Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

Abstract

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361---366, 1976, Arch. Ration. Mech. Anal. 62:295---301, 1976, Arch. Ration. Mech. Anal. 69:37---52, 1979, Pac. J. Math. 104:77---83, 1980; Knightly in SIAM J. Math. Anal. 3:506---511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361---366, 1976, Arch. Ration. Mech. Anal. 62:295---301, 1976, Arch. Ration. Mech. Anal. 69:37---52, 1979, Pac. J. Math. 104:77---83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633---652, 2012, Quad. Mat. 4:27---68, 1999, Nonlinear Anal. 47:4151---4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159---175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719---732, 2008; Maremonti in Mat. Ves. 61:81---91, 2009, Appl. Anal. 90:125---139, 2011).