I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

I shall present some recent results obtained with F. de la Hoz about the selfsimilar solutions of the Binormal Flow, also known as the Vortex Filament Equation. Some connections with the transfer of energy in the case when the filament is a regular polygon will be also made.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

By combining the Kenig-Merle approach with a suitable inequality proved by Tao we deduce that solutions to gKdV, in the L^2-supercitical regime, scatter to free waves for large times.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We discuss some essential features of solitons for the energy-critical half-wave maps equation. Furthermore, we will present a Lax pair structure and explain its applications to understanding the dynamics. The talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

In this talk we will discuss some recent improvements on well-known decay estimates for nonlinear dispersive and wave equations in 1D with supercritical decay, or no decay at all. Using Virial estimates, we will get local decay where standard dispersive techniques are not available yet. These are joint works with M.-A. Alejo, M. Kowalczyk, Y. Martel, F. Poblete, and J.-C. Pozo.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We will see various notion of nondispersive solution in the case of the energy criticl wave equation and applications.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider the $L^2$ critical gKdV equation with a saturated perturbation. In this case, all $H^1$ solution are global in time. Our goal is to classify the asymptotic dynamics for solutions with initial data near the ground state. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $gamma^{-2/(q-1)}$, as $gammarightarrow0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+infty$; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves. This extends the result of classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $gamma=0$) by Martel, Merle and Rapha"el. It also provides a way to consider the continuation properties after blow-up time for $L^2$ -crtitical gKdV equations.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider two non-variational semilinear parabolic systems, with different diffusion constants between the two components. The reaction terms are of power type in the first system. They are of exponential type in the second. Using a formal approach, we derive blow-up profiles for those systems. Then, linearizing around those profiles, we give the rigorous proof, which relies on the two-step classical method: (i) the reduction of the problem to a finite-dimensional one, then, (ii) the proof of the latter thanks to Brouwer's lemma. In comparison with the standard semilinear heat equation, several technical problems arise here, and new ideas are needed to overcome them. This is a joint work with T. Ghoul and V.T. Nguyen from NYU Abu Dhabi.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

This talk is about singularity formation for solutions to $$ (*) partial_{t}u+upa_x u-pa_{yy}u=0, (x,y) in mathbb R^2 $$ which is a simplified model of Prandtl's boundary layer equation. Note that it reduces to Burgers equation for $y$-independent solutions $u(t,x,y)=v(t,x)$. We will first recast the well-known shock formation theory for Burgers equation using the framework of self-similar blow-up. This will provide us with an analytic framework to study the effect of the transversal viscosity. The main result (still work in progress) is the construction and precise description of singular solutions to $(*)$. This is joint work with T.E. Ghoul and N. Masmoudi.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We solve the cubic nonlinear Schrödinger equation on $mathbb R$ with initial data a sum of Diracs. Then we describe some consequences for a class of singular solutions of the binormal flow, that is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. This is a joint work with Luis Vega.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider the linear wave equation outside a compact, strictly convex obstacle in R^3 with smooth boundary and we show that the linear wave flow satisfies the dispersive estimates as in R^3 (which is not necessarily the case in higher dimensions).

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)