This work connects the idea of a “blow-up” of a quiver with that of injectivity, showing that for a class of monic maps Φ, a quiver is Φ-injective if and only if all blow-ups of it are as well. This relationship is then used to characterize all quivers that are injective with respect to the natural embedding of Pn into Cn.

We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.

Radial blow-up, including inhomogeneous versions, of boundary faces of a manifold (always with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of generalized boundary blow-up in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial dat...

Self-similar blow-up behaviour for the fourth-order quasilinear p-Laplacian equation with source, ut = −(|uxx| uxx)xx + |u| u in R × R+, where n > 0, p > 1, is studied. Using variational setting for p = n+1 and branching techniques for p 6= n+1, finite and countable families of blow-up patterns of the self-similar form uS(x, t) = (T − t) − 1 p−1 f(y), where y = x/(T − t) , β = − p−(n+1) 2(n+2)(...

The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested in the situation where, starting from a smooth initial configuration, and after a first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in finite time due to the cumulative effec...

The structure of the blow-up in nite time of a solution of the Generalized Korteweg-de Vries equation arising from a perturbed unstable solitary wave is studied numerically. The computed solution is observed to blow-up in the L 1-norm in nite time by forming a spike of innnite height at x = x and at t = t. Scaled coordinates are introduced to examine the detailed structure of the solution in th...