Nonhomogeneous boundary value problem for "Equation missing"  similar solutions of incompressible two-dimensional Euler equations

Wenjing Song,Hua Li,Ganshan Yang,George Xianzhi Yuan

doi:10.1186/1029-242x-2014-277

Abstract

Abstract In this paper we introduce the ( I , J ) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit ( I , J ) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave solution and an affine solution to the two-dimensional incompressible Euler equations are, respectively, a plane wave and constant vector. We prove that the initial boundary value problem of the incompressible two-dimensional Euler equations admits a unique solution and discuss the stability of the solution. Finally, we supply some explicit piecewise smooth solutions to the incompressible three-dimensional Euler case and an example of the incompressible three-dimensional Navier-Stokes equations which indicates that the viscosity limit of a solution to the Navier-Stokes equations does not need to be a solution to the Euler equations. MSC:35Q30, 76D05, 76D10.

Highlights

In this paper we consider the Euler equations (σ = ) or the Navier-Stokes equations below: ⎧ ⎨ ∂u ∂t + (u · ∇)u ∇p = σ u, in
In this paper we find a so-called (I, J) similar method which can give some explicit smooth solutions to two-dimensional incompressible Euler equations
In Section we discuss the method of determining the nonexistence of a non-constant affine solution to the two-dimensional Euler equations, which we can correctly obtain due to the (I, J) similar method

Summary

Introduction

In this paper we find a so-called (I, J) similar method which can give some explicit smooth solutions to two-dimensional incompressible Euler equations (see Section ). In Section we discuss the method of determining the nonexistence of a non-constant affine solution to the two-dimensional Euler equations, which we can correctly obtain due to the (I, J) similar method. In Section , we prove that incompressible two-dimensional Euler equations under a class of initial boundary values has a unique solution u(x, t) ∈ C∞([ , ∞); L ( )) for every bounded domain ⊂ Rn+, and we discuss the stability of solutions in Section .

Results

Conclusion