Nonlinear Hyperbolic-Elliptic Systems in the Bounded Domain

TL;DR

This paper proves the global solvability of a hyperbolic-elliptic PDE system modeling magnetic vortices in superconductors and cell movement, using a viscous approximation and kinetic formulation to handle boundary conditions.

Contribution

It introduces a novel approach to establish global solutions for a hyperbolic-elliptic system with boundary flux conditions via viscous approximation and limit analysis.

Findings

Proved global solvability of the hyperbolic-elliptic system.

Established strong convergence of viscous solutions to the hyperbolic-elliptic system.

Addressed the boundary layer problem with a kinetic formulation.

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.