Some Analytic Questions in Mathematical Physic Problems

Abstract

In this thesis we study two different problems of mathematical physics. In the first part of the thesis we discuss some questions related to the partial regularity theory of the Navier-Stokes equations. In particular, we obtain some summabilities properties of the pressure field associated to a Hopf weak solution and we give an existence theorem for suitable weak solutions; moreover, we prove that a Hopf weak solution which satisfies a suitable extra-condition, is a suitable weak one too and we consider the question of Hausdorff dimension of the possible singular set S of the weak solution. In the second part of the thesis we study the asymptotic stability of solitary waves solutions for the Maxwell-Schrödinger system