Journal of Hyperbolic Differential Equations

This paper is devoted to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions or, more generally, nonconstant monotonic bounded functions as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multi-type version of the sticky particle dynamics and we obtain the existence of global weak solutions via a compactness argument. We then derive a ${\mathrm{\text{L}}}^p$ stability estimate on the particle system which is uniform in the number of particles. This allows us to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161(1) (2005) 223–342]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all order, which extends the classical ${\mathrm{\text{L}}}^1$ estimate and generalizes to diagonal systems a result by Bolley, Brenier and Loeper [Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ. 2(1) (2005) 91–107] in the scalar case. Our results are established without any smallness assumption on the variation of the data, and we only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.

We give a proof of the non-blow-up of the Yang–Mills curvature on arbitrary curved space-times using the Klainerman–Rodnianski parametrix combined with suitable Grönwall type inequalities. While the Chruściel–Shatah argument requires a control on two derivatives of the Yang–Mills curvature, we can get away by controlling only one derivative instead, and we propose a new gauge-independent proof on sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds.

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable $t\to +\infty$. Our results are then interpreted in the framework of conservation laws on the Bohr compact.