Journal of Hyperbolic Differential Equations

Conical shock wave is generated when a sharp conical projectile flies supersonically in the air. We study the linear stability and existence of steady conical shock waves in supersonic flow for the equations of complete Euler system in 3D non-isentropic gas-dynamics.

We investigate the system of compressible Navier–Stokes equations with hyperbolic heat conduction, i.e. replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time $\tau$, global smooth solution exists for small initial data. Moreover, as $\tau$ goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier–Stokes equations.

We consider the Cauchy problem for the 3D Euler system with damping coupled to radiation through two singular limits. Assuming suitable smallness hypotheses for the data, we prove that each of these two problems admits a unique smooth solution.

We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $({ℝ}^{3+1},g)$ with time dependent inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{(t,x)|\left|x\right|\le R\}$ and approaches the Minkowski metric slowly as $\left|x\right|\to \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data.

The Cauchy problem for the Chern–Simons–Higgs system in the $(2+1)$-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an $L_x^6{L}_t^2$-estimate for solutions of the wave equation, and also takes advantage of a null condition.

We compute explicitly the peakon–antipeakon solution of the Camassa–Holm (CH) equation $u_t-u_{txx}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ in the non-symmetric and $\alpha$-dissipative case. The solution experiences wave breaking in finite time, and the explicit solution illuminates the interplay between the various variables.

In this paper we study the flow of an inviscid fluid composed by three different phases. The model is a simple hyperbolic system of three conservation laws, in Lagrangian coordinates, where the phase interfaces are stationary. Our main result concerns the global existence of weak entropic solutions to the initial-value problem for large initial data.

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is