Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces

A note on critical point and blow-up rates for singular and degenerate parabolic equations

In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...

Blow up of the critical norm for some radial L super critical non linear Schrödinger equations

We consider the nonlinear Schrödinger equation iut = −∆u−|u|p−1u in dimension N ≥ 3 in the L super critical range N+3 N−1 ≤ p < N+2 N−2 . The corresponding scaling invariant space is Ḣc with 1 2 ≤ sc < 1 and this covers the physically relevant case N = 3, p = 3. The existence of finite time blow up solutions is known. Let u(t) ∈ Ḣc ∩ Ḣ be a radially symmetric blow up solution which blows up at ...

Slow Blow up Solutions for Certain Critical Wave Equations

We describe in this article two recent results [11], [12], obtained by the author jointly with W. Schlag and D. Tataru, about singular solutions for the critical wave maps equation, as well as the critical focussing semilinear wave equation. Specifically, the first result [11] establishes for the first time the conjectured formation of singularities for co-rotational wave maps into the sphere S...

Fuchsian Equations in Sobolev Spaces and Blow-up

We construct solutions of u = e which blow-up precisely on a given space-like hypersurface of class H. For this purpose, we prove a general existence theorem for Fuchsian PDE in Sobolev spaces. The precise relation between the regularity of the data and that of the solution is shown to involve logarithmic symbols, in a model situation. A few further results on power nonlinearities are also incl...

Regularity criteria for the 3D magneto-micropolar fluid equations in Besov spaces with negative indices