Dynamics of a system of sticking particles of a finite size on the line

The continuous limit of large systems of particles of finite size on the line is described. The particles are assumed to move freely and stick under collision, to form compound particles whose mass and size is the sum of the masses and sizes of the particles before collision, and whose velocity is determined by conservation of linear momentum.Comment: 15 page

A Simplified Mathematical Model for the Formation of Null Singularities Inside Black Holes II

We study a simple system of two hyperbolic semi-linear equations, inspired by the Einstein equations. The system, which was introduced in gr-qc/0612136, is a model for singularity formation inside black holes. We show for a particular case of the equations that the system demonstrates a finite time blowup. The singularity that is formed is a null singularity. Then we show that in this particular case the singularity has features that are analogous to known features of models of black-hole interiors - which describe the inner-horizon instability. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes.Comment: 25 pages, 10 figure

From optimal transportation to optimal teleportation

The object of this paper is to study estimates of $\epsilon^{-q}W_p(\mu+\epsilon\nu, \mu)$ for small $\epsilon>0$. Here $W_p$ is the Wasserstein metric on positive measures, $p>1$, $\mu$ is a probability measure and $\nu$ a signed, neutral measure ($\int d\nu=0$). In [W1] we proved uniform (in $\epsilon$) estimates for $q=1$ provided $\int \phi d\nu$ can be controlled in terms of the $\int|\nabla\phi|^{p/(p-1)}d\mu$, for any smooth function $\phi$. In this paper we extend the results to the case where such a control fails. This is the case where if, e.g. $\mu$ has a disconnected support, or if the dimension of $\mu$ , $d$ (to be defined) is larger or equal $p/(p-1)$. In the later case we get such an estimate provided $1/p+1/d\not=1$ for $q=\min(1, 1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate. As an application we obtain H\"{o}lder estimates in $W_p$ for curves of probability measures which are absolutely continuous in the total variation norm . In case the support of $\mu$ is disconnected (corresponding to $d=\infty$) we obtain sharp estimates for $q=1/p$ ("optimal teleportation"): $$\lim_{\epsilon\rightarrow 0}\epsilon^{-1/p}W_p(\mu, \mu+\epsilon\nu) = \|\nu\|_{\mu}$$ where $\|\nu\|_{\mu}$ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of $\mu$, and the weights of the measure $\nu$ in each connected component of this support.Comment: 24 pages, 3 figure