Existence and uniqueness of solutions for the two-dimensional Euler and Navier-Stokes equations with initial data in $ H^1 $

Abstract

In this paper, we consider the incompressible Euler and Navier-Stokes equations in $ \mathbb{R}^2 $. It is well known that the Euler and Navier-Stokes equations are globally well-posed for initial data in $ H^s(s > 2) $. The main purpose of the present paper is to consider the case $ s = 1 $. We prove that, for initial data in $ H^1 $, the Euler and Navier-Stokes equations both have global solutions, and the solutions are uniformly bounded with respect to time. Moreover, the solution for the Navier-Stokes equations is unique. We also prove that, as the viscosity tends to zero, the solution of the Navier-Stokes equations converges to the one of the Euler equations.

Keywords

• euler equations
• navier-stokes equations
• global existence
• weak solutions