Dirac equations

The Maxwell-Dirac equation

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

This equation essentially reads

$D_{A}y=-y$ $\Box A+\nabla (\nabla _{x,t}A)={\underline {y}}y$

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 $H^{s_{2}}\times H^{s_{2}-1}$.

• 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

$Dy=fy-y$ $Boxf={\underline {y}}y$

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 $H^{s_{2}}\times H^{s_{2}-1}$.

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

$Dy-my=\lambda (\gamma y,y)y$

where $y$ is a spinor field, $m>0$ is the mass, $\lambda$ is a complex parameter, $\gamma$ 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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