101. Local uniqueness for vortex patch problem in incompressible planar steady flow
- Author
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Daomin Cao, Shusen Yan, Shuangjie Peng, and Yuxia Guo
- Subjects
Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,Vorticity ,01 natural sciences ,Domain (mathematical analysis) ,Vortex ,010101 applied mathematics ,Flow (mathematics) ,Bounded function ,Stream function ,Uniqueness ,0101 mathematics ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.
- Published
- 2019