I will address the orbital stability problem for a class of dispersive Hamiltonian systems arising in nonlinear quantum media. In particular, I will elaborate on the construction of ground states as well as the threshold dynamics for the bi-harmonic NLS with a second-order perturbation. The proof relies on the profile decomposition that might be an efficient tool treating other systems like the magnetic NLS, fractional NLS and Dirac equations.
Abstract: We present recent results on time-scales separation in fluid
mechanics. The fundamental mechanism to detect in a precise
quantitative manner is commonly referred to as fluid mixing. Its
interaction with advection, diffusion and nonlocal effects produces a
variety of time-scales which explain many experimental and numerical
results related to hydrodynamic stability and turbulence theory.
Abstract: The Patlak-Keller-Segel models chemotactic aggregation, and is a nonlocal non-
linear reaction diffusion equation. In dimension two, the problem is said to be critical as the conservation law (the mass of the solution), is left invariant by the scaling symmetry of the equation. Solutions do not exist for all times for this equation, and singularities may form. As in other critical settings (nonlinear SchrÃ¶dinger, waves, wave maps etc...) the blow up phenomenon is not a truly self-similar one, but displays degenerate self-similarity: solutions concentrate at a point in finite time a bubble consisting of a stationary state, at a speed that is to be determined. After formal computations based essentially on matched asymptotics [Herrero-Velazquez, Dejak-Lushnikov-Yu-Ovchinnikov-Sigal, Dyachenko-Lushnikov-Vladimirova], a rigorous proof was provided [Raphael-Schweyer] relying on the precise control near the stationary state and the so-called tail dynamics. We propose a new proof of this result, conciliating the two approaches, which has at its heart a perturbative spectral problem encoding precisely the behaviour of the solution both near the stationary state and in the parabolic zone. This enables us to show the full nonradial stability of the dynamics, and to obtain refined asymptotics for the scaling law. This is joint work with T. Ghoul, N. Masmoudi and V.-T. Nguyen.
In this talk I will present some results concerning the analysis of the existence of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which model the dynamic of a viscous compressible fluid with degenerate viscosity and capillarity tensor. These kind of model are useful to study the dynamic of fluid near vacuum regions. A general theory of global existence is still missing, however for some particular cases of physical interest, it is possible to prove global existence of weak solutions. In particular, I will present two results regarding the case when the capillarity coefficient is constant and when the capillarity coefficient gives the Bohm potential. Moreover, in the cases under consideration the viscosity coefficient in the stress tensor are degenerating at the vacuum. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, LâAquila)
Abstract: If we want the solution to the Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer's distance set conjecture, etc. All these problems essentially ask how to control Schrodinger solutions on sparse and spread-out sets, which can be partially answered by several recent results derived from induction on scales and Bourgain-Demeter's decoupling theorem.
Abstract: Elliptic integrands are used to model anisotropic energies in
variational problems. These energies are employed in a variety of
applications, such as crystal structures, capillarity problems and
gravitational fields, to account for preferred inhomogeneous and
directionally dependent configurations. After a brief introduction to
variational problems involving elliptic integrands, I will present an
overview of the techniques I have developed to prove existence,
regularity and uniqueness properties of the critical points of
anisotropic energies. In particular, I will present the anisotropic
extension of Allard's rectifiability theorem and its applications to the
Plateau problem. Furthermore, I will describe the anisotropic
counterpart of Alexandrov's characterization of volume-constrained
critical points. Finally, I will mention some of my ongoing and future
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