Asymmetric Traveling Vortex Pairs Of The Generalized Surface Quasi-geostrophic Equation

This thesis studies concentrated vortices for the generalized surface quasigeostrophic equation(hereinafter abbreviated as gSQG equation)in fluid dynamics.The main contents include the construction of traveling-wave solutions for the gSQG equation with sharply concentrated vorticities and the study of asymptotic behavior of traveling-wave solutions constructed.In Chapter 1,we briefly introduce how to generalize two-dimensional incompressible Euler equations to obtain gSQG equations,and the research background and current situation of gSQG equations.Then we derive the equations satisfied by traveling wave solutions and the main results of this paper.Finally,ideas of proofs for the main results and the structure of the whole paper are given.In Chapter 2,for the general nonlinear function,the variational method is used to transform problem into maximizing problem of variational functional by constructing appropriate variational functional and admissible set.However,since the variational functional constructed for the original equation satisfied by the traveling-wave solution may not be C1.By adding a perturbation term to the original equation,we make the energy functional corresponding to the disturbed equation is C1.Then it is proved that when the perturbation approaches 0,the limit of the maximizers of the energy functional after perturbation is the maximizers of the original variational problem,which is also the traveling wave solution we hope to obtain.In order to prove this family of solution is the desingularization of a translation point vortex pair,we give precise estimates of the energy lower bound of the vortices and its related quantities,Lagrange multipliers,and the diameter of support sets of vorticity.In Chapter 3,we define the scaling transformations of vorticity and study their asymptotic behavior.In the limit of a certain parameter,the scaling transformation of vorticity converges to the function of radial decreasing according to the weak*topology of L∞.The main result of this thesis is the generalization of Cao et al.’s results of the incompressible Euler equation.Moreover,we have used assumptions different from those used by Gravejat,Smets and Godard-Cadillac where family of smooth traveling wave solutions for SQG equation and gSQG equation were obtained respectively,so our method is different from theirs.