Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

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Methods Of Modern Mathematical Physics

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)^s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabre and Sire , we show that the linear equation 
(−Δ)^s u + Vu=0 in R^N
has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrodinger operator H = (−Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space R^N_+^(+1), we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation 
(−Δ)^s Q + Q- │Q│^ɑ Q=0 in R^N 
for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland.

/pdf/uniqueness-of-radial-solutions-for-the-fractional-laplacian-3qutbu7q5f.pdf

Uniqueness of Radial Solutions for the Fractional Laplacian

The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s, John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called geons, but none were found. Instead, particle-like solutions were found in the late 1960s with the addition of a scalar field, and these were given the name boson stars. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single Killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.

/pdf/dynamical-boson-stars-2qp0jhfpoi.pdf

Dynamical boson stars

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation (−Δ)^sQ+Q−Q^(α+1)=0inR,
( − Δ ) s Q + Q − Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 s < 1 2 and 0 < α < ∞ for s≥12 s ≥ 1 2 . Here (−Δ)^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s=12 s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation.
As a technical key result in this paper, we show that the associated linearized operator L_+ = (−Δ)^s +1−(α+1)Q^α is non-degenerate; 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 non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

/pdf/uniqueness-of-non-linear-ground-states-for-fractional-2yfuffqk3l.pdf

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

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ 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 $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. 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 and Nondegeneracy of Ground States for $(-\Delta)^s Q + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}$

We prove local and global well-posedness for semi-relativistic, nonlinear Schrodinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with initial data in H s (ℝ3), $s \geqslant 1/2$ 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 L 2-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

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

We prove uniqueness of ground states Q∈H1∕2(ℝ3) for the pseudorelativistic Hartree equation,
 −
 Δ
 +
 m
 2
 Q
 −
 (
 x
 −
 1
 ∗
 Q
 2
 )
 Q
 =
 −
 μ
 Q
 ,
 in the regime of Q with sufficiently small L2-mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N= ∫ |Q|2≪1 except for at most countably many N.
 Our proof combines variational arguments with a nonrelativistic limit, leading 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 nonrelativistic Hartree equations.

/pdf/uniqueness-of-ground-states-for-pseudorelativistic-hartree-3md5m8kx5p.pdf

Enno Lenzmann

Papers

Uniqueness of Radial Solutions for the Fractional Laplacian

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

Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}$

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

Uniqueness of ground states for pseudorelativistic Hartree equations