Related papers: Dispersion and collapse of wave maps

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1 + n}$ to the unit sphere $\mathbb{S}^2$, which obey a system of non-linear wave equations. The nonlinearity enjoys…

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be…

In this paper, for compressible Euler equations in multiple space dimensions, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including…

We study finite-time blowup for a nonlinear wave equation for maps from the Minkowski space $\mathbb{R}^{1+d}$ into the 1-sphere $\mathbb{S}^1$, whose nonlinearity exhibits a null-form structure. We construct, for every dimension $d \geq…

We construct a family of asymptotically flat Cauchy initial data for the Einstein vacuum equations that contain no trapped surfaces, yet whose future development admits multiple causally independent trapped surfaces. Assuming the weak…

We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $d\geq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on…

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$…

The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant…

We study the vacuum Cauchy problem for K-essence, i.e. cosmologically relevant scalar-tensor theories that involve first-order derivative self-interactions, and which pass all existing gravitational wave bounds. We restrict to spherical…

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately…

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the…

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the…

We study corotational wave maps from $(1+4)$-dimensional Minkowski space into the $4$-sphere. We prove the stability of an explicitly known self-similar wave map under perturbations that are small in the critical Sobolev space.

We present the first results for Cauchy nonlinear evolution of 3D, nonaxisymmetric distorted black holes. We focus on the extraction and verification of 3D waveforms determined by numerical relativity. We show that the black hole evolution…

We study an internal structure of (2+1)-dimensional black hole with the neutral scalar matter in the spherically symmetric geometry by using a quantum theory of gravity which holds in the both vicinities of the singularity and the apparent…

The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schr\"{o}dinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist…

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 $t_0$, where…

This work is on the Cauchy problem for critical wave maps coupled to Einstein's equations of general relativity. The main result of this work is the proof that the energy of the Einstein-equivariant wave map system does not concentrate…

In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to…