On the global regularity for the supercritical SQG equation

Michele Coti Zelati, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation (Equation presented) on T2 = [0,1]2 with γ ∈ (0,1). The coefficient of the dissipative term Λγ = (-Δ)γ/2 is normalized to 1. We show that, given a smooth initial datum with ∥θ0L2γ/2 ∥θ0∥H 2γ/2 ≤ R, where R is arbitrarily large, there exists γ1 = γ1(R) ∈ (0,1) such that, for γ ≥ γ1, the solution of the supercritical SQG equation with dissipation λγ 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, which relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

Original languageEnglish (US)
Pages (from-to)535-552
Number of pages18
JournalIndiana University Mathematics Journal
Volume65
Issue number2
DOIs
StatePublished - Jan 1 2016

Fingerprint

Global Regularity
Fractional Laplacian
Quasi-geostrophic Equations
Dissipative Equations
Maximum Principle
Blow-up
Initial Value Problem
Dissipation
Regularity
Lower bound
Coefficient
Term

Keywords

  • Eventual regularity
  • Global regularity
  • Lower bounds for fractional laplacian
  • Supercritical SQG

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the global regularity for the supercritical SQG equation. / Zelati, Michele Coti; Vicol, Vlad.

In: Indiana University Mathematics Journal, Vol. 65, No. 2, 01.01.2016, p. 535-552.

Research output: Contribution to journalArticle

Zelati, Michele Coti ; Vicol, Vlad. / On the global regularity for the supercritical SQG equation. In: Indiana University Mathematics Journal. 2016 ; Vol. 65, No. 2. pp. 535-552.
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