Narrow Results

We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n≥3, including Hartree equations with potential V(x)= λ|x|^-γ with γ<1. For 1/2<γ<1 we prove the existence of modified wave operators with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.

We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.

We consider nonlinear Choquard equation

By $I$-method, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation

We investigate the well-posedness in the generalized Hartree equation $i{u}_t+\mathrm{\Delta }u+(|x{|}^{-(N-\gamma )}\ast |u{|}^p)|u{|}^{p-2}u=0$, $x\in {ℝ}^N$, $0<\gamma <N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

In the 2-d setting, given an H¹ solution v(t) to the linear Schrödinger equation i∂_tv + Δ v = 0, we prove the existence (but not uniqueness) of an H¹ solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂_t u + Δu -|u|^p-1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H¹ solution u(t) to the defocusing Hartree equation i∂_t u + Δu -(|x|^-γ⋆|u|²)u = 0 for interaction powers 1 < γ < 2, such that ‖u(t) - v(t)‖_H¹ → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H¹ → H¹ for these equations. For NLS in 2-d, such wave operators are known to exist for p ≥ 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of Colliander–Grillakis–Tzirakis to a strategy devised by Nakanishi. For the Hartree equation, we prove a new correlation estimate following the method of Colliander–Grillakis–Tzirakis.

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential $i{u}_t+\mathrm{\Delta }u+(|x{|}^{-b}\ast |u{|}^p)|u{|}^{p-2}u=0,x\in {ℝ}^N$. We establish the local well-posedness at the nonconserved critical regularity ${Ḣ}^{s_c}$ for $s_c\ge 0$, which also includes the energy-supercritical regime $s_c>1$ (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the $H^1$ well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

We consider the problems of local and global in time existence of the solution to the Schrödinger-Poisson system (or the Hartree equation) in ℝ³. We prove that if the solution at t = 0 has a bounded H^1/2-norm, then it remains bounded in some time interval [0, T], which together with the charge conservation law immediately imply global existence. Thus, we extend the previous results for this type of nonlinear Schrödinger equation.

We review some recent results concerning the derivation of effective evolution equations from first principle quantum dynamics. In particular, we discuss the derivation of the Hartree equation for mean field systems and the derivation of the Gross-Pitaevskii equation for the time evolution of Bose-Einstein condensates.