We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in half space. The whole space analogue, due to Jia and Šverák [21], is central tool two of authors' recent work on quantitative Lx3 blow-up criteria [7]. main difficulty that non-local effects pressure are much stronger than As an application, we demonstrate critical norm must concentrate at scales ∼T⁎−t ...

We study obstructions for a quasi-regular mapping f : M → N of finite degree between Riemannian manifolds to blow up on or collapse on a non-trivial part of the boundary of M .

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source, (0.1) ut = −(|u|u)xxxx + |u|u in R × R+, n > 0, p > 1, is studied. Countable and finite families of similarity blow-up patterns of the form uS(x, t) = (T − t)− 1 p−1 f(y), where y = x/(T − t) , β = p−(n+1) 4(p−1) , which blow-up as t → T− < ∞, are described. These solutions explain key features of regional (f...

The tight transverse emittance budget for the LHC beam imposes severe constraints on the blow-up produced by injection errors. The modifications to the beam distributions and their possible parameterisation in terms of blow-up factors are discussed for betatron and dispersion mismatch as well as for injection offsets. The cases of initial uniform or gaussian distributions are considered in deta...

Let $$G=(V,E)$$ be a locally finite, connected and weighted graph. We prove that, for graph satisfying curvature dimension condition $$CDE'(n,0)$$ uniform polynomial volume growth of degree m, all non-negative solutions the equation $$\partial _tu=\Delta u+u^{1+\alpha }$$ blow up in finite time, provided that $$\alpha =\frac{2}{m}$$ . also consider blow-up problem under certain conditions initi...