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Decay estimates for the wave and Dirac equations with a magnetic potential
Author(s) -
D'Ancona Piero,
Fanelli Luca
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20152
Subject(s) - massless particle , dirac equation , wave equation , mathematics , norm (philosophy) , mathematical physics , dirac (video compression format) , mathematical analysis , physics , quantum mechanics , law , neutrino , political science
We study the electromagnetic wave equation and the perturbed massless Dirac equation on ℝ t × ℝ 3 :$$u_{tt}-(\nabla+iA(x))^{2}u+B(x)u=0, \;\;\;\; iu_{t}-{\cal D}u+V(x)u=0,$$ where the potentials A ( x ), B ( x ), and V ( x ) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution$$|u(t,x)|\leq {1 \over {t}}\|f\|_X$$ where the norm ‖ f ‖ X can be expressed as the weighted L 2 ‐norm of a few derivatives of the data f . © 2006 Wiley Periodicals, Inc.