Bournaveas, Nikolaos

Carbery, A.

Candy, Timothy Lars

Engineering and Physical Sciences Research Council (EPSRC)

2013-10-22T13:41:18Z

2013-10-22T13:41:18Z

2012-11-28

We investigate the local and global well-posedness of a variety of nonlinear Dirac type equations with null structure on R1+1. In particular, we prove global existence in L2 for a nonlinear Dirac equation known as the Thirring model. Local existence in Hs for s > 0, and global existence for s > 1/2 , has recently been proven by Selberg-Tesfahun where they used Xs,b spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara-Nakanishi-Tsugawa, we prove local existence in the scale invariant class L2 by using null coordinates. Moreover, again using null coordinates, we prove almost optimal local wellposedness for the Chern-Simons-Dirac equation which extends recent work of Huh. To prove global well-posedness for the Thirring model, we introduce a decomposition which shows the solution is linear (up to gauge transforms in U(1)), with an error term that can be controlled in L∞. This decomposition is also applied to prove global existence for the Chern-Simons-Dirac equation. This thesis also contains a study of bilinear estimates in Xs,b± (R2) spaces. These estimates are often used in the theory of nonlinear Dirac equations on R1+1. We prove estimates that are optimal up to endpoints by using dyadic decomposition together with some simplifications due to Tao. As an application, by using the I-method of Colliander-Keel-Staffilani-Takaoka-Tao, we extend the work of Tesfahun on global existence below the charge class for the Dirac-Klein- Gordon equation on R1+1. The final result contained in this thesis concerns the space-time Monopole equation. Recent work of Czubak showed that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in Hs(R2) for s > 1/4 . Here we show that the Monopole equation has null structure in Lorenz gauge, and use this to prove local well-posedness for large initial data in Hs(R2) with s > 1/4.

http://hdl.handle.net/1842/7962

en

The University of Edinburgh

Nikolaos Bournaveas and Timothy Candy, Local well-posedness for the spacetime Monopole equation in Lorenz gauge, Nonlinear Diff. Equations and Applications 19 (2012), no. 1, 67–78.

Nikolaos Bournaveas, Timothy Candy, and Shuji Machihara, Local and global well-posedness for the Chern-Simons-Dirac system in one dimension, Differential Integral Equations (2012).

Timothy Candy, Bilinear estimates and applications to global well-posedness for the Dirac-Klein- Gordon equation, preprint (2011).

Timothy Candy, Global existence for an L2 critical nonlinear Dirac equation in one dimension, Adv. Differential Equations 16 (2011), no. 7-8, 643–666.

global well-posedness

Dirac equations

dispersive PDE

Local and global well-posedness for nonlinear Dirac type equations

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy