Data di Pubblicazione:
2012
Citazione:
A posteriori estimates from approximate
solutions of the Euler or Navier-Stokes equations / L. Pizzocchero. ((Intervento presentato al 14. convegno International Conference on
Hyperbolic Problems: Theory Numerics and Applications tenutosi a Padova nel 2012.
Abstract:
This communication deals with the Cauchy problem for the incompressible Euler or Navier-Stokes (NS) equations
on a d-dimensional torus T^d, in a setting based on the Sobolev spaces H^n(T^d) (n > d/2 + 1; typically, d = 3).
Following [Morosi and Pizzocchero, Nonlinear Analysis, 2012], an approach will be presented to obtain fully quantitative information on the exact solution u of the Euler or NS Cauchy problem from a posteriori analysis of any approximate solution u_a.
This approach allows to derive estimates on the interval of existence [0, T) of the exact solution u, and
on the Sobolev distance between the exact and the approximate solution. The latter estimate has the form
||u(t) − u_a(t)||n ≤ R_n(t) where R_n(t) is a real, nonnegative function of time t, obtained solving a differential
“control inequality”. In particular, the exact solution u of the Cauchy problem is granted to be global in time
if the control inequality has a global solution R_n : [0,+∞) → [0,+∞).
The quantitative implementation of the above setting requires accurate estimates on the constants in a
number of inequalities, in the Sobolev setting for the Euler/NS equations. For example, it is necessary to use
estimates [Morosi and Pizzocchero, CPAA, 2012] on the constants in the celebrated Kato inequality for < (v•∇)w|w>_n (with v, w two velocity fields).
The above scheme will be compared with the setting proposed by [Chernyshenko et al, J. Math. Phys., 2007] for the approximate solutions of the Euler or NS equations (and with other works on this subject by Morosi and Pizzocchero [Rev. Math. Phys. 2008; Nonlinear Analysis, 2011]).
Finally, as an application, some results will be presented on the Euler or NS equations on T^3 with the
Behr-Neˇcas-Wu initial datum [ESAIM: M2AN, 2001]; such a datum was proposed by the cited authors as a candidate for finite-time blow-up of the Euler equations.
Tipologia IRIS:
14 - Intervento a convegno non pubblicato
Keywords:
Euler and Navier-Stokes equations ; existence and regularity theory ; theoretical
approximation.
Elenco autori:
L. Pizzocchero
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