The main result of [C. Morosi and L. Pizzocchero, Nonlinear Analysis, 2012] is presented in a variant, based on a C^infinity formulation of the Cauchy problem; in this approach, the a posteriori analysis of an approximate solution gives a bound on th
e Sobolev distance of any order between the exact and the approximate solution.
We consider the Cauchy problem for the incompressible homogeneous Navier-Stokes (NS) equations on a d-dimensional torus, in the C^infinity formulation described, e.g., in [25]. In [22][25] it was shown how to obtain quantitative estimates on the exac
t solution of the NS Cauchy problem via the a posteriori analysis of an approximate solution; such estimates concern the interval of existence of the exact solution and its distance from the approximate solution. In the present paper we consider an approximate solutions of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where R is the mathematical Reynolds number (the reciprocal of the kinematic viscosity) and the coefficients u_j(t) are determined stipulating that the NS equations be satisfied up to an error O(R^{N+1}). This subject was already treated in [24], where, as an application, the Reynolds expansion of order N=5 in dimension d=3 was considered for the initial datum of Behr-Necas-Wu (BNW). In the present paper, these results are enriched regarding both the theoretical analysis and the applications. Concerning the theoretical aspect, we refine the approach of [24] following [25] and use the symmetries of the initial datum in building up the expansion. Concerning the applicative aspect we consider two more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and Kida-Murakami (KM); the Reynolds expansions for the BNW, TG and KM data are performed via a Python program, attaining orders between N=12 and N=20. Our a posteriori analysis proves, amongst else, that the solution of the NS equations with anyone of the above three data is global if R is below an explicitly computed critical value. Our critical Reynolds numbers are below the ones characterizing the turbulent regime; however these bounds have a sound theoretical support, are fully quantitative and improve previous results of global existence.
We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to
the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001) 229-238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius tau_3 in the H^3 Sobolev space, with 0.32 < tau_3 < 0.35; they regarded this as an indication that the solution of the Euler equation blows up. We have repeated the calculations of Behr, Necas and Wu, using again computer algebra; the order has been increased from 35 to 52, using the symmetries of the initial datum to speed up computations. As for tau_3, our results agree with the original computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 < tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a) The finiteness of tau_3 is not at all an indication of a possible blow-up. (b) There is a strong indication that the solution of the Euler equation does not blow up at a time close to tau_3. In fact, the solution is likely to exist, at least, up to a time theta_3 > 0.47. (c) Pade analysis gives a rather weak indication that the solution might blow up at a later time.
