Solutions Of Quasilinear Hyperbolic Systems Research Articles

Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems

This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states in d≥3 space dimensions. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by algebraic conditions implying the uniform decay of all Fourier modes at the reference state. On a technical level, the theory developed herein uses para-differential operators as its main tool. Apparently being the first to apply such operators in the context of global-in-time existence of small solutions for quasi-linear hyperbolic systems, the present work contains a new version of the strong Gårding inequality. In the context of theoretical physics, the theorem applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids such as notably that of Bemfica, Disconzi and Noronha (2018) [2].

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Singularities of Multivalued Solutions of Quasilinear Hyperbolic Systems

Исследуются особенности многозначных решений квазилинейной гиперболической системы с двумя независимыми и двумя зависимыми переменными, удовлетворяющей условию сильной нелинейности. Получена лево-правая локальная классификация проектирований таких решений на плоскость независимых переменных в конечнократных точках ранга $1$.

Analytical Study of Weak Shock Waves in Gas with dust particles

In the present paper, the exact solution of quasilinear hyperbolic system of equations governing the propagation of weak shock waves in a one-dimensional dusty adiabatic gas flow with generalized geometries is derived. Here, the density behind the shock front is assumed to vary according to a power law of the distance. An analytical expression for the total energy carried by weak shock wave in dusty gas is also derived.

Singularities of Multivalued Solutions of Quasilinear Hyperbolic Systems

We study the singularities of multivalued solutions of a quasilinear hyperbolic system with two independent and two dependent variables that satisfies the strong nonlinearity condition. For such solutions we obtain a local left-right classification of their projections onto the plane of independent variables at points of finite multiplicity of rank 1.

Exact Solution of the Weak Shock Wave in Non-ideal Gas

In the present paper the exact solution of quasilinear hyperbolic system of equations governing the propagation of weak shock waves in a one dimensional non-ideal adiabatic gas flow with generalized geometries is derived. Here the density ahead of the shock front is assumed to vary according to a power law of the distance. The effect of van der Waals parameter on the radius of weak shock wave is analyzed. An analytical expression for the total energy carried by weak shock wave in non-ideal gas is also derived.

Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms

We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.

Exact boundary synchronization for a coupled system of 1-D quasilinear wave equations

Based on the theory of semi-global classical solutions for quasilinear hyperbolic systems, under suitable hypotheses, an iteration procedure given by a unified constructive method is presented to establish the exact boundary synchronization for a coupled system of 1-D quasilinear wave equations with boundary conditions of various types.

Decay Estimates of Solutions for Quasi-Linear Hyperbolic Systems of Viscoelasticity

This paper is devoted to the study of the sharp decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity in the whole space. We develop the time-weighted energy method for our system, which can yield the decay estimate of solutions for small initial data in $L^2$, provided that $n\geq 2$. Also, we discuss the fundamental solutions to the linearized system and study the decay properties for the corresponding solution operators. Then, by employing the same time-weighted energy method together with the semigroup argument, we show the optimal decay estimate of solutions for small initial data in $L^2\cap L^1$ and for all $n\geq 1$.

Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary: Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the piecewise C1 solution u = u(t, x) of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U(x/t) of the corresponding Riemann problem. The piecewise C1 solution u = u(t, x) to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.

The Tychonoff-Schauder theorem and the existence of bounded solutions of quasilinear hyperbolic systems of differential equations

The Tychonoff-Schauder fixed point theorem is applied to prove the existence of bounded solutions of systems of ordinary differential equations that are C0-close to linear hyperbolic systems. Bibliography: 3 titles.

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C 1 solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C 1 traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space–time R 1 + n .

Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves

This work is a continuation of our previous work (Kong, J. Differential Equations 188 (2003) 242–271) “Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities”. In the present paper we prove the global structure instability of the Lax's Riemann solution u = U ( x t ) , containing rarefaction waves, of general n × n quasilinear hyperbolic system of conservation laws. Combining the results in (Kong, 2003), we prove that the Lax's Riemann solution of general n × n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.

Weak solutions of quasilinear functional differential systems in the second canonical form with unbounded delay

Weak solutions of quasilinear hyperbolic systems in the second canonical form with unbounded delay are investigated. The phase space is defined as an abstract functional space satisfying some suitable axioms. A theorem on the existence, uniqueness and continuous dependence upon initial data is given. The method of bicharacteristics and the Banach fixed-point principle are used.

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution u=U( x t ) , containing only shocks and contact discontinuities, of general n× n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C 1 solution u= u( t, x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution u=U( x t ) . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n× n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.

Asymptotic Behavior of Global Classical Solutions of Quasilinear Hyperbolic Systems

We study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with weakly linearly degenerate characteristic fields. Based on the existence results on the global classical solution, we prove that, when ttends to infinity, the solution approaches a combination of C 1travelling wave solutions at algebraic rate , provided that the initial data decay at the rate as xtends to ± ∞, where μis a positive constant.

Generalized Rankine–Hugoniot Condition and Shock Solutions for Quasilinear Hyperbolic Systems

For a quasilinear hyperbolic system, we use the method of vanishing viscosity to construct shock solutions. The solution consists of two regular regions separated by a free boundary (shock). We use Melnikov's integral to obtain a system of differential/algebraic equations that governs the motion of the shock. For Lax shocks in conservation laws, these equations are equivalent to the Rankine–Hugoniot condition. For under compressive shocks in conservation laws, or shocks in non-conservation systems, the Melnikov-type integral obtained in this paper generalizes the Rankine–Hugoniot condition. Under some generic conditions, we show that the initial value problem of shock solutions can be solved as a free boundary problem by the method of characteristics.

Generalized Solutions of Quasi-linear Hyperbolic Systems of Partial Differential-Functional Equations

Generalized Solutions of Quasi-linear Hyperbolic Systems of Partial Differential-Functional Equations

Separation of solutions of quasilinear hyperbolic systems in the neighborhood of a large discontinuity

We consider solutions of quasilinear hyperbolic systems of any dimension in one space variable. Locally, our solutions are differentiable excapt for a single jump discontinuity, either an entropy shock or a contacr discontinuity, which may be of any strengrth. With some additional assumptions. we show that the map of the initrial into the solutions at some later time is continuous in L1