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

  • Edited by Charles L. Fefferman, Princeton University, New Jersey, James C. Robinson, University of Warwick, José L. Rodrigo, University of Warwick
  • Book:
    Partial Differential Equations in Fluid Mechanics
    Published online:
    15 August 2019
    Print publication:
    27 September 2018, pp 204-223

  • 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 $$C^0 (J, L^r (\Omega))$$, $$1 < r < \infty$$.