Nonlinear Hyperbolic-Elliptic Systems in the Bounded Domain

Abstract

In the article we study a hyperbolic-elliptic system of PDE. The system can describe two different physical phenomena: 1st one is the motion of magnetic vortices in the II-type superconductor and 2nd one \ is the collective motion of cells. Motivated by real physics, we consider this system with boundary conditions, describing the flux of vortices (and cells, respectively) through the boundary of the domain. We prove the global solvability of this problem.\ To show the solvability result we use a "viscous" parabolic-elliptic system. Since the viscous solutions do not have a compactness property, we justify the limit transition on a vanishing viscosity, using a kinetic formulation of our problem. As the final result of all considerations we have solved a very important question related with a so-called "boundary layer problem", showing the strong convergence of the viscous solutions to the solution of our hyperbolic-elliptic system.