Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

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Abstract

For the generalized surface quasi-geostrophic equation {equation presented} 0 < s < 1, we consider for k ≥ 1 the problem of finding a family of k-vortex solutions Θ ϵ(x, t) such that as ϵ → 0 {equation presented} for suitable trajectories for the vortices x = ζ j(t). We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem {equation presented} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].

Original languageEnglish
Pages (from-to)6665-6689
Number of pages25
JournalTransactions of the American Mathematical Society
Volume374
Issue number9
Early online date9 Jun 2021
DOIs
Publication statusPublished - 31 Dec 2021

Bibliographical note

Funding Information:
Received by the editors August 28, 2020, and, in revised form, February 1, 2021. 2020 Mathematics Subject Classification. Primary 35Q35, 35J61; Secondary 35Q31. The first author was partially supported by NSF of China. The second author was supported by a Royal Society Wolfson Fellowship, UK and Fondecyt grant 1170224, Chile. The third author was supported by a Royal Society Research Professorship, UK. The fourth author was supported by EPSRC research Grant EP/T008458/1. The research of the fifth author was partially supported by NSERC of Canada.

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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