Growth-in-time of higher Sobolev norms of solutions to the 1D Dirac–Klein–Gordon system

2019 ◽  
Vol 16 (02) ◽  
pp. 313-332 ◽  
Author(s):  
Achenef Tesfahun
Keyword(s):  
Space Dimension ◽  
Higher Order ◽  
Sobolev Norms ◽  
Klein Gordon ◽  
One Space Dimension

We study the growth-in-time of higher order Sobolev norms of solutions to the Dirac–Klein–Gordon (DKG) equations in one space dimension. We show that these norms grow at most polynomially-in-time. The main ingredients in the proof are the upside-down [Formula: see text]-method which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and bilinear null-form estimates.

scholarly journals LOW REGULARITY WELL-POSEDNESS OF THE DIRAC–KLEIN–GORDON EQUATIONS IN ONE SPACE DIMENSION

2008 ◽  
Vol 10 (02) ◽  
pp. 181-194 ◽  
Author(s):  
SIGMUND SELBERG ◽  
ACHENEF TESFAHUN
Keyword(s):  
Dirac Operator ◽  
Space Dimension ◽  
One Dimension ◽  
Well Posedness ◽  
Low Regularity ◽  
Klein Gordon ◽  
One Space Dimension

We extend recent results of Machihara and Pecher on low regularity well-posedness of the Dirac–Klein–Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections for the Dirac operator. We also show that the result is best possible up to endpoint cases, if one iterates in Bourgain–Klainerman–Machedon spaces.

Slow motion in higher-order systems and Γ-convergence in one space dimension

2001 ◽  
Vol 44 (1) ◽  
pp. 33-57 ◽  
Author(s):  
William D. Kalies ◽  
Robert C.A.M. VanderVorst ◽  
Thomas Wanner
Keyword(s):  
Space Dimension ◽  
Slow Motion ◽  
Higher Order ◽  
Γ Convergence ◽  
One Space Dimension
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