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

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.