On the global regularity for the supercritical SQG equation

By Michele Coti Zelati, Vlad Vicol. 2014

75 citations recorded by the source corpus.

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t \theta + \mathcal{R}^\perp \theta \cdot \nabla \theta + \Lambda^\gamma \theta = 0, \qquad \theta(\cdot,0) =\theta_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $\gamma \in (0,1)$. The coefficient in front of the dissipative term $\Lambda^\gamma = (-\Delta)^{\gamma/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|\theta_0\|_{L^2}^{\gamma/2} \|\theta_0\|_{\dot{H}^2}^{1-\gamma/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $\gamma_1 = \gamma_1(R) \in (0,1)$ such that for $\gamma \geq \gamma_1$, the solution of the supercritical SQG equation with dissipation $\Lambda^\gamma$ does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88a3d5c42629ce7796f896ec82260fa2e7305794

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In Applied Mathematics

  1. Vlad Vicol
  2. T. Buckmaster
  3. Steve Shkoller
  4. Maria Colombo
  5. Nader Masmoudi
  6. Martina Hofmanov'a
  7. Elia Brué
  8. Xiangchan Zhu

In Mathematical Physics

  1. Vlad Vicol
  2. T. Buckmaster
  3. Steve Shkoller
  4. Nader Masmoudi
  5. Maria Colombo
  6. Peter Constantin
  7. Elia Brué
  8. Martina Hofmanov'a

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