Dirac equations: Difference between revisions

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* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.
* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.
* When <math>n=1</math>, there is GWP for small smooth data [[Bibliography#Chd1973|Chd1973]]
* When <math>n=1</math>, there is GWP for small smooth data [[Chd1973]]
* When <math>n=3</math> there is LWP for <math>(s_1, s_2) = (1, 1)</math> in the Coulomb gauge [[Bibliography#Bou1999|Bou1999]], and for <math>(s_1, s_2) = (1/2+, 1+)</math> in the Lorentz gauge [[Bibliography#Bou1996|Bou1996]]
* When <math>n=3</math> there is LWP for <math>(s_1, s_2) = (1, 1)</math> in the Coulomb gauge [[Bou1999]], and for <math>(s_1, s_2) = (1/2+, 1+)</math> in the Lorentz gauge [[Bou1996]]
** For <math>(s_1, s_2) = (1,2)</math> in the Coulomb gauge this is in [[Bibliography#Bou1996|Bou1996]]
** For <math>(s_1, s_2) = (1,2)</math> in the Coulomb gauge this is in [[Bou1996]]
** This has recently been improved by Selberg to <math>(1/4+, 1)</math>. Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. <math>A</math>) is kept fixed.
** This has recently been improved by Selberg to <math>(1/4+, 1)</math>. Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. <math>A</math>) is kept fixed.
** LWP for smooth data was obtained in [[Bibliography#Grs1966|Grs1966]]
** LWP for smooth data was obtained in [[Grs1966]]
** GWP for small smooth data was obtained in [[Bibliography#Ge1991|Ge1991]]
** GWP for small smooth data was obtained in [[Ge1991]]
* When <math>n=4</math>, GWP for small smooth data is known (Psarelli?)
* When <math>n=4</math>, 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 [[Bibliography#MasNa2003|MasNa2003]].
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]].
 
----  [[Category:Equations]]
   
   


Line 31: Line 28:
This equation essentially reads
This equation essentially reads


<center><math>D y = f y - y </math></center>
<center><math>D \psi = \phi \psi - \psi </math></center>
<center><math>\Box  f = \underline{y} y </math></center>
<center><math>\Box  \phi = \overline{\psi} \psi </math></center>


where  <math>y</math>  is a spinor field (solving a coupled massive Dirac equation), <math>D</math> is the Dirac operator and  <math>f</math>  is a scalar (real) field. We put <br /> <math>y</math>  in <math>H^{s_1}</math> and <math>( f ,  f _t)</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.
where  <math>\psi</math>  is a spinor field (solving a coupled massive Dirac equation), <math>D</math> is the Dirac operator and  <math>\phi</math>  is a scalar (real) field. We put <math>\psi</math>  in <math>H^{s_1}</math> and <math>( \phi,  \phi_t)</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.


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


* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.
* Scaling is <math>(s_1, s_2) = (d/2-3/2, d/2-1)</math>.
* When <math>n=1</math> there is GWP for <math>(s_1,s_2) = (1,1)</math> [[Bibliography#Chd1973|Chd1973]], [[Bibliography#Bou2000|Bou2000]] and LWP for <math>(s_1, s_2) = (0, 1/2)</math> [[Bibliography#Bou2000|Bou2000]].
* When <math>n=1</math> there is GWP for <math>(s_1,s_2) = (1,1)</math> [[Chd1973]], [[Bou2000]] and LWP for <math>(s_1, s_2) = (0, 1/2)</math> [[Bou2000]].
* When <math>n=2</math> there are some LWP results in [[Bibliography#Bou2001|Bou2001]]
* When <math>n=2</math> there are some LWP results in [[Bou2001]]
 
----  [[Category:Equations]]
   
   


Line 50: Line 44:
This equation essentially reads
This equation essentially reads


<center><math>D y -  m y  = \lambda (\gamma y, y) y </math></center>
<center><math>D \psi -  m \psi = \lambda (\gamma \psi, \psi) \psi</math></center>


where  <math>y</math>  is a spinor field, <math>m > 0</math> is the mass, <math>\lambda</math> is a complex parameter,  <math>\gamma</math> is the zeroth Pauli matrix, and <math>(,)</math> is the spinor inner product.
where  <math>\psi</math>  is a spinor field, <math>m > 0</math> is the mass, <math>\lambda</math> is a complex parameter,  <math>\gamma</math> is the zeroth Pauli matrix, and <math>(,)</math> is the spinor inner product.


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


   
   
----  [[Category:Equations]]
[[Category:wave]]
[[Category:Equations]]

Revision as of 03:20, 4 August 2006

The Maxwell-Dirac equation

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

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_A y = - y }
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 \Box A + \nabla (\nabla_{x,t} A)= \underline{y} y }

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

  • 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_1, s_2) = (n/2-3/2, n/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 small smooth data Chd1973
  • 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=3} there is 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) = (1, 1)} in the Coulomb gauge Bou1999, and 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) = (1/2+, 1+)} in the Lorentz gauge Bou1996
    • 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) = (1,2)} in the Coulomb gauge this is in Bou1996
    • This has recently been improved by Selberg to 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 (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. 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 A} ) is kept fixed.
    • LWP for smooth data was obtained in Grs1966
    • GWP for small smooth data was obtained in Ge1991
  • 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=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

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 = \phi \psi - \psi }
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 \Box \phi = \overline{\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 (solving a coupled massive Dirac equation), 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} is the Dirac operator 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 \phi} is a scalar (real) field. We put 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} 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^{s_1}} 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 ( \phi, \phi_t)} 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^{s_2} \times H^{s_2 - 1}} .

The energy class is essentially 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) = (1/2,1)} , but the energy density is not positive. However, the 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 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 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) = (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 MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.
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