GLOBAL EXISTENCE FOR AN L-2 CRITICAL NONLINEAR DIRAC EQUATION IN ONE DIMENSION

Timothy Candy

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Abstract

We prove global existence from L-2 initial data for a nonlinear Dirac equation known as the Thirring model [12]. Local existence in H-s for s > 0, and global existence for s > 1/2, has recently been proven by Selberg and Tesfahun in [9] where they used X-s,X-b spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, [7] we first prove local existence in L-2 by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of L-2 norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all s >= 0.

Original languageEnglish
Pages (from-to)643-666
Number of pages24
JournalAdvances in Differential Equations
Volume16
Issue number7-8
Publication statusPublished - 2011

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