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32 pages
ON THE NON STATIONARY NAVIER-STOKES EQUATIONS WITH AN EXTERNAL FORCE Marco Cannone and Fabrice Planchon Abstract. We present two dierent existence and uniqueness algorithms for con- structing global mild solutions in C([0; T );L 3 (R 3 )) to the Cauchy problem for the Navier-Stokes equations with an external force. INTRODUCTION There is a rich literature concerning the existence and uniqueness of mild strong local or global solutions of the Cauchy problem for the Navier-Stokes equations in R 3 . The basic approach to tackling the problem is, in principle, always the same. One rst transforms the Navier-Stokes equations, with the unknown velocity v and pressure p and initial velocity v 0 and external force , 8 > > < > > : @v @t v = (v r)v rp+ r v = 0 v(0) = v 0 (0:1) into the mild integral equation v(t) = S(t)v 0 + B(v; v)(t) + Z t 0 S(t s)P(s)ds; (0:2) where B(v; u)(t) = Z t 0 S(t s)Pr (v u)(s)ds; (0:3

  • limit spaces

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  • mild solutions

  • navier stokes equations

  • limit space

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  • space variable

  • structing global

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