Journal of Hyperbolic Differential Equations

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and "hyperbolic" L^∞ weak-⋆ compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, …). Our results cover the case of non-Lipschitz nonlinearities.

Global existence of solutions is proved and asymptotic behavior is investigated, in the case of a positive cosmological constant and positive initial velocity of the cosmological expansion factor.

This paper is devoted to a hyperbolic 2-phase model for traffic flow on a network. The model is rigorously described and the existence of solutions is proved, without any restriction on the network geometry.

For a convex conservation law

In the 2-d setting, given an H¹ solution v(t) to the linear Schrödinger equation i∂_tv + Δ v = 0, we prove the existence (but not uniqueness) of an H¹ solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂_t u + Δu -|u|^p-1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H¹ solution u(t) to the defocusing Hartree equation i∂_t u + Δu -(|x|^-γ⋆|u|²)u = 0 for interaction powers 1 < γ < 2, such that ‖u(t) - v(t)‖_H¹ → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H¹ → H¹ for these equations. For NLS in 2-d, such wave operators are known to exist for p ≥ 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of Colliander–Grillakis–Tzirakis to a strategy devised by Nakanishi. For the Hartree equation, we prove a new correlation estimate following the method of Colliander–Grillakis–Tzirakis.

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic system appears naturally in the modeling of the dynamics of dislocation densities.

The initial-boundary value problem for the (viscous) nonlinear scalar conservation law is considered,