The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity

In \cite{C1996}, Chemin shows that solutions of the Navier-Stokes equations in $\R^2$ for an incompressible fluid whose initial vorticity lies in $L^2 \cap L^\iny$ converge in the zero-viscosity limit in the $L^2$–norm to a solution of the Euler equations, convergence being uniform over any finite time interval. In \cite{Y1995}, Yudovich assumes an initial vorticity lying in $L^p$ for all $p \ge p_0$, and establishes the uniqueness of solutions to the Euler equations for an incompressible fluid in a bounded domain of $\R^n$, assuming a particular bound on the growth of the $L^p$–norm of the initial vorticity as $p$ grows large. We combine these two approaches to establish, in $\R^2$, the uniqueness of solutions to the Euler equations and the same zero-viscosity convergence as Chemin, but under Yudovich’s assumptions on the vorticity with $p_0 = 2$. The resulting bounded rate of convergence can be arbitrarily slow as a function of the viscosity $\nu$.

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Published 1 January 2004