Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations
The paper addresses systems of nonlinear one-dimensional parabolic equations in the form of \[ U_t + (F(U))_x = DU_{xx}, \] where \(U\) is a set of unknown functions, \(F(U)\) represents the nonlinearity, and \(D\) is a constant matrix of diffusion coefficients. The objective is to develop an analysis in a vicinity of critical constant solutions of the system, at which the Jacobian of the flux vector has a simple or multiple zero eigenvalue. It is rigorously demonstrated that, on a sufficiently large time scale, that in a vicinity of the critical point with the single zero eigenvalue the system may be asymptotically reduced to a Burger equation, the rest being governed by a system enslaved to the dynamics determined by this equation. The analysis is performed in an appropriate functional space (called local spaces), which makes it possible to do it for solutions with finite and infinite energy, the latter being essential for the description of shock waves. In the case of a multiple zero eigenvalue, a system of Burger equations, coupled by a matrix of constant diffusion coefficients, is derived instead of the single equation. The general theory is illustrated by an example from hydrodynamics: the flow of viscous shallow water near the criticality, and a model of a three-layer shallow fluid.
scientific article; zbMATH DE number 6363076
scientific article