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