LPstability for the strong solutions of the NavierStokes equations in the whole space
Abstract
We consider the motion of a viscous fluid filling the whole space R3, governed by the classical NavierStokes equations (1). Existence of global (in time) regular solutions for that system of nonlinear partial differential equations, is still an open problem. From either the mathematical and the physical point of view, an interesting property is the stability (or not) of the (eventual) global regular solutions. Here, we assume that v1(t,x) is a solution, with initial data a1(x). For small perturbations of a1, we want the solution v1(t,x) being slightly perturbed, too. Due to viscosity, it is even expected that the perturbed solution v2(t,x) approaches the unperturbed one, as time goes to + infinity. This is just the result proved in this paper. To measure the distance between v1(t,x) and v2(t,x), at each time t, suitable norms are introduced (LPnorms). For fluids filling a bounded vessel, exponential decay of the above distance, is expected. Such a strong result is not reasonable, for fluids filling the entire space.
 Publication:

Technical Summary Report Wisconsin Univ
 Pub Date:
 October 1985
 Bibcode:
 1985wisc.reptU....B
 Keywords:

 Differential Equations;
 NavierStokes Equation;
 Partial Differential Equations;
 Perturbation;
 Stability;
 Viscous Fluids;
 Filling;
 Nonlinear Equations;
 Ships;
 Solutions;
 Viscous Flow;
 Fluid Mechanics and Heat Transfer