In a previous paper of ours (Rev.Math.Phys.34 (2006) 319-363), a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of the cited paper is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus of any dimension d. In this way, a number of results are obtained in the setting of the Sobolev spaces H^n on the torus, choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces or global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e., giving the values of all the necessary constants; this makes a difference with most of the existing literature on this topic). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for heir H^n distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).
On approximate solutions for semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations
MOROSI, CARLO;
2008-01-01
Abstract
In a previous paper of ours (Rev.Math.Phys.34 (2006) 319-363), a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of the cited paper is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus of any dimension d. In this way, a number of results are obtained in the setting of the Sobolev spaces H^n on the torus, choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces or global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e., giving the values of all the necessary constants; this makes a difference with most of the existing literature on this topic). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for heir H^n distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).File | Dimensione | Formato | |
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