Searcharxiv⌕ Search

arXiv subjects

Benjamin Dodson

Publications and source records attributed to Benjamin Dodson.

53 records · Page 3Linked to original sources
arXiv2016

Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{ö}dinger equation when $d = 2$

In this paper we prove that the defocusing, cubic nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{2})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in \cite{CKSTT4}. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.

math.AP↗
arXiv2015

The defocusing quintic NLS in four space dimensions

We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

math.AP↗
arXiv2014

Scattering for the radial 3d cubic wave equation

Consider the Cauchy problem for the radial cubic wave equation in 1+3 dimensions with either the focusing or defocusing sign. This problem is critical in $\dot{H}^{\frac{1}{2}} \times \dot{H}^{-\frac{1}{2}}$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time-interval of existence, then the solution must in fact be global-in-time and scatter to free waves as $t \to \pm \infty$.

math.AP↗
arXiv2014

Global well - posedness and scattering for the focusing, energy - critical nonlinear Schrödinger problem in dimension $d = 4$ for initial data below a ground state threshold

In this paper we prove global well - posedness and scattering for the focusing, energy - critical nonlinear Schrödinger initial value problem in four dimensions. Previous work proved this in five dimensions and higher using the double Duhamel trick. In this paper, using long time Strichartz estimates we are able to overcome the logarithmic blowup in four dimensions.

math.AP↗
arXiv2014

Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm

In this paper we study the focusing cubic wave equation in 1+5 dimensions with radial initial data as well as the one-equivariant wave maps equation in 1+3 dimensions with the model target manifolds $\mathbb{S}^3$ and $\mathbb{H}^3$. In both cases the scaling for the equation leaves the $\dot{H}^{\frac{3}{2}} \times \dot{H}^{\frac{1}{2}}$-norm of the solution invariant, which means that the equation is super-critical with respect to the conserved energy. Here we prove a conditional scattering result: If the critical norm of the solution stays bounded on its maximal time of existence, then the solution is global in time and scatters to free waves both forwards and backwards in infinite time. The methods in this paper also apply to all supercritical power-type nonlinearities for both the focusing and defocusing radial semi-linear equation in 1+5 dimensions, yielding analogous results.

math.AP↗
arXiv2013

A controlling norm for energy-critical Schrödinger maps

We consider energy-critical Schroedinger maps with target either the sphere S^2 or hyperbolic plane H^2 and establish that a unique solution may be continued so long as a certain space-time L^4 norm remains bounded. This reduces the large data global wellposedness problem to that of controlling this norm.

math.AP↗
arXiv2012

Bilinear Strichartz estimates for the Schr{ö}dinger map problem

In this paper we prove bilinear Strichartz estimates for a solution to the Schr{ö}dinger map problem whose size is small in the critical Strichartz space $| |\nabla|^{\frac{d - 2}{2}} ψ_{x} |_{L_{t,x}^{\frac{2(d + 2)}{d}}}$. These estimates will be useful in an upcoming paper in proving a local well - posedness result. Bilinear estimates make use of an argument similar to the argument found in Planchon and Vega (2009). We use the same gauges as in Bejenaru, Ionescu and Kenig (2007), Bejenaru et al. (2011), and Smith.

math.AP↗
arXiv2011

Global well-posedness and scattering for the mass critical nonlinear Schr{ö}dinger equation with mass below the mass of the ground state

In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$, $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q \|_{L^{2}(\mathbf{R}^{d})}$, where $Q$ is the ground state, and $d \geq 1$. We first establish an interaction Morawetz estimate that is positive definite when $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q \|_{L^{2}(\mathbf{R}^{d})}$, and has the appropriate scaling. Next, we will prove a frequency localized interaction Morawetz estimate similar to the estimates made in \cite{D2}, \cite{D3}, \cite{D4}. See also \cite{CKSTT4} for the energy critical case. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.

math.AP↗
arXiv2011

Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{ö}dinger equation when $d \geq 3$

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$ and $d \geq 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [10]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.

math.AP↗
arXiv2011

Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 1$

In this paper we prove that the defocusing, quintic nonlinear Schrödinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.

math.AP↗
arXiv2009

Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension

In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. We show that a unique solution exists for $u_{0} \in H^{s}(\mathbf{R})$, $s > {8/29}$. This improves the result in [13], which proved global well-posedness for $s > {1/3}$. The main new argument is that we obtain almost Morawetz estimates with improved error.

math.AP↗
arXiv2009

Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{ö}dinger equation in higher dimensions

In this paper, we consider the global well-posedness of the defocusing, $L^{2}$ - critical nonlinear Schr{ö}dinger equation in dimensions $n \geq 3$. Using the I-method, we show the problem is globally well-posed in $n = 3$ when $s > {2/5}$, and when $n \geq 4$, for $s > \frac{n - 2}{n}$. We combine energy increments for the I-method, interaction Morawetz estimates, and almost Morawetz estimates to prove the result.

math.AP↗
arXiv2004

Infrared Photometry of NGC 6791

We present deep JHK photometry of the old and metal-rich open cluster NGC 6791. The photometry reaches below the main sequence turn-off to K = 16.5 mag. We combine our photometry with that from Stetson, Bruntt, & Grundahl (2003) to provide color-magnitude diagrams showing K vs. J-K, K vs. V-K, and V vs. V-K. We study the slope of the red giant branch in the infrared, but find that it is not a useful metallicity indicator for the cluster, nor any metal-rich cluster that lacks a well-populated red giant branch, because it is not linear, as has often been assumed, in K vs. J-K. The mean color of the red horizontal branch/red clump stars provide an estimate the cluster reddening, E(B-V) = 0.14 +/- 0.04 mag for [Fe/H] = +0.4 +/- 0.1. The mean magnitudes of these stars also provide a good distance estimate, (m-M)_0 = 13.07 +/- 0.04. Finally, we find that the isochrones of Yi, Kim, & Demarque (2003) provide optimal fits in V vs. B-V and V-K and K vs. J-K and V-K for such values if [Fe/H] lies between +0.3 and +0.5 (with a slight preference for +0.5) and ages between 9 Gyrs ([Fe/H] = +0.3) and 7.5 Gyrs ([Fe/H] = +0.5).

astro-ph↗