A posteriori estimates for Euler and Navier-Stokes equations

Abstract

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 u_a. The next two sections present an application to the Euler Cauchy problem, where u_a 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-Necas-Wu datum, and using for u_a 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).