Journal of Hyperbolic Differential Equations

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.

We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrödinger equation in high dimensions n ≥ 5, for general L² initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the ground state in the focusing case.

We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is analogous to what is found in certain Vaidya spacetimes. The analysis is carried out in Eddington–Finkelstein coordinates.

In this paper, a framework for the analysis of measure-valued solutions of the nonlinear structured population model is presented. Existence and Lipschitz dependence of the solutions on the model parameters and initial data are shown by proving convergence of a variational approximation scheme, defined in the terms of a suitable metric space. The estimates for a corresponding linear model are used based on the duality formula for transport equations. An extension of a Wasserstein metric to the measures with integrable first moment is proposed to cope with the nonconservative character of the model. This metric is compared with a bounded Lipschitz distance, also called a flat metric, and the results are discussed in the context of applications to biological data.

A suitably modified front-tracking algorithm is used to solve a class of Cauchy problems for a pair of conservation laws known as the model system for singular shocks or as the Keyfitz–Kranzer system. Initial data of large oscillation and including a finite number of positive Dirac masses in one of the dependent variables is permitted.

For such initial data, a distribution solution is constructed up to an arbitrary given finite time T. At any given time up to T, the constructed solution is uniformly bounded in space except possibly at a finite number of points. With respect to time, our solutions are weakly Lipshitz in the space of measures on ℝ. However, our solutions are not traditional weak solutions, because of the unavoidable appearance of singular shocks carrying finite Dirac mass. Furthermore, the bounded parts of our solutions are of lower than customary regularity, being of bounded 2-variation in space, pointwise but not uniformly with respect to time.