Related papers: Finite-time singularity formation for scalar stret…
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the…
In this paper, we examine a flat FLRW spacetime with a scalar field potential and show by applying Osgood's criterion to the Einstein field equations that all such models, irrespective of the particular choice of potential develop…
In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. In particular, the flow…
In this article we study a class of generalised linear systems of difference equations with given non-consistent initial conditions and infinite many solutions. We take into consideration the case that the coefficients are square constant…
One or two negative mass singularities are found to occur in static inhomogeneous spatially closed solutions to the Einstein equations. The singularities produce a positive Komar mass, and this decreases the size of the cosmological…
In this paper, we consider the singularity formation of smooth solutions for the compressible radially symmetric Euler equations. By applying the characteristic method and the invariant domain idea, we show that, for polytropic ideal gases…
We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u -…
T. Tao constructed an averaged Navier-Stokes equations which obey an energy identity. Nevertheless, he proved that smooth solutions can blow up in finite time. This demonstrates that any proposed positive solution to the famous regularity…
The conditions are found that all solutions of a systems dynamic equations on time scales tends to finite limits as $t\to\infty$.
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^\alpha$…
Consider a spherically symmetric spacelike slice through a spherically symmetric spacetime. One can derive a universal bound for the optical scalars on any such slice. The only requirement is that the matter sources satisfy the dominant…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show…
We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to…
The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the…
We prove finite-time singularity formation for Lipschitz continuous solutions of the inviscid porous medium equation which vanish on the boundary of the domain. As the density vanishes on the boundary of the domain, the full regularizing…
One way to understand more about spacetime singularities is to construct solutions of the Einstein equations containing singularities with prescribed properties. The heuristic ideas of the BKL picture suggest that oscillatory singularities…
We show that all known naked singularities in spherically symmetric self-similar spacetimes arise as a result of singular initial matter distribution. This is a result of the peculiarity of the coordinate transformation that takes these…
We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational…
Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…