi-geostrophic equation
• An Onsager-type theorem for SQG