The first two sections of this work review the framework of [Morosi and Pizzocchero, Nonlinear Analysis 2012] 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 casen d=3, with an initial datum for which Behr, Necas and Wu have conjectured a finite time blowup [ESAIM:M2AN 2001]. These sections combine the general approach of [Morosi and Pizzocchero, Nonlinear Analysis, 2012] with the computer algebra methods developed in [Morosi, Pernici and Pizzocchero, ESAIM:M2AN 2013]; 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_(t).

A posteriori estimates for Euler and Navier-Stokes equations / C. Morosi, M. Pernici, L. Pizzocchero (AIMS SERIES ON APPLIED MATHEMATICS). - In: Hyperbolic problems : theory, numerics, applications : proceedings of the 14. international conference on hyperbolic problems held in Padova, June 25-29, 2012 / [a cura di] F. Ancona, A. Bressan, P. Marcati, A. Marson. - Springfield : American Institute of Mathematical Sciences, 2014. - ISBN 978-1-60133-017-8. - pp. 847-855 (( Intervento presentato al 14. convegno International Conference on Hyperbolic Problems tenutosi a Padova nel 2012.

A posteriori estimates for Euler and Navier-Stokes equations

L. Pizzocchero
2014

Abstract

The first two sections of this work review the framework of [Morosi and Pizzocchero, Nonlinear Analysis 2012] 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 casen d=3, with an initial datum for which Behr, Necas and Wu have conjectured a finite time blowup [ESAIM:M2AN 2001]. These sections combine the general approach of [Morosi and Pizzocchero, Nonlinear Analysis, 2012] with the computer algebra methods developed in [Morosi, Pernici and Pizzocchero, ESAIM:M2AN 2013]; 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_(t).
Navier-Stokes equations ; existence and regularity theory ; theoretical approximation ; symbolic computation
Settore MAT/07 - Fisica Matematica
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/252649
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