Dispersive Equations with Multiple Potentials /

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Bibliographic Details
Author / Creator:Chen, Gong, author.
Ann Arbor : ProQuest Dissertations & Theses, 2017
Description:1 electronic resource (369 pages)
Format: E-Resource Dissertations
Local Note:School code: 0330
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/11715118
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Other authors / contributors:University of Chicago. degree granting institution.
Notes:Advisors: Wilhelm Schlag Committee members: Wilhelm Schlag; Luis E. Silvestre.
Dissertation Abstracts International, Volume: 78-12(E), Section: B.
Summary:In this thesis, we study dispersive equations with several moving potentials, a.k.a. charge transfer Hamiltonians. We mainly focus on two models: the Schödinger equation and the wave equation. We prove linear estimates and analyze nonlinear models based on them.
In Chapter 1, we briefly survey the historical backgrounds and motivation for the main results in this thesis.
Then, in Chapter 2, we prove Strichartz estimates for scattering states of the scalar charge transfer models in R³. More precisely, we study the time-dependent charge transfer Hamiltonian [special characters omitted]
with rapidly decaying smooth potentials Vj( x), say, exponentially decaying and a set of mutually non-parallel constant velocities v¯j. We prove Strichartz estimates for the evolution [special characters omitted].
Based on the idea of the proof of Strichartz estimates which follows [22, 51], we also show the energy of the whole evolution is bounded independent of time without using the phase space method, for example, in [27]. One can easily generalize our argument to Rⁿ for n ≥ 3. We also discuss the extension of above results to matrix charge transfer models in R³.
Next, in Chapter 3, we prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in R³: [special characters omitted].
where Vj(x)'s are rapidly decaying smooth potentials and {v¯j] is a set of distinct constant velocities such that [special characters omitted] .
The energy estimate and the local energy decay of a scattering state are also established. In order to study nonlinear multisoltion systems, we will present the inhomogeneous generalizations of Strichartz estimates. As an application of our results, we show that scattering states indeed scatter to solutions to the free wave equation.
In Chapter 4, we study the endpoint reversed Strichartz estimates along general time-like trajectories for wave equations in R³. We also discuss some applications of the reversed Strichartz estimates and the structure formula of wave operators to the wave equation with one potential. These techniques are useful to analyze the stability problem of traveling solitons.
In Chapter 5, lastly, we construct multisoliton solutions to the defocusing energy critical wave equation with potentials in R³ based on regular and reversed Strichartz estimates developed in Chapter 3 for wave equations with charge transfer Hamiltonians. We also show the asymptotic stability of multisoliton solutions. The multisoliton structures with both stable and unstable solitons are covered. Since each soliton decays slowly with rate [special characters omitted], the interactions among the solitons are strong. Some reversed Strichartz estimates and local decay estimates for the charge transfer model are established to handle strong interactions.