Non-stationary Navier-Stokes Equations with Mixed Boundary Conditions

In this paper we are concerned with the initial boundary value problem of the 2, 3-D Navier-Stokes equations with mixed boundary conditions including conditions for velocity, static pressure, stress, rotation and Navier slip condition together. Under a compatibility condition at the initial instance it is proved that for the small data there exists a unique solution on the given interval of time. Also, it is proved that if a solution is given, then there exists a unique solution for small perturbed data satisfying the compatibility condition. Our smoothness condition for initial functions in the compatibility condition is weaker than one in such a previous result.Comment: 22 page

Local uniqueness of vortices for 2D steady Euler flow

We study the steady planar Euler flow in a bounded simply connected domain, where the vortex function is $f=t_+^p$ with $p>0$ and the vorticity strength is prescribed. By studying the location and local uniqueness of vortices, we prove that the vorticity method and the stream function method actually give the same solution. We also show that if the vorticity of flow is located near an isolated minimum point and non-degenerate critical point of the Kirchhoff-Routh function, it must be stable in the nonlinear sense.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1703.0986