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Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension. (English) Zbl 1229.35225
Summary: We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the {\it W. E. Thirring} model [Ann. Phys. 3, 91--112 (1958; Zbl 0078.44303)]. Local existence in $H^s$ for $s>0$, and global existence for $s>\frac12$, has recently been proven by {\it S. Selberg} and {\it A. Tesfahun} [Differ. Integral Equ. 23, No. 3--4, 265--278 (2010; Zbl 1240.35362)] where they used $X^{s,b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of {\it S. Machihara, K. Nakanishi} and {\it K. Tsugawa} [Kyoto J. Math. 50, No. 2, 403--451 (2010; Zbl 1248.35170)] 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 $s\ge 0$.

35Q41Time-dependent Schrödinger equations, Dirac equations
35A01Existence problems for PDE: global existence, local existence, non-existence
Full Text: arXiv