Tai-Ping Liu

Hyperbolic conservation laws with relaxation

The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman-Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.

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 Laxs shock inequalities when the system is “genuinely nonlinear.” These general observations are then applied to the equations of gas dynamics without the convexity condition Pvv(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).

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcys law time-asymptotically. Our model may also be viewed as an elastic model with damping.

Quasilinear hyperbolic systems

We construct global solutions for quasilinear hyperbolic systems and study their asymptotic behaviors. The systems include models of gas flows in a variable area duct and flows with a moving source. Our analysis is based on a numerical scheme which generalizes the Glimm scheme for hyperbolic conservation laws.

The deterministic version of the Glimm scheme

The Glimm scheme for solving hyperbolic conservation laws has a stochastic feature; it depends on a random sequence. The purpose of this paper is to show that the scheme converges for any equidistributed sequence. Thus the scheme becomes deterministic.

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 Lp 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.

The entropy condition and the admissibility of shocks

Abstract The author proposed (Trans. Amer. Math. Soc. 199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations 18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [Uspehi Mat. Nauk 14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [Comm. Pure Appl. Math. 23 (1970), 867–884] proved that a shock satisfies Laxs shock inequalities [cf. Comm. Pure Appl. Math. 14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.

Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations

It is shown that expansion waves for the compressible Navier-Stokes equations are nonlinearly stable. The expansion waves are constructed for the compressible Euler equations based on the inviscid Burgers equation. Our result shows that Navier-Stokes equations and Euler equations are time-asymptotically equivalent on the level of expansion waves. The result is proved using the energy method, making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas.

Well-posedness theory for hyperbolic conservation laws

The paper presents a well-posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimms existence theory and discuss the principle of nonlinear through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L1(x) distance between the two solutions and is time-decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L1(x) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional.

On the vacuum state for the isentropic gas dynamics equations

p(0) =p’(O) = 0, and p’ > 0,p” > 0 in P > 0. (4 For example, one can take p(p) = py, y > 1. By definition, a vacuum state is any portion of the x t plane in which p = 0. In Section 2 we show that a vacuum state must be bounded by rarefaction waves. We then study the ways in which it is possible to go from a nonvacuum to a vacuum state (see [IS]). There are, in fact two distinct types of vacuums which we call compression and rarefaction vacuums; the compression vacuums differ from the rarefaction vacuums in that they give rise to compression waves which eventually form shock