Infinite Energy Solutions for Navier-Stokes Equations in a Strip.

This thesis deals with infinite energy solutions of the Navier-Stokes and Boussi-nesq equations in a strip. Here, the properly chosen Uniformly local Sobolev Spaces of functions are used as the phase spaces for the problem considered. The global well-posedness and dissipativity of the Navier-Stokes equations was first established in a paper by Zelik [37] on Spatially Nondecaying Solutions of the 2D Navier-Stokes equations in a strip. However, the proof given there contains error which emmanated from wrong estimation of the solutions of the auxiliary non-autonomous linear Stokes problem with non-homogeneous divergence. In this thesis, we correct the aforementioned error and show that the main results of [37], i. e the well-posedness of the Navier-Stokes problem in uniformly local spaces, remains true. Albeit, only a weaker version of the postulated results in [37] was amenable; therefore, we reworked most part of the non-linear theory as well as to show that they are sufficient for the well-posedness of the non-linear system. We also extended these results to the thermal convection problem in a strip and associated Boussinesq equations. We considered the temperature equation and proved, using maximum principle, the well-posedness of the full Boussinesq system in a Strip