Blowup, solitary waves and scattering for the fractional nonlinear Schrödinger equation

In this thesis we are concerned with the rigorous analysis for an evolution problem arising in mathematical physics: the nonlinear Schrödinger equation with power-type nonlinearity involving the fractional Laplace operator (fractional NLS). We are particularly interested in the long-time dynamics of this nonlocal equation, and study three basic problems of fundamental importance.
First, we shall deduce sufficient criteria for blowup of radial solutions of the focusing problem in the mass-supercritical and mass-critical cases. The conditions are given in terms of inequalities between a combination of the (kinetic) energy and mass of the initial datum, and that of the ground state for the corresponding elliptic equation. Using a new method to deal with the nonlocality of the fractional Laplacian, a localized virial argument enables us to conclude blowup in finite and infinite time, respectively.
Second, we consider a special class of nondispersive solutions of the focusing fractional NLS: the traveling solitary waves. Introducing an appropriate variational problem, we establish the existence of their stationary profiles (boosted ground states). In order to deal with the lack of compactness, we use the technique of compactness modulo translations adapted to the fractional Sobolev spaces. In the case of algebraic (even integer-order) nonlinearities, we derive symmetries of boosted ground states with respect to the boost direction, relying on symmetric decreasing rearrangements in Fourier space. Moreover, we show a non-optimal spatial decay of these profiles at infinity.
Third and finally, we concentrate on the asymptotics of global solutions of the defocusing problem. To have a full range of Strichartz estimates available, we restrict to the radially symmetric case. We construct the wave operator on the radial subclass of the energy space, and show asymptotic completeness. Thus we infer that any radial solution scatters to a linear solution in infinite time. Similarly to the blowup theory, this is done in the spirit of monotonicity formulae: taking a suitable virial weight and using the favourable sign of the defocusing nonlinearity, we develop a lower bound for the Morawetz action. The resulting decay estimates permit us to build a satisfactory scattering theory in the radial case.