We consider a general Euler-Korteweg-Poisson system in R 3 , supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density -the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum.
The aim of this manuscript is to analyze an intracranic fluid model from a mathematical point of view. By means of an iterative process we are able to prove the existence and uniqueness of a local solution and the existence and uniqueness of a global solution under some restriction conditions on the initial data. Moreover the last part of the paper is devoted to present numerical simulations for the analyzed cerebrospinal model. In particular, in order to assess the reliability of the stated theoretical results, we carry out the numerical simulations in two different cases: first, we fix initial data which satisfy the conditions for the global existence of solutions, then, we choose initial data that violate them.
In this paper we investigate a quasineutral type limit for the Navier-Stokes-Poisson system. We prove that the projection of the approximating velocity fields on the divergence-free vector field is relatively compact and converges to a Leray weak solution of the incompressible Navier-Stokes equation. By exploiting the wave equation structure of the density fluctuation we achieve the convergence of the approximating sequences by means of a dispersive estimate of the Strichartz type.
Abstract.In this paper we investigate the Cauchy Problem for coupled NavierStokes-Poisson equation. The global existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation.
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