Large time well-posedness for a Dirac–Klein–Gordon system

In this paper we prove well posedness for a system coupling a nonlinear Dirac with a Klein-Gordon equation that represents a toy model for the Helium atom with relativistic corrections: the wave function of the electrons interacts with an electric field generated by a nucleus with a given charge density. One of the main ingredients we need is a new family of Strichartz estimates for time dependent perturbations of the Dirac equation: these represent a result of independent interest.

[1]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[2]  K. Nakanishi,et al.  Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation , 2005 .

[3]  L. Fanelli,et al.  Virial identity and weak dispersion for the magnetic Dirac equation , 2009, 0911.1287.

[4]  F. Cacciafesta.  Global small solutions to the critical radial Dirac equation with potential , 2011, 1103.0186.

[5]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[6]  L. Hörmander,et al.  Remarks on the Klein-Gordon equation , 1987 .

[7]  Seungly Oh,et al.  The Kato-Ponce Inequality , 2013, 1303.5144.

[8]  Dispersive estimates for principally normal pseudodifferential operators , 2004, math/0401234.

[9]  N. Bournaveas,et al.  Global well-posedness for the massless cubic Dirac equation , 2014, 1407.0655.

[10]  K. Tsutaya,et al.  Scattering theory for the Dirac equation with a non-local term , 2009, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  F. Cacciafesta,et al.  A Dirac field interacting with point nuclear dynamics , 2017, Mathematische Annalen.

[12]  S. Herr.  The Cubic Dirac Equation: Small Initial Data in H 1 ( R 3 ) , 2014 .

[13]  K. Nakanishi,et al.  Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation , 2003 .

[14]  Lucie Baudouin,et al.  Existence and Regularity of the Solution of a Time Dependent Hartree-Fock Equation Coupled with a Classical Nuclear Dynamics , 2005 .

[15]  L. Fanelli,et al.  Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials , 2007, math/0702362.

[16]  S. Herr,et al.  CONDITIONAL LARGE INITIAL DATA SCATTERING RESULTS FOR THE DIRAC–KLEIN–GORDON SYSTEM , 2017, Forum of Mathematics, Sigma.

[17]  A. Kiselev,et al.  Maximal Functions Associated to Filtrations , 2001 .

[18]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[19]  Eric S'er'e,et al.  LOCAL SMOOTHING ESTIMATES FOR THE MASSLESS DIRAC-COULOMB EQUATION , 2015, 1503.00945.

[20]  F. Cacciafesta,et al.  Endpoint estimates and global existence for the nonlinear Dirac equation with potential , 2011, 1103.4014.

[21]  Claude Le Bris,et al.  ON THE TIME-DEPENDENT HARTREE–FOCK EQUATIONS COUPLED WITH A CLASSICAL NUCLEAR DYNAMICS , 1999 .