The zero diffusion limit of 2-D Navier-Stokes equations with \(L^1\) initial vorticity (Q1576841)

The limiting process of vanishing kinematic viscosity \(\nu\) is considered for the pure initial value problem for the incompressible Navier-Stokes equations in \(\mathbb{R}^2\). It is shown that a subsequence of the family of solutions \((v^\nu)_{\nu\in]0,1]}\) converges weakly in \(L^\infty(]0,\infty[, L^2(\mathbb{R}^2))\) to a weak solution \(v^0\) of the limit problem with \(\nu = 0\) (incompressible Euler equation) as \(\nu\to 0+\), provided that the initial data \(v^\nu(0+) =v_0\in L^2(\mathbb{R}^2)\) are such that the vorticity curl \(v_0\) is in \(L^1(\mathbb{R}^2)\).