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Benjamin Dodson

Publications and source records attributed to Benjamin Dodson.

At least 37 records · Page 2Linked to original sources
arXiv2020

The profile decomposition for the hyperbolic Schrödinger equation

In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot H}^{\frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.

math.AP↗
arXiv2020

The nonlinear Schrodinger equation on Z and R with bounded initial data: examples and conjectures

We study the nonlinear Schrödinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

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arXiv2019

Global well-posedness and scattering for nonlinear Schr{ö}dinger equations with algebraic nonlinearity when $d = 2, 3$, $u_{0}$ radial

In this paper we discuss global well - posedness and scattering for some initial value problems that are $L^{2}$ supercritical and $\dot{H}^{1}$ subcritical, with radial data. We prove global well - posedness and scattering for radial data in $H^{s}$, $s > s_{c}$, where the problem is $\dot{H}^{s_{c}}$ - critical. We make use of the long time Strichartz estimates of \cite{D2} to do this.

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arXiv2019

Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation

We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.

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arXiv2019

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{3} < s < 1$. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schrödinger equation, which is inspired by certain function spaces used in the study of the Schrödinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{2} < s < 1$.

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arXiv2018

Scattering for defocusing energy subcritical nonlinear wave equations

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.

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arXiv2018

Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions. In an appendix, we establish a corresponding quantitative result for the energy-critical problem.

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arXiv2018

Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation with scaling super-critical initial data. The proof is based on the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data.

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arXiv2017

Global well - posedness for the defocusing, cubic, nonlinear wave equation in three dimensions for radial initial in $\dot{H}^{s} \times \dot{H}^{s - 1}$, $s > \frac{1}{2}$

In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $(\dot{H}^{s} \times \dot{H}^{s - 1}) \cap (\dot{H}^{1/2} \times \dot{H}^{-1/2})$ for some $s > \frac{1}{2}$, then the cubic initial value problem is globally well - posed. We use the I - method and the long time Strichartz estimates. This method is quite similar to the method used in [D2].

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arXiv2017

On Scattering for Small Data of 2+1 Dimensional Equivariant Einstein-Wave Map System

We consider the Cauchy problem of 2+1 equivariant wave maps coupled to Einstein's equations of general relativity and prove that two separate (nonlinear) subclasses of the system disperse to their corresponding linearized equations in the large. Global asymptotic behaviour of 2+1 Einstein-wave map system is relevant because the system occurs naturally in 3+1 vacuum Einstein's equations.

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