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We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time t0, where both the density and the velocity gradient become unbounded. Aim of this paper is to provide an asymptotic description of the solution beyond the time of singularity formation. For t > t_0 we show that a singular curve is formed, where the mass has positive density w.r.t. 1-dimensional Hausdorff measure. The system of equations describing the behavior of the singular curve is not hyperbolic. Working within a class of analytic data, local solutions can be constructed using a version of the Cauchy-Kovalevskaya theorem. For this purpose, by a suitable change of variables we rewrite the evolution equations as a first order system of Briot-Bouquet type, to which a general existence-uniqueness theorem can then be applied. 

In this paper, we consider the Cauchy problem for pressureless gases in two space dimensions with generic smooth initial data (density and velocity). These equations give rise to singular curves, where the mass has positive density w.r.t. 1-dimensional Hausdorff measure. We observe that the system of equations describing these singular curves is not hyperbolic. For analytic data, local solutions are constructed using a version of the Cauchy-Kovalevskaya theorem. We then study the interaction of two singular curves, in generic position. Finally, for a generic initial velocity field, we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed. 

Abstract Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small BV functions which are global solutions of this equation. For any small BV initial data, such global solutions are known to exist. Moreover, they are known to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even BV ) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak- BV uniqueness result, where only one of the solutions is supposed to be small BV , and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with 2 unknowns. The method is $$L^2$$ L 2 based, and builds up from the theory of a-contraction with shifts, where suitable weight functions a are generated via the front tracking method. 

We report an experimental realization of a modified counterfactual communication protocol that eliminates the dominant environmental trace left by photons passing through the transmission channel. Compared to Wheeler’s criterion for inferring past particle paths, as used in prior protocols, our trace criterion provides stronger support for the claim of the counterfactuality of the communication. We verify the lack of trace left by transmitted photons via tagging the propagation arms of an interferometric device by distinct frequency-shifts and finding that the collected photons have no frequency shift which corresponds to the transmission channel. As a proof of principle, we counterfactually transfer a quick response code image with sufficient fidelity to be scanned with a cell phone.