@article{Yuan2025, 
author = {Shaoliang Yuan and Lin Cheng and Liangyong Lin},
title = {Existence and uniqueness of solutions for the two-dimensional Euler and Navier-Stokes equations with initial data in        H    1},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {4},
pages = {9310-9321},
keywords = {Euler equations, Navier-Stokes equations, global existence, weak solutions},
url = {https://www.sciopen.com/article/10.3934/math.2025428},
doi = {10.3934/math.2025428},
abstract = {In this paper, we consider the incompressible Euler and Navier-Stokes equations in              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.}
}