• crystal
• liquid

This thesis focuses on the Beris-Edward system whose equations are meant to describe the evolution in time of liquid crystals. The system couples an energy equation for Q-tensors, which describe the alignment of the crystal molecules, with a forced Navier-Stokes equation for incompressible fluid.The first chapter is devoted to the presentation of the model: on first we set the central notion of Q-tensor and introduced the so called Landau-De Gennes energy associated to this tensor; we then proceeded to present the Beris-Edward equations and show that it is possible to associate to the system an energy which is also non increasing along solutions.The final section of the first chapter is about the topic of ground states and their existence. Since no literature is available concerning these aspects, at least at our knowledge, we presented a first attempt to overcome the difficulties of the general case restricting ourselves to a simpler situation werewe could prove existence and regularity for such ground states.In view of the application to the Beris-Edward system, in chapter two wepresented a model semilinear parabolic equation the analysis of which con-stitute a good bases for the treatment of the general case. Through theusage of Strichartz type estimates we were able to prove existence of globalsolution and their decay in time.As a way to deepen the understanding of dispersive estimates, we also stud-ied a variant of the aforementioned equation obtained by adding a smallparameter counter acting the strong dispersive properties possessed by the original problem. As before we proved global in time existence of solutions and managed to show their decay in time.The last part of the chapter is devoted to the second component of the system which is the Navier-Stokes equation for incompressible fluid. By means of the Leary projection we reduced to an equation similar to the one faced in the treatment of the model problem, we were then able to apply some of the techniques already used in that case and prove local existence in time of mild solutions.In the last chapter we concluded our work introducing the analysis of the stability for the steady states of the system: perturbing with a small error the ground states found in the first chapter we derived two new equations the analysis of which is expected to carry some insight on the stability of unperturbed steady states, thus presenting a perspective for future work.