Global weak solutions for generalized SQG in bounded domains

Nguyen, Huy Quang

doi:10.2140/apde.2018.11.1029

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Global weak solutions for generalized SQG in bounded domains

Huy Quang Nguyen

Vol. 11 (2018), No. 4, 1029–1047

DOI: 10.2140/apde.2018.11.1029

Abstract

We prove the existence of global $L^{2}$ weak solutions for a family of generalized inviscid surface quasigeostrophic (SQG) equations in bounded domains of $ℝ^{2}$ . In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.

Keywords

generalized SQG, global weak solutions, bounded domains

Mathematical Subject Classification 2010

Primary: 35Q35, 35Q86

Milestones

Received: 7 April 2017

Revised: 28 July 2017

Accepted: 16 October 2017

Published: 12 January 2018

Authors

Huy Quang Nguyen
Program in Applied and Computational Mathematics Princeton University Princeton, NJ United States

Vol. 11, No. 4, 2018