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On the weak-strong uniqueness of the dissipative surface quasi-geostrophic equation. (English) Zbl 1241.35158

Summary: The weak-strong uniqueness of the dissipative surface quasi-geostrophic equation is studied. It is proved that if \(\theta\) and \(\widetilde\theta\) are two weak solutions of the quasi-geostrophic equation initially from the same function \(\theta(0)=\widetilde\theta(0)\in L^2(\mathbb{R}^2)\) and the weak solution \(\theta\) is in the regular class \[ \nabla\theta\in L^r(0,T;B^0_{p,\infty}(\mathbb{R}^2))\quad\text{for }\frac 2p+\frac \alpha r=\alpha,\;\frac 2\alpha<p<\infty,\;0<\alpha\leq 2, \] then \(\theta=\widetilde\theta\) on \(\mathbb{R}^2\times[0,T]\).

MSC:

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35D30 Weak solutions to PDEs
35Q86 PDEs in connection with geophysics
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