Conservation vs. Dissipation for Weak Solutions in Fluid Dynamics

Abstract

In this thesis, we consider several problems related to conservation laws of fluid dynamics. It is well-known that for classical sufficiently smooth solutions to time-dependent partial differential equations (PDEs) some quantities such as the energy are conserved, for example, for the Euler equations. However, in physical applications, solutions to the PDEs in connection with fluid dynamics are not always smooth. Therefore, we should consider weak or distributional solutions of the PDEs. Then, it is not clear whether the energy or entropy is conserved for these weak solutions. We mainly focus our attention on the transport equation and Euler equations in this thesis which contains three main results. First, we consider the transport equation. We show that the transport equation can be renormalized. We employ methods of complex analysis in order to obtain this conservation laws. Secondly, we deal with energy conservation for the compressible Euler equations. By using classical commutator methods of Constantin-E-Titi, we obtain sufficient conditions under which the energy is conserved. The main problem of the considered cases is related to the physical interesting case of vacuum. Finally by using convex integration methods, we show that if the density satisfies some compatibility condition then the Euler equations admit infinitely many localized solutions. Moreover, the solutions that we generate satisfy the entropy condition in some finite time interval.