Dirac equations

The Maxwell-Dirac equation

[More info on this equation would be greatly appreciated. - Ed.]

This equation essentially reads

D_A y <nowiki\geq</nowiki> - y
\Box A + Ñ (\div_{x,t} A)= y y

where y is a spinor field (solving a coupled massive Dirac equation), and D is the Dirac operator with connection A. We put y in H^{s_1} and A in H^{s_2} x H^{s_2 - 1}.

• Scaling is (s_1, s_2) = (n/2-3/2, n/2-1).
• When n=1, there is GWP for small smooth data Chd1973
• When n=3 there is LWP for (s_1, s_2) = (1, 1) in the Coulomb gauge Bou1999, and for (s_1, s_2) = (1/2+, 1+) in the Lorentz gauge Bou1996
  • For (s_1, s_2) = (1,2) in the Coulomb gauge this is in Bou1996
  • This has recently been improved by Selberg to (1/4+, 1). Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. A) is kept fixed.
  • LWP for smooth data was obtained in Grs1966
  • GWP for small smooth data was obtained in Ge1991
• When n=4, GWP for small smooth data is known (Psarelli?)

In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [BecMauSb-p2]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in MasNa2003.

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Dirac-Klein-Gordon equation

[More info on this equation would be greatly appreciated. - Ed.]

This equation essentially reads

D y = f y - y
Box f = y y

where y is a spinor field (solving a coupled massive Dirac equation), D is the Dirac operator and f is a scalar (real) field. We put
y in H^{s_1} and ( f , f _t) in H^{s_2} x H^{s_2 - 1}.

The energy class is essentially (s_1,s_2) = (1/2,1), but the energy density is not positive. However, the L^2 norm of y is also positive and conserved..

• Scaling is (s_1, s_2) = (n/2-3/2, n/2-1).
• When n=1 there is GWP for (s_1,s_2) = (1,1) Chd1973, Bou2000 and LWP for (s_1, s_2) = (0, 1/2) Bou2000.
• When n=2 there are some LWP results in Bou2001

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Nonlinear Dirac equation

This equation essentially reads

D y - m y = l(g y, y) y

where y is a spinor field, m > 0 is the mass, l is a complex parameter, g is the zeroth Pauli matrix, and (,) is the spinor inner product.

• Scaling is s_c =1 (at least in the massless case m=0).
• In R^3, LWP is known for H^s when s > 1 EscVe1997
  • This can be improved to LWP in H^1 (and GWP for small H^1 data) if an epsilon of additional regularity as assumed in the radial variable [MacNkrNaOz-p]; in particular one has GWP for radial H^1 data.
• In R^3, GWP is known for small H^s data when s > 1 [MacNaOz-p2].Some results on the nonrelativistic limit of this equation are also obtained in that paper.

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