2013/10/21 by Carlo Morosi, Mario Pernici, Livio Pizzocchero · 4 citations
Mathematics · #math.AP #msc:35Q31 #msc:35Q30 #msc:76B03 #msc:76D03 #msc:68W30
published as AIMS Series on Applied Mathematics 8 (2014), 847-855 · AUTHORS' NOTE. In Sect.s 1 and 2, some overlap with our previous works on the Euler/NS equations (arXiv:1203.6865, arXiv:0709.1670, arXiv:0909.3707, arXiv:1009.2051, arXiv:1104.3832, arXiv:1007.4412, arXiv:1304.2972). These overlaps aim to make the present paper self-contained, and do not involve the main results of Sect.s 3, 4. To appear in the Proceedings of Hyp 2012
arxiv created 2013/10/21 · arxiv updated 2014/11/21
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 Td, in a Sobolev setting. This approach starts from an approximate solution ua 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-Necas-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 H3 Sobolev distance between u(t) and ua(t).