# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 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 $s>\frac{1}{2}$, has recently been proven by S. Selberg and A. Tesfahun [Differ. Integral Equ. 23, No. 3–4, 265–278 (2010; Zbl 05944745)] where they used ${X}^{s,b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of S. Machihara, K. Nakanishi and K. Tsugawa [Kyoto J. Math. 50, No. 2, 403–451 (2010; Zbl 05735955)] 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 $s\ge 0$.
##### MSC:
 35Q41 Time-dependent Schrödinger equations, Dirac equations 35A01 Existence problems for PDE: global existence, local existence, non-existence