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.
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