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- Tai-Ping Liu
- 1992

We study the limiting behavior of systems of hyperbolic conservation laws with stii relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N N systems that ensures the hyperbolicity of the reduced inviscid system. The… (More)

- TAI-PING LIU
- 2007

where u = u{x,t) E R , the flux f(u) is a smooth n-vector-valued function, and the viscosity B(u) is a smooth n x n matrix. We are interested in the stability of traveling waves, the "viscous shock waves", for (1). It is shown that when the initial data are a perturbation of viscous shock waves, then the solution converges to these viscous shock waves,… (More)

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation… (More)

Let u t + f(u) x = 0 be a strictly hyperbolic n n system of conservation laws, each characteristic eld being linearly degenerate or genuinely nonlinear. In this paper we explicitly deene a functional = (u; v), equivalent to the L 1 distance, which is \almost decreasing" i.e. ? u(t); v(t) ? ? u(s); v(s) O(") (t ? s) for all t > s 0; for every couple of… (More)

We consider the problem of 2D supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A long-standing natural conjecture is that… (More)

- TAI-PING LIU, TONG YANG
- 1999

We assume that the system is strictly hyperbolic, i.e. the matrix ∂f(u) ∂u has real and distinct eigenvalues λ1(u) < λ2(u) for all u under consideration, with the corresponding right eigenvectors ri(u), i = 1, 2. Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear [11], i.e. ri(u) · 5λi(u) ≡ 0 or ri(u) · 5λi(u) 6= 0,… (More)

- Tai-Ping Liu, Zhouping Xin
- 2005

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 timeasymptotically equivalent on the level of expansion waves. The result… (More)

We prove the ellipticity principle for selfsimilar potential flows for gas dynamics. We show that the interior of a pseudo-subsonic-sonic-region of a smooth solution must be pseudo-subsonic. In fact, the pseudo-Mach number is below that of a domain-dependent function which is < 1 in the interior and ≤ 1 on the boundary. Therefore the interior must stay… (More)

- Tai-Ping Liu, Shih-Hsien Yuy
- 1998

We show that the weak detonation waves for a combustion model of Rosales-Majda are nonlinearly stable. Because of the strongly non-linear nature of the wave, usual stability analysis of weakly nonlinear nature does not apply. The chemical switch on-oo is the main feature of nonlinearity. In particular, the propagation of the wave depends sensitively on the… (More)

We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear… (More)