Boundary Conditions For First-order Hyperbolic Relaxation Systems

Hyperbolic systems of partial differential equations with stiff source terms describe various non-equilibrium phenomena.This kind of equations has wide applications in many fields including non-equilibrium thermodynamics,gas kinetic theory,traffic flows,chemically reactive flows,compressible viscoelastic flows and so on.On the other hand,the non-equilibrium phenomena generally occur in a spatial domain with boundaries.Therefore,proper boundary conditions(BCs)should be given in studying such equations.This thesis systematically studies the BCs for the hyperbolic relaxation systems and consists of two parts: 1)BCs for hyperbolic relaxation systems with characteristic boundaries.2)construction of BCs for relaxation approximations.The first part focuses on two types of characteristic initial-boundary value problems.Type I: the boundary is characteristic for the relaxation system but is non-characteristic for the equilibrium system.In this case,we propose a modified General Kreiss Condition.Under this condition,the reduced BC is derived and its validity is verified by combining the Laplace transform with an energy method.Moreover,the existence of boundary-layers is shown for nonlinear relaxation systems.Type II: the boundary is non-characteristic for the relaxation system and is characteristic for the equilibrium system.For this type characteristic boundaries,we introduce a three-scale asymptotic expansion,analyze the boundary-layer behaviors of the general multi-dimensional linear relaxation systems and derive the reduced BCs.Similar to the first type,the validity of the reduced BCs is verified by combining the Fourier-Laplace transform and an energy method.The second part focuses on the application of the BC theory in the relaxation modeling.The main goal of this part is to construct proper BCs for the given relaxation systems.Physically,such BCs are not always available.Our construction is based on the assumption that the equilibrium systems and their well-posed BCs are given.To identify the main issues in the construction,this part focuses on a specific model—the linearized Suliciu model.We obtain strictly dissipative and compatible BCs for different non-characteristic boundaries.Moreover,the effectiveness of the constructed BCs is shown by resorting to formal asymptotic solutions and energy estimates.