Related papers: Blowups and Resolution
We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably…
We construct examples of blowup from smooth data for complex-valued solutions to linear uniformly parabolic equations in dimension $n \geq 2$, which are exactly as irregular as parabolic energy estimates allow.
In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new…
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…
In this short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using comparison principle and the results of deterministic parabolic equations, we obtain blowup results of solutions for stochastic…
We study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive…
Langmuir waves take place in a quasi-neutral plasma and are modeled by the Zakharov system. The phenomenon of collapse, described by blowing up solutions plays a central role in their dynamics. We present in this article a review of the…
The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of…
Blowing up a point p in a manifold M builds a new manifold M' in which p is replaced by the projectivization of the tangent space of M at p. This well-known operation also applies to fixed points of diffeomorphisms, yielding continuous…
For Liouville equation with quantized singular sources, the non-simple blowup phenomenon has been a major difficulty for years. It was conjectured by the first two authors that the non-simple blowup phenomenon does not occur if the equation…
An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…
We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up…
This paper deals with the blow-up properties of positive solutions to a system of two heat equations.
We analyze the blowup (finite-time singularity) in inviscid shell models of convective turbulence. We show that the blowup exists and its internal structure undergoes a series of bifurcations under a change of shell model parameter. Various…
This paper gives a general nonsingularity condition for a blowup along an ideal, related to Gaussian curvature.
A new proof for the embedded resolution of surface singularities in a three-dimensional smooth ambient space over algebraically closed fields of arbitrary characteristic. The proof makes use of an upper semicontinuous resolution invariant…
We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincar\'e inequality, might be employed to completely linearize blow-ups along at least one…
The paper contains several regularity results and blow-up criterions for a surface growth model, which seems to have similar properties to the 3D Navier-Stokes, although it is a scalar equation. As a starting point we focus on energy…
An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime…
Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be…