Duality solutions for pressureless gases,monotone scalar conservation laws,and uniqueness

We introduce for the system of pressureless gases a new notion of solution, which consist in interpreting the system as two nonlinearly coupled linear equations. We prove In this setting existence of solutions for the Cauchy Problem, as well as uniqueness under optimal conditions on initlaffata. The proofs rely on the detailed study of the relations between pressureless gases, tie dynamics of sticky particles and nonlinear scalar conservation laws with monotone initial data. We prove for the latter problem that monotonicit implies uniqueness. and a generalization of Oleinik's entropy condition     

Delta Shock Waves as Limits of Vanishing Viscosity for 2-D Steady Pressureless Isentropic Flow

The Riemann problem for the two-dimensional steady pressureless isentropic flow in gas dynamics is solved completely. The Riemann solutions contain two kinds: delta-shock solutions and vacuum solutions. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions is established. Moreover, the stability of delta-shock solution to a reasonable viscous perturbation is proven. The numerical results coinciding with the theoretical solutions are also presented.     

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 , 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 C¹ 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 . 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.     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.     

Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al., which consists of a constrained pressureless gas dynamics system and can be derived from the Aw-Rascle model under the constraint condition ρ?ρ^? by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form

     

Higher order shadow waves and delta shock blow up in the Chaplygin gas

The introductory part of this paper contains an overview of known results about elementary and delta shock solutions to Riemann problem for well known Chaplygin gas model (nowadays used in cosmological theories for dark energy) in terms of entropic shadow waves. Shadow waves are introduced in [17] and they are represented by shocks depending on a small parameter ε with unbounded amplitudes having a distributional limit involving the Dirac delta function. In a search for admissible solutions to all possible cases of mutual interactions of waves arising from double Riemann initial data we found same cases that cannot be resolved with already known types of elementary or shadow wave solutions. These cases are resolved by introducing a sequence of higher order shadow waves depending on integer powers of ε. It is shown that such waves have a distributional limit but only until some finite time T.     

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe δ-shock wave generation from continuous initial data in the case of triangular conservation law system arising from “generalized pressureless gas dynamics model.” We use smooth approximations in the weak sense that are more general than small viscosity approximations.     

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 , 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.     

Elliptic regularization for the semi-linear telegraph system

The solution of the semi-linear telegraph system is compared with the solution of an elliptic regularization, to which one associates two-point boundary conditions. An asymptotic approximation for the solution of the elliptic regularization is constructed. The method employed here is the boundary function method due to Vishik and Lyusternik. The problem is singularly perturbed of elliptic-hyperbolic type. To conduct this analysis, high regularity with respect to t for the solutions of both problems is required. Finally, the order of this approximation is found in different spaces of functions.     

A complete global solution to the pressure gradient equation

We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants. The global existence of a smooth solution was established in Dai and Zhang [Z. Dai, T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal. 155 (2000) 277-298] up to the free boundary of vacuum. We prove that the vacuum boundary is the coordinate axes.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

Vanishing viscosity for hemivariational inequalities modeling dynamic problems in elasticity

In this paper we consider a mathematical model describing a dynamic linear elastic contact problem with nonmonotone skin effects. The subdifferential multivalued and multidimensional reaction–displacement law is employed. We treat an evolution hemivariational inequality of hyperbolic type which is a weak formulation of this mechanical problem. We establish a result on the existence of solutions to the Cauchy problem for the hemivariational inequality. This result is a consequence of an existence theorem for second order evolution inclusion. For the latter, using the parabolic regularization method, we obtain the solution as a limit when the viscosity term tends to zero.     

Asymptotic stability of viscous contact wave for the one-dimensional compressible viscous gas with radiation

In this paper, we study the large-time behavior of solutions of the Cauchy problem to a one-dimensional Navier-Stokes-Poisson coupled system, modeling the dynamics of a viscous gas in the presence of radiation. When the far field states are suitably given, and the corresponding Riemann problem for the Euler system admits only a contact discontinuity wave solution with the far field states as Riemann initial data. Then, we can define a “viscous contact wave” for such a Navier-Stokes-Poisson coupled system. Based on elementary energy methods and ellipticity of the equation of the radiation flux, we can prove the “viscous contact wave” is stable provided the strength of the contact discontinuity wave and the perturbation of the initial data are suitably small.     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

A stochastic model of AIDS and condom use

In this paper we introduce stochasticity into a model of AIDS and condom use via the technique of parameter perturbation which is standard in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.