Nonlocal interactions by repulsive-attractive potentials: radial ins/stability

In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins/stability are sharp.Comment: 42 pages, 7 figure

Geometric Analysis and General Relativity

This article discusses methods of geometric analysis in general relativity, with special focus on the role of "critical surfaces" such as minimal surfaces, marginal surface, maximal surfaces and null surfaces.Comment: to appear in Elsevier Encyclopedia of Mathematical Physics, 200

Remarks on LpL^{p}-vanishing results in geometric analysis

We survey some $L^{p}$-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schr\"{o}dinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed.Comment: 18 pages. Some oversights corrected. Accepted for publication in International Journal of Mathematic

The region with trapped surfaces in spherical symmetry, its core, and their boundaries

We consider the region $\mathscr{T}$ in spacetime containing future-trapped closed surfaces and its boundary \B, and derive some of their general properties. We then concentrate on the case of spherical symmetry, but the methods we use are general and applicable to other situations. We argue that closed trapped surfaces have a non-local property, "clairvoyance", which is inherited by \B. We prove that \B is not a marginally trapped tube in general, and that it can have portions in regions whose whole past is flat. For asymptotically flat black holes, we identify a general past barrier, well inside the event horizon, to the location of \B under physically reasonable conditions. We also define the core $\mathscr{Z}$ of the trapped region as that part of $\mathscr{T}$ which is indispensable to sustain closed trapped surfaces. We prove that the unique spherically symmetric dynamical horizon is the boundary of such a core, and we argue that this may serve to single it out. To illustrate the results, some explicit examples are discussed, namely Robertson-Walker geometries and the imploding Vaidya spacetime.Comment: 70 pages, 14 figures. Figure 6 has been replaced, and corrected. Minor changes around Propositions 10.3 and 10.4, and some typos correcte