REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW

In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations{-△u=λ∑1Bδ（x0,j）（u-kj）p＋,in Ω,u=0,onΩ is a bounded simply-connected smooth domain, ki （i = 1,… , k） is prescribed positive constant. The result we prove is that for any given non-degenerate critical pointX0=（x0,1,…,x0,k of the Kirchhoff-Routh function defined on Ωk corresponding to （ k1,……kk ）there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ→＋∞ shrinks to {x05}, and the local vorticity strength near each x0,j approaches kj, j = 1,… , k. This result makes the study of the above problem with p _〉 0 complete since the cases p 〉 1, p = 1, p = 0 have already been studied in [11, 12] and [13] respectively.

Steady vortex patches near a nontrivial irrotational flow

In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function q corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extreme points of q on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of q corresponds to a family of steady multiple vortex patches approaching it.

Preface

The Silkroad Mathematics Center（SMC）was established in September 2016 by the Chinese Mathematical Society under the support of the China Association for Science and Technology.The main task of the center is to promote mathematics exchanges and cooperation among the countries along the Belt and Road.Professor Ya-xiang Yuan is the current director of SMC.The founding member societies of SMC include Chinese Mathematical Society,

MULTIPLE VORTICES FOR THE SHALLOW WATER EQUATION IN TWO DIMENSIONS

In this paper,we construct stationary classical solutions of the shallow water equation with vanishing Froude number Fr in the so-called lake model.To this end we need to study solutions to the following semilinear elliptic problem -ε^2div(△u/b)=b(u-qlog1/ε)^p+,in Ω,u=0,on aΩfor small ε>0,where p>1,div(△q/b)=0 and ΩcR^2 is a smooth bounded domain.We show that if q^2/b has m strictly local minimum(maximum)points Zi,i=1,...,m,then there is a stationary classical solution approximating stationary m points vortex solution of shallow water equations with vorticity m∑i=1π2q(zi)/b(zi).Moreover,strictly local minimum points of q^2/b on the boundary can also give vortex solutions for the shallow water equation.