Enno Lenzmann

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) > 0 for the nonlinear equation (−∆)Q+Q−Q = 0 in R, where 0 < s < 1 and 0 < α < 4s 1−2s for s < 1/2 and 0 < α < ∞ for s > 1/2. Here (−∆) denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s = 1/2 and α = 1 in [Acta Math., 167 (1991), 107–126]. As a technical key result in this paper, we show that the associated linearized operator L+ = (−∆)+1− (α+1)Q is nondegenerate; i. e., its kernel satisfies kerL+ = span {Q}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation [image] modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ Cmath image (ℝ3), with negative energy, we prove blowup of u(t, x) in the H1/2-norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0.

Well-posedness for Semi-relativistic Hartree Equations of Critical Type

We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations

Uniqueness of ground states for pseudorelativistic Hartree equations

i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)

On the continuum limit for discrete NLS with long-range lattice interactions

with initial data in Hs(ℝ3),

MINIMIZERS FOR THE HARTREE-FOCK-BOGOLIUBOV THEORY OF NEUTRON STARS AND WHITE DWARFS

s \geqslant 1/2

Effective dynamics for boson stars

. Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing F(u), which arise in the quantum theory of boson stars, we derive global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L2-norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class.

On Blowup for Time-Dependent Generalized Hartree-Fock Equations

We prove uniqueness of ground states Q ∈ H 1/2 (R 3 ) for the pseudo-relativistic Hartree equation, p −� + m 2 Q − ` |x| 1 ∗ |Q| 2 ´ Q = −µQ, in the regime of Q with sufficiently smallL 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau in (CMP 112 (1987), 147-174) holds true at least for N = R |Q| 2 ≪ 1 except for at most countably many N. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard- Pekard or Schrodinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativis- tic Hartree equations.

ON SINGULARITY FORMATION FOR THE L 2 -CRITICAL BOSON STAR EQUATION

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice