Related papers: Singularity Formation in 2+1 Wave Maps
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…
A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes…
We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized…
This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional $\mathcal W_{\lambda_1, \lambda_2}$, the sum of the Willmore energy, $\lambda_1$ times the area, and $\lambda_2$ times…
The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation.…
In this paper, we study the formation of finite time singularities in the form of super norm blowup for a spatially inhomogeneous hyperbolic system. The system is related to the variational wave equations as those in [18]. The system posses…
To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent…
Spacetime singularities have been discovered which are physically much weaker than those predicted by the classical singularity theorems. Geodesics evolve through them and they only display infinities in the derivatives of their curvature…
We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold,…
We consider the late-time asymptotic behavior for solutions of Einstein's equations with the wave map matter. Solutions starting from small compactly supported $\ell$-equivariant initial data with $\ell\geq 1$ are shown to decay as…
The origin of Big Bang singularity in 3+1 dimensions can be understood in an exact string theory background obtained by an analytic continuation of a cigar like geometry with a nontrivial dilaton. In a T-dual conformal field theory picture…
The dynamics of singularity formation on the interface between two ideal fluids is studied for the Kelvin-Helmholtz instability development within the Hamiltonian formalism. It is shown that the equations of motion derived in the small…
In this article we consider large data Wave-Maps from $\mathbb{R}^{2+1}$ into a compact Riemannian manifold $(\mathcal{M},g)$, and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive)…
In this paper we study the Burgers equation with a nonlocal term of the form $Hu$ where $H$ is the Hilbert transform. This system has been considered as a quadratic approximation for the dynamics of a free boundary of a vortex patch. We…
In this article we initiate the study of 1+ 2 dimensional wave maps on a curved spacetime in the low regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical…
The propagation of a localized wave packet in the conical space-time created by a pointlike massive source in 2+1 dimensional gravity is analyzed. The scattering amplitude is determined and shown to be finite along the classical scattering…
We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…
The formation of singularities in the three-dimensional Euler equation is investigated. This is done by restricting the number of Fourier modes to a set which allows only for local interactions in wave number space. Starting from an initial…
This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps…
We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove…