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A determining form for the subcritical surface quasi-geostrophic equation. (English) Zbl 1420.35042

Summary: We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35R11 Fractional partial differential equations
35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
35G25 Initial value problems for nonlinear higher-order PDEs
35B10 Periodic solutions to PDEs
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