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On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. (English) Zbl 1496.76161

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \(\theta\) on the whole plane \(\mathbb{R}^2\) whose velocities have been mildly regularized. More precisely, they study \[ \left\{\begin{array}{l}\partial_{t} \theta+u \cdot \nabla \theta=0 \\ u=\nabla^{\perp} \psi:=\left(-\partial_{x_{2}} \psi, \partial_{x_{1}} \psi\right), \quad \Delta \psi=\Lambda^{\beta} p(\Lambda) \theta, \quad \\ \theta(0, x)=\theta_{0}(x)\end{array}\right. \] where \(\psi\) is the stream function. It is well know that the well-posedness of these regularized models in the borderline Sobolev space \(H^{\beta+1}\) has previously been studied by D. Chae and J. Wu [J. Math. Phys. 53, No. 11, 115601, 15 p. (2012; Zbl 1329.76027)] in the case when the velocity \(u\) is of lower singularity, \(p\) is a logarithmic smoothing and \(0\leq \beta \leq 1\). Here, the authors studied the well-posedness, in the Hadamard sense, in the case when \( 0 \leq \beta<2 \). To prove this, the authors consider, as a main tool, the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. They remark the fact that the main obstruction for proving the well-posedness for these models lies not in establishing existence and uniqueness, but rather in proving continuity of the data-to-solution map.

MSC:

76U60 Geophysical flows
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 1329.76027
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References:

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