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.

Dirac-Klein-Gordon equation

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

This equation essentially reads

$D\psi =\phi \psi -\psi$ $\Box \phi ={\overline {\psi }}\psi$

where $\psi$ is a spinor field (solving a coupled massive Dirac equation), $D$ is the Dirac operator and $\phi$ is a scalar (real) field. We put $\psi$ in $H^{s_{1}}$ and $(\phi ,\phi _{t})$ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y} is also positive and conserved..

• Scaling is $(s_{1},s_{2})=(d/2-3/2,d/2-1)$.
• When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1} there is GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1,s_2) = (1,1)} Chd1973, Bou2000 and LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1, s_2) = (0, 1/2)} Bou2000.
• When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=2} there are some LWP results in Bou2001

Nonlinear Dirac equation

This equation essentially reads

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D \psi - m \psi = \lambda (\gamma \psi, \psi) \psi}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \psi} is a spinor field, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m > 0} is the mass, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} is a complex parameter, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} is the zeroth Pauli matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (,)} is the spinor inner product.

• Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c =1} (at least in the massless case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m=0} ).
• In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3} , LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1} EscVe1997
  • This can be improved to LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} (and GWP for small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} data) if an epsilon of additional regularity as assumed in the radial variable MacNkrNaOz-p; in particular one has GWP for radial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} data.
• In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3} , GWP is known for small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} data when $s>1$ MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.