51 research outputs found
Blowup for Biharmonic NLS
We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS
with focusing nonlinearity given by for , where for and for ; and is some parameter to include a possible lower-order dispersion.
In the mass-supercritical case , we prove a general result on
finite-time blowup for radial data in in any dimension . Moreover, we derive a universal upper bound for the blowup rate for
suitable . In the mass-critical case , we
prove a general blowup result in finite or infinite time for radial data in
. 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 where takes values on the
two-dimensional unit sphere and (real line
case) or (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 (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 for the -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 .
In a previous version of this paper, we also stated uniqueness and
nondegeneracy of ground states for the -critical boson star equation in
, 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 space dimension.Comment: Replaced version; see also http://arxiv.org/abs/1009.404
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 for and for . We study traveling solitary waves of the
form with frequency ,
velocity , and some finite-energy profile ,
. We prove that traveling solitary waves for speeds 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 .
Finally, we discuss the energy-critical case when in dimensions
.Comment: 17 page
A Short Primer on the Half-Wave Maps Equation
We review the current state of results about the half-wave maps equation on
the domain with target . In particular, we focus
on the energy-critical case , where we discuss the classification of
traveling solitary waves and a Lax pair structure together with its
implications (e.\,g.~invariance of rational solutions and infinitely many
conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also
comment on the one-dimensional space-periodic case. Finally, we list some open
problem for future research.Comment: To appear in the proceeding of the Journ\'ees EDP 2018 (Obernai
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
Uniqueness of ground states for the L^2-critical boson star equation
We establish uniqueness of ground states for the -critical
boson star equation in
. The proof blends variational arguments with the harmonic extension to
the halfspace . Apart from uniqueness, we also show radiality of ground
states (up to translations) and the nondegeneracy of the linearization. Our
results provide an indispensable basis for the blowup analysis of the
time-dependent -critical boson star equation. The uniqueness proof can be
generalized to different fractional Laplacians and space
dimensions.Comment: Research announcement note; submitted for publicatio
Semi-Classical Dynamics in Quantum Spin Systems
We consider two limiting regimes, the large-spin and the mean-field limit, for the dynamical evolution of quantum spin systems. We prove that, in these limits, the time evolution of a class of quantum spin systems is determined by a corresponding Hamiltonian dynamics of classical spins. This result can be viewed as a Egorov-type theorem. We extend our results to the thermodynamic limit of lattice spin systems and continuum domains of infinite size, and we study the time evolution of coherent spin states in these limiting regime
Boson Stars as Solitary Waves
We study the nonlinear equation which is known to describe the dynamics of pseudo-relativistic boson stars in the mean-field limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, , for some and with speed |v| < 1, where c=1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments. In addition to their existence, we prove orbital stability of travelling solitary waves and pointwise exponential decay of in
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