Journal of Hyperbolic Differential Equations

A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for $(1+1)$-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.

It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.

In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying $𝕌(1)$ symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the $H^3$ level. In addition, we prove a blow-up criterium at the $H^2$ level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized $𝕌(1)$ symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.

We study the Ericksen–Leslie hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity and scattering for small and smooth initial data near equilibrium are proved for the case that the system is a nonlinear coupling of compressible Navier–Stokes equations with wave map to ${𝕊}^2$. The main strategy relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. Unlike the incompressible model, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role, which is a particular feature of compressible model.

We consider the follow-the-leader particle approximation scheme for a $1d$ scalar conservation law with non-negative compactly supported $L^{\infty }$ initial datum and with a $C^1$ concave flux, which is known to provide convergence towards the entropy solution $\rho$ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ${\rho }^n$ is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on $f$ (which imply concavity of $f$) we prove that the continuum version ${(f(\rho )/\rho )}_x\le 1/t$ of said condition allows to select a unique weak solution, despite ${(f(\rho )/\rho )}_x\le 1/t$ is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition $f^{\prime }{(\rho )}_x\le 1/t$. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case $f(\rho )=\rho (A-{\rho }^{\gamma })$ the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.