7 - Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations

Summary

By their use of mild solutions, Fujita-Kato and later on Giga-Miyakawa opened the way to solving the initial-boundary value problem for the Navier-Stokes equations with the help of the contracting mapping principle in suitable Banach spaces, on any smoothly bounded domain $$\Omega \subset \R^n, n \ge 2$$, globally in time in case of sufficiently small data. We will consider a variant of these classical approximation schemes: by iterative solution of linear singular Volterra integral equations, on any compact time interval J, again we find the existence of a unique mild Navier-Stokes solution under smallness conditions, but moreover we get the stability of each (possibly large) mild solution, inside a scale of Banach spaces which are imbedded in some $$C^0 (J, L^r (\Omega))$$, $$1 < r < \infty$$.