×

A vector field method for some nonlinear Dirac models in Minkowski spacetime. (English) Zbl 1464.35276

The authors study global existence and asymptotic decay properties for nonlinear massless Dirac equations.
By using the Lie derivatives of spinors with respect to Killing vector fields and conformal Killing vector fields, the authors first obtain the generalized commuting vector field set adapted to the Dirac operator. By the conservation law of the charge current and total angular momentum, as well as the weighted Sobolev estimates, the generalized commuting vector field set gives the optimal decay for the linear massless spinor fields.
Furthermore, by exploiting the Dirac null structure, the authors prove the global existence and asymptotic decay properties for small-norm solutions to the nonlinear Dirac equations with cubic nonlinearities in space dimension two and with quadratic nonlinearities in space dimension three.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
15A66 Clifford algebras, spinors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alinhac, Serge, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145, 3, 597-618 (2001) · Zbl 1112.35341
[2] Alinhac, Serge, The null condition for quasilinear wave equations in two space dimensions, II, Am. J. Math., 123, 6, 1071-1101 (2001) · Zbl 1112.35342
[3] Bachelot, Alain, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. IHP, Phys. Théor., 48, 4, 387-422 (1988) · Zbl 0672.35071
[4] Bournaveas, Nikolaos; Candy, Timothy, Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not., 2016, 22, 6735-6828 (2015) · Zbl 1404.35380
[5] Christodoulou, Demetrios, Global solutions of nonlinear hyperbolic equations for small initial data, Commun. Pure Appl. Math., 39, 2, 267-282 (1986) · Zbl 0612.35090
[6] Christodoulou, Demetrios; Klainerman, Sergiu, Asymptotic properties of linear field equations in Minkowski space, Commun. Pure Appl. Math., 43, 2, 137-199 (1990) · Zbl 0715.35076
[7] D’Ancona, Piero; Foschi, Damiano; Selberg, Sigmund, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc., 9, 4, 877-899 (2007) · Zbl 1187.35191
[8] Fatibene, Lorenzo; Ferraris, Marco; Francaviglia, Mauro; Godina, Marco, A geometric definition of Lie derivative for spinor fields, (Proceedings of “6th International Conference on Differential Geometry and Its Applications (1995)), 549 · Zbl 0858.53035
[9] Godina, Marco; Matteucci, Paolo, The Lie derivative of spinor fields: theory and applications, Int. J. Geom. Methods Mod. Phys., 2, 02, 159-188 (2005)
[10] Katayama, Soichiro; Kubo, Hideo, Global existence for quadratically perturbed massless Dirac equations under the null condition, (Fourier Analysis (2014), Springer), 253-262 · Zbl 1318.35095
[11] Klainerman, Sergiu, Global existence for nonlinear wave equations, Commun. Pure Appl. Math., 33, 1, 43-101 (1980) · Zbl 0405.35056
[12] Klainerman, Sergiu, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math., 38, 3, 321-332 (1985) · Zbl 0635.35059
[13] Klainerman, Sergiu, The null condition and global existence to nonlinear wave equations, Lect. Appl. Math., 23, 293-326 (1986) · Zbl 0599.35105
[14] Klainerman, Sergiu, Remarks on the global Sobolev inequalities in the Minkowski space \(\mathbb{R}^{n + 1} \), Commun. Pure Appl. Math., 40, 1, 111-117 (1987) · Zbl 0686.46019
[15] Klainerman, Sergiu; Machedon, Matei, Finite energy solutions of the Yang-Mills equations in \(\mathbb{R}^{3 + 1} \), Ann. Math., 142, 1, 39-119 (1995) · Zbl 0827.53056
[16] Klainerman, Sergiu; Machedon, Matei, On the regularity properties of a model problem related to wave maps, Duke Math. J., 87, 3, 553-589 (1997) · Zbl 0878.35075
[17] Kosmann, Yvette, Dérivées de Lie des spineurs, C. R. Hebd. Séances Acad. Sci., Sér. A, 262, 5, 289 (1966) · Zbl 0136.18402
[18] Kosmann, Yvette, Dérivées de Lie des spineurs, Ann. Mat. Pura Appl., 91, 1, 317-395 (1971) · Zbl 0231.53065
[19] Jiongyue Li, Yunlong Zang, Asymptotic properties of the spinor field and the application to nonlinear Dirac models, preprint, 2019.
[20] Michael, Beals; Bézard, Max, Low regularity local solutions for field equations, Commun. Partial Differ. Equ., 21, 1-2, 79-124 (1996) · Zbl 0852.35098
[21] Pecher, Hartmut, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13, 2 (2014) · Zbl 1296.35174
[22] Pecher, Hartmut, Corrigendum of “Local well-posedness for the nonlinear Dirac equation in two space dimensions”, Commun. Pure Appl. Anal., 14, 2, 737-742 (2015) · Zbl 1314.35123
[23] Penrose, Roger; Rindler, Wolfgang, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry, vol. 2 (1986), Cambridge University Press · Zbl 0591.53002
[24] Shu, Wei-Tong, Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space, Commun. Math. Phys., 140, 3, 449-480 (1991) · Zbl 0735.53060
[25] Soler, Mario, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1, 10, 2766 (1970)
[26] Thaller, Bernd, The Dirac Equation (1992), Springer Berlin Heidelberg · Zbl 0765.47023
[27] Thirring, Walter E., A soluble relativistic field theory, Ann. Phys., 3, 1, 91-112 (1958) · Zbl 0078.44303
[28] Tzvetkov, Nickolay, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22, 1, 193-211 (1998) · Zbl 0945.35075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.