Related papers: Finite-time singularity formation for scalar stret…
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down…
The study of dynamic singularity formation in spacetime, focusing on scalar field collapse models, is analysed. We revisit key findings regarding open spatial topologies, concentrating on minimal conditions necessary for singularity and…
We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(\Delta u+|\nabla u|^2u) -b u\wedge \Delta u &\ \mbox{ in }\ {\mathbb…
In this paper we construct smooth, non-radial solutions of the defocusing nonlinear Schr\"odinger equation that develop an imploding finite time singularity, both in the periodic setting and the full space.
The final fate of the spherically symmetric collapse of a perfect fluid which follows the $\gamma$-law equation of state and adiabatic condition is investigated. Full general relativistic hydrodynamics is solved numerically using a retarded…
We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{\"o}dinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law''…
The future finite-time singularities emerging in alternative gravity dark energy models are classified and studied in Jordan and Einstein frames. It is shown that such singularity may occur even in flat spacetime for the specific choice of…
In this paper, the space-fractional Schr\"{o}dinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened'…
We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth…
A global extension theorem is established for isotropic singularities in polytropic perfect fluid Bianchi space-times. When an extension is possible, the limiting behaviour of the physical space-time near the singularity is analysed.
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…
We investigate the role of convection on its large time behavior of 3D incompressible Euler equations. In \cite{HL09a}, we constructed a new 3D model by neglecting the convection term from the reformulated axisymmetric Navier-Stokes…
In this paper we study the compressible magnetohydrodynamics equations in three dimensions, which offer a good model for plasmas. Formation of singularity for C1-solution in finite time is proved with axisymmetric initial data. The key…
We investigate gradient flows of some homogeneous functionals in R^N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong…
The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…
This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to…
It is shown that there are static spacetimes with timelike curvature singularities which appear completely nonsingular when probed with quantum test particles. Examples include extreme dilatonic black holes and the fundamental string…
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - \Delta u = u^{p}$ in three space dimensions. We show that for generic large initial…
Numerical simulations of the approach to the singularity in vacuum spacetimes are presented here. The spacetimes examined have no symmetries and can be regarded as representing the general behavior of singularities. It is found that the…
This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to…