A posteriori estimates from approximate solutions of the Euler or Navier-Stokes equations

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.