Show simple item record

On some nonlinear partial differential equations for classical and quantum many body systems

dc.contributorMarkowich, Peter A.
dc.contributorSparber, Christof
dc.creatorMarahrens, Daniel
dc.date.accessioned2018-11-24T23:17:43Z
dc.date.available2013-01-22T11:20:11Z
dc.date.available2018-11-24T23:17:43Z
dc.date.issued2012-11-13
dc.identifierhttp://www.dspace.cam.ac.uk/handle/1810/244203
dc.identifierhttps://www.repository.cam.ac.uk/handle/1810/244203
dc.identifier.urihttp://repository.aust.edu.ng/xmlui/handle/123456789/3152
dc.description.abstractThis thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.
dc.languageen
dc.publisherUniversity of Cambridge
dc.publisherDepartment of Applied Mathematics and Theoretical Physics
dc.publisherClare Hall
dc.rightshttp://creativecommons.org/licenses/by-nc-sa/2.0/uk/
dc.rightsAttribution-NonCommercial-ShareAlike 2.0 UK: England & Wales
dc.subjectNonlinear Schrödinger equation
dc.subjectGlobal existence
dc.subjectOptimal control
dc.subjectHydrodynamic limit
dc.subjectZero range process
dc.subjectNonlinear diffusion equation
dc.titleOn some nonlinear partial differential equations for classical and quantum many body systems
dc.typeThesis


Files in this item

FilesSizeFormatView
thesis.pdf2.250Mbapplication/pdfView/Open

This item appears in the following Collection(s)

Show simple item record