These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^1$, construction of various exotic blow up rates in $H^1$, including grow up in infinite time.

We consider the critical nonlinear Schrödinger equation $i{u}_t=-\Delta u-{\left|u\right|}^{\frac{4}{N}}u$ with initial condition $u(0,x)=u_0$ in dimension $N$. For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is $u_0$ such that ${lim}_{t\to T\<+\infty }{\left|u_x\left(t\right)\right|}_{L^2}=+\infty$. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data...

We consider the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$. We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

Si considera il problema di Cauchy per l'equazione (cf. [1]): $${\varphi }_{tt}-{\varphi }_{xx}-{\varphi }_x^2{\varphi }_{xx}+\sin \varphi =0\left(x,t\right)\in \mathbb{R}\times {\mathbb{R}}_{+}.$$ Nella prima parte di questo articolo si dimostra, per dati iniziali particolari, un risultato di «blow-up» della soluzione classica locale (in tempo), seguendo le idee introdotte in [8], [2] ed [4]. Nella seconda parte, viene utilizzato il metodo di compattezza per compensazione (cf. [13], [10] ed [5]) ed una estensione del principio delle regioni invarianti (cf. [12]) per dimostrare l'esistenza di una soluzione debole globale entropica....

We study compressible isentropic Navier-Stokes-Poisson equations in ${ℝ}^3$. With some appropriate assumptions on the density, velocity and potential, we show that the classical solution of the Cauchy problem for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing will blow up in finite time. The proof is based on a contradiction argument, which relies on proving the conservation of total mass and total momentum.

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 singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that...

We will show the blow-up of smooth solutions to the Cauchy problems for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing and compressible bipolar isentropic Navier-Stokes-Poisson equations in arbitrary dimensions under some restrictions on the initial data. The key of the proof is finding the relations between the physical quantities and establishing some differential inequalities.

We give a sufficient condition under which the solutions of the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential blow up. The method is a modified variational approach, in the spirit of the work by Ibrahim et al. [Anal. PDE 4 (2011), 405-460].

The focusing nonlinear Schrödinger equation (NLS) with confining harmonic potential $i{\partial }_{t}u+1/2\Delta u-1/2\left|x\right|²u=-{\left|u\right|}^{4/(d-2)}u,x\in {ℝ}^d$, is considered. By modifying a variational technique, we shall give a sufficient condition under which the corresponding solution blows up.

One of the most interesting phenomena exhibited by ultracold quantum gases is the appearance of vortices when the gas is put in rotation. The talk will bring a survey of some recent progress in understanding this phenomenon starting from the many-body ground state of a Bose gas with short range interactions. Mathematically this amounts to describing solutions of a linear Schrödinger equation with a very large number of variables in terms of a nonlinear equation with few variables and analyzing the...