Journal of Hyperbolic Differential Equations

In this note, we show that small-data asymptotically flat spherically-symmetric solutions to the Einstein–Vlasov system for massless particles are causally geodesically complete and decay appropriately. This extends previous results of Rein and Rendall in the massive case.

Let be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form . In particular, we establish results on the distribution kernels and mapping properties of e^-tL and (μ + L)^s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the W^s,p-Sobolev spaces on M and E. We also prove that L satisfies maximal L^p–L^q-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂_tu + Lu = F(t, x, u, ∇ u).

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

We study a nonlinear hyperbolic system of balance laws that arises from an entropy-based moment closure of a kinetic equation. We show that the corresponding homogeneous Riemann problem can be solved without smallness assumption, and we exhibit invariant regions.

The local decay of energy is established for solutions to certain linear, multidimensional symmetric hyperbolic systems, with constraints. The key assumptions are isotropy and nondegeneracy of the associated symbols. Examples are given, including Maxwell's equations and linearized elasticity. Such estimates prove useful in treating nonlinear perturbations.

We consider the timelike minimal surface problem in Minkowski spacetimes and show local and global existence of such surfaces having arbitrary dimension ≥ 2 and arbitrary co-dimension, provided they are initially close to a flat plane.

We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L¹ distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks — which are recognized as a source for non-uniqueness and instability. Our Haar–Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It generalizes the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L¹ norm upon their initial data, by exhibiting an L¹ functional controlling the distance between two solutions.

We develop a class of Hamiltonian-preserving numerical schemes for high frequency elastic waves in heterogeneous media. The approach is based on the high frequency approximation governed by the Liouville equations with singular coefficients due to material interfaces. As previously done by Jin and Wen [10, 12], we build into the numerical flux the wave scattering information at the interface, and use the Hamiltonian preserving principle to couple the wave numbers at both sides of the interface. This gives a class of numerical schemes that allows a hyperbolic CFL condition, is positive and l^∞ stable, and captures correctly wave scattering at the interface with a sharp numerical resolution. We also extend the method to curved interfaces. Numerical experiments are carried out to study the numerical convergence and accuracy.