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.
Citation: Shaoliang Yuan, Lin Cheng, Liangyong Lin. Existence and uniqueness of solutions for the two-dimensional Euler and Navier-Stokes equations with initial data in $ H^1 $[J]. AIMS Mathematics, 2025, 10(4): 9310-9321. doi: 10.3934/math.2025428
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.
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Shaoliang Yuan, Lin Cheng, Liangyong Lin. Existence and uniqueness of solutions for the two-dimensional Euler and Navier-Stokes equations with initial data in $ H^1 $[J]. AIMS Mathematics, 2025, 10(4): 9310-9321. doi: 10.3934/math.2025428
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