Blowup for Biharmonic NLS

We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS with focusing nonlinearity given by $i \partial_t u = \Delta^2 u - \mu \Delta u -|u|^{2 \sigma} u$ for $(t,x) \in [0,T) \times \mathbb{R}^d$, where $0 < \sigma <\infty$ for $d \leq 4$ and $0 < \sigma \leq 4/(d-4)$ for $d \geq 5$; and $\mu \in \mathbb{R}$ is some parameter to include a possible lower-order dispersion. In the mass-supercritical case $\sigma > 4/d$, we prove a general result on finite-time blowup for radial data in $H^2(\mathbb{R}^d)$ in any dimension $d \geq 2$. Moreover, we derive a universal upper bound for the blowup rate for suitable $4/d < \sigma < 4/(d-4)$. In the mass-critical case $\sigma=4/d$, we prove a general blowup result in finite or infinite time for radial data in $H^2(\mathbb{R}^d)$. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems. In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.Comment: Revised version. Corrected some minor typos, added some remarks and included reference [12

A Lax Pair Structure for the Half-Wave Maps Equation

We consider the half-wave maps equation $$\partial_t \vec{S} = \vec{S} \wedge |\nabla| \vec{S},$$ where $\vec{S}= \vec{S}(t,x)$ takes values on the two-dimensional unit sphere $\mathbb{S}^2$ and $x \in \mathbb{R}$ (real line case) or $x \in \mathbb{T}$ (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in \cite{LS,Zh}, which formally arises as an effective evolution equation in the classical and continuum limit of Haldane-Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target $\mathbb{H}^2$ (hyperbolic plane).Comment: Included an explicit calculation of the Lax operator for a single speed soliton. Corrected some minor typo

Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs

We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the well-studied Hartree-Fock theories in atomic physics.Comment: 43 pages. Third and final version. Section 5 revised and main result extended. To appear in Duke Math. Journal

On ground states for the L^2-critical boson star equation

We consider ground state solutions $u \geq 0$ for the $L^2$-critical boson star equation \sqrt{-\Delta} \, u - \big (|x|^{-1} \ast |u|^2 \big) u = -u \quad {in $\R^3$}. We prove analyticity and radial symmetry of $u$. In a previous version of this paper, we also stated uniqueness and nondegeneracy of ground states for the $L^2$-critical boson star equation in $\R^3$, but the arguments given there contained a gap. However, we refer to our recent preprint \cite{FraLe} in {\tt arXiv:1009.4042}, where we prove a general uniqueness and nondegeneracy result for ground states of nonlinear equations with fractional Laplacians in $d=1$ space dimension.Comment: Replaced version; see also http://arxiv.org/abs/1009.404

Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system

An analysis is given of particlelike nonlinear bound states in the Newtonian limit of the coupled Einstein-Dirac system introduced by Finster, Smoller and Yau. A proof is given of existence of these bound states in the almost Newtonianian regime, and it is proved that they may be approximated by the energy minimizing solution of the Newton-Schr\"odinger system obtained by Lieb

On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

We consider nonlinear half-wave equations with focusing power-type nonlinearity i \pt_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} with exponents $1 < p < \infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the form $$u(t,x) = e^{i\omega t} Q_v(x-vt)$$ with frequency $\omega \in \R$, velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$, $Q_v \not \equiv 0$. We prove that traveling solitary waves for speeds $|v| \geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator \sqrt{-\DD+m^2} and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d \geq 2$.Comment: 17 page

Blow-Up for Nonlinear Wave Equations describing Boson Stars

We consider the nonlinear wave equation $i \partial_t u= \sqrt{-\Delta + m^2} u - (|x|^{-1} \ast |u|^2) u$ on \RR^3 modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, u_0(x) \in C^\infty_{\mathrm{c}}(\RR^3), with negative energy, we prove blow-up of $u(t,x)$ in $H^{1/2}$-norm within a finite time. Physically, this phenomenon describes the onset of "gravitational collapse" of a boson star. We also study blow-up in external, spherically symmetric potentials and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability for ground state solitary waves at rest if $m=0$.Comment: final version; to appear in Comm. Pure Appl. Math; 14 page

Solitary waves for the Hartree equation with a slowly varying potential

We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $\epsilon \le \sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\log h|/h$ and errors of size $\epsilon + h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer-Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer-Zworski to more general inital conditions.Comment: 28 page

On Blowup for time-dependent generalized Hartree-Fock equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.Comment: 24 page

Uniqueness and Nondegeneracy of Ground States for (−Δ)^sQ+Q−Q^(α+1)=0 in R

We prove uniqueness of ground state solutions Q = Q(|x|)≥0 for the nonlinear equation (−Δ)^sQ + Q − Q^(α+)1 = 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 (−Δ)^s 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_+ = (−Δ)^s + 1− (α+1)Q^α is nondegenerate; i.,e., its kernel satisfies ker L_+ = 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