Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

Abstract

The existence and uniqueness of solutions to the Euler equations for initial vorticity in B Γ ∩ L p 0 ∩ L p 1 was proved by Misha Vishik, where B Γ is a borderline Besov space parameterized by the function Γ and 1 < p 0 < 2 < p 1 . Vishik established short time existence and uniqueness when Γ ( n ) = O ( log n ) and global existence and uniqueness when Γ ( n ) = O ( log 1 2 n ) . For initial vorticity in B Γ ∩ L 2 , we establish the vanishing viscosity limit in L 2 ( R 2 ) of solutions of the Navier–Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ ( n ) = O ( log n ) and uniform over any finite time when Γ ( n ) = O ( log κ n ) , 0 ⩽ κ < 1 , and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include B Γ ∩ L 2 when Γ ( n ) = O ( log κ n ) for 0 < κ < 1 .