Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws

This paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws. Under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws. The proof is due to the energy principle, using the weighted norms, the interaction of waves from various modes is treated by imposing suitable weight matrix.     

Asymptotic stability of viscous shock wave profiles for viscous conservation laws

This paper is concerned with the asymptotic stability of travelling wave solution to the two-dimensional steady isentropic irrotational flow with artificial viscosity. We prove that there exists a unique travelling wave solution up to a shift to the system if the end states satisfy both the Rankine–Hugoniot condition and Lax's shock condition, and that the travelling wave solution is stable if the initial disturbance is small.     

Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type

Riemann problems with initial data inside elliptic regions are quite different from those for hyperbolic systems. First, we have found that approximate solutions may present persistent oscillations, giving rise to a new type of (measure-valued) waves besides the usual (distributional) ones, shocks and rarefaction waves. Second, any local disturbance of a constant state inside the elliptic region will result in a non-trivial (distributional or, more generally, measure-valued) solution, which is independent of any particular choice of disturbance. For our numerical experiments, we establish two analytical results for testing convergence of finite difference schemes, and for determining expectation values of state functions with respect to the measure-valued solutions when oscillation waves occur. Numerical examples are presented to illustrate those interesting aspects, including the appearance of oscillation waves together with the analysis of the corresponding Young measures.     

Time-varying Riemann solvers for conservation laws on networks

We consider a conservation law on a network and generic Riemann solvers at nodes depending on parameters, which can be seen as control functions. Assuming that the parameters have bounded variation as functions of time, we prove existence of solutions to Cauchy problems on the whole network.     

The Riemann problem for general systems of conservation laws

An extended entropy condition (E) has previously been proposed, by which we have been able to prove uniqueness and existence theorems for the Riemann problem for general 2-conservation laws. In this paper we consider the Riemann problem for general n-conservation laws. We first show how the shock are related to the characteristic speeds. A uniqueness theorem is proved subject to condition (E), which is equivalent to Lax's shock inequalities when the system is “genuinely nonlinear.” These general observations are then applied to the equations of gas dynamics without the convexity condition P_vv(v, s) > 0. Using condition (E), we prove the uniqueness theorem for the Riemann problem of the gas dynamics equations. This answers a question of Bethe. Next, we establish the relation between the shock speed σ and the entropy S along any shock curve. That the entropy S increases across any shock, first proved by Weyl for the convex case, is established for the nonconvex case by a different method. Wendroff also considered the gas dynamics equations without convexity conditions and constructed a solution to the Riemann problem. Notice that his solution does satisfy our condition (E).     

Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate m|x|. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function.We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.     

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.     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts

A pair of kinematical conservation laws (KCL) in a ray coordinatesystem (,t) are the basic equations governing the evolutionof a moving curve in two space dimensions. We first study elementarywave solutions and then the Riemann problem for KCL when themetric g, associated with the coordinate designating differentrays, is an arbitrary function of the velocity of propagationm of the moving curve. We assume that m>1 (m is appropriatelynormalized), for which the system of KCL becomes hyperbolic.We interpret the images of the elementary wave solutions inthe (,t)-plane to the (x,y)-plane as elementary shapes of themoving curve (or a nonlinear wavefront when interpreted in aphysical system) and then describe their geometrical properties.Solutions of the Riemann problem with different initial datagive the shapes of the nonlinear wavefront with different combinationsof elementary shapes. Finally, we study all possible interactionsof elementary shapes.     

Asymptotic states for hyperbolic conservation laws with a moving source

We construct noninteracting wave patterns (i.e., asymptotic states) for a conservation law with a general moving source term. When nonlinear resonance occurs, which is the case when the characteristic speed is near the speed of the source, instability may result. We identify a stability criterion which is independent of the flux function. This is so, even if composite wave patterns exist, as may be the case for nonconvex flux functions. We study the general scalar model as well as transonic gas flows through a duct with varying cross section. For the latter case, noninteracting wave patterns for such a flow are constructed for arbitrary equations of state. It is shown that the stability of a wave pattern depends on the geometry of the duct, and not on the equation of the state. In particular, transonic steady shock waves along a converging duct are unstable, and flow along a diverging duct is always stable.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Pointwise convergence to shock waves for viscous conservation laws

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal L_p convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.