A POSTERIORI ESTIMATES FOR EULER AND NAVIER-STOKES EQUATIONS

The first two sections of this work review the framework of [6] for approximate solutions of the incompressible Euler or Navier-Stokes (NS) equations on a torus T-d, in a Sobolev setting. This approach starts from an approximate solution u(a) of the Euler/NS Cauchy problem and, analyzing it a posteriori, produces estimates on the interval of existence of the exact solution u and on the distance between u and ua. The next two sections present an application to the Euler Cauchy problem, where ua is a Taylor polynomial in the time variable t; a special attention is devoted to the case d = 3, with an initial datum for which Behr, Necas and Wu have conjectured a finite time blowup [1]. These sections combine the general approach of [6] with the computer algebra methods developed in [9] choosing the Behr-Neeas-Wu datum, and using for ua a Taylor polynomial of order 52, a rigorous lower bound is derived on the interval of existence of the exact solution u, and an estimate is obtained for the H-3 Sobolev distance between u(t) and u(a)(t).