Inviscid limit for axisymmetric flows without swirl in a critical Besov space
The author considers the Cauchy problem for the Navier-Stokes equations assuming zero force and the initial condition axisymmetric without swirl (i.e. in cylindrical coordinates the velocity depends only on \(r\), \(z\) and \(t\), and the angular component is zero). It is well-known that if the initial conditions are sufficiently smooth, the regular axisymmetric solution without swirl exists globally in time. The author studies the problem with initial conditions in the critical Besov space \(B_{2,1}^{5/2}(R^3)\) and shows that the regular solution exists in the space \(C(R^+; B_{2,1}^{5/2}(R^3))\) together with the bound \(\|v(t)\|_{B_{2,1}^{5/2}(R^3)} \leq C_0 e^{\exp(\exp(\exp(C_0 t)))}\). Moreover, if the viscosity \(\nu \to 0^+\), then these solutions converge to \(\overline{v}\), a regular solution to the Euler equations, with the convergence rate \(\|v(t)-\overline{v}(t)\|_{L^2(R^3)} \leq C_0 \nu t e^{\exp(\exp(C_0 t))}\), \(C_0\) independent of \(t\) and \(\nu\).
scientific article; zbMATH DE number 5728541
scientific article