TY - JOUR

T1 - Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

AU - Ao, Weiwei

AU - Davila Bonczos, Juan

AU - Del Pino, Manuel

AU - Musso, Monica

AU - Wei, Juncheng

N1 - 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.

PY - 2021/12/31

Y1 - 2021/12/31

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=85106081548&partnerID=8YFLogxK

U2 - 10.1090/tran/8406

DO - 10.1090/tran/8406

M3 - Article

VL - 374

SP - 6665

EP - 6689

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -