# Difference between revisions of "Yang-Mills equations"

m (bibliographic references) |
m |
||

Line 1: | Line 1: | ||

====The Yang-Mills equation==== | ====The Yang-Mills equation==== | ||

− | Let <math>A</math> be a connection on R^{d+1} which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the ''Yang-Mills equation'' if it is a critical point for the Lagrangian functional | + | Let <math>A</math> be a connection on <math>R^{d+1}</math> which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the ''Yang-Mills equation'' if it is a critical point for the Lagrangian functional |

<center><math>\int F^{ a b } F_{ a b }</math></center> | <center><math>\int F^{ a b } F_{ a b }</math></center> | ||

Line 18: | Line 18: | ||

<center><math>A -> U^{-1} dU + U^{-1} A U</math></center> | <center><math>A -> U^{-1} dU + U^{-1} A U</math></center> | ||

− | < | + | <center><math>F -> U^{-1} F U</math></center> |

in the equation, where U is an arbitrary function taking values in <math>G</math>. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones: | in the equation, where U is an arbitrary function taking values in <math>G</math>. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones: | ||

Line 52: | Line 52: | ||

where <math>D_ a = \partial_ a + [A_a, .]</math> are covariant derivatives and <math>V</math> is some potential function (e.g. <math>V( f ) = | f |^{k+1})</math>. The corresponding Euler-Lagrange equations have the schematic form | where <math>D_ a = \partial_ a + [A_a, .]</math> are covariant derivatives and <math>V</math> is some potential function (e.g. <math>V( f ) = | f |^{k+1})</math>. The corresponding Euler-Lagrange equations have the schematic form | ||

− | <center><math>\Box A + \nabla (\nabla_{x,t} A) = [A, \nabla A] + [A, [A,A]] + [ f , D f ] | + | <center><math>\Box A + \nabla (\nabla_{x,t} A) = [A, \nabla A] + [A, [A,A]] + [ f , D f ], D_ a D^ a f = V'( f )</math></center> |

and are generally known as the ''Yang-Mills-Higgs system of equations''. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [#mkg The Maxwell-Klein-Gordon system] is a special case of Yang-Mills-Higgs. | and are generally known as the ''Yang-Mills-Higgs system of equations''. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [#mkg The Maxwell-Klein-Gordon system] is a special case of Yang-Mills-Higgs. | ||

Line 58: | Line 58: | ||

The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise. | The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise. | ||

− | Attention has mostly focussed on the [#YM_on_R^3 three] and [#YM_on_R^4 four dimensional] cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to | + | Attention has mostly focussed on the [#YM_on_R^3 three] and [#YM_on_R^4 four dimensional] cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to <math>A_{tt} = 0</math>). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data [[CaSaTv1998]] (see also [[Biz-p]]). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension [[BizTb2001]], [[Biz-p]]. |

The Yang-Mills equations can also be coupled with a spinor field. In the <math>U(1)</math> case this becomes the Maxwell-Dirac equation. | The Yang-Mills equations can also be coupled with a spinor field. In the <math>U(1)</math> case this becomes the Maxwell-Dirac equation. | ||

Line 72: | Line 72: | ||

* Scaling is <math>s_c = 0</math>. | * Scaling is <math>s_c = 0</math>. | ||

− | * One can use the method of descent and finite speed of propagation to infer R<sup>2</sup> results from [#YM_on_R^3 the R<sup>3</sup> results]. Thus, for instance, one has LWP for s>3/4 in the temporal gauge and GWP in the temporal gauge for s\geq 1. These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure). | + | * One can use the method of descent and finite speed of propagation to infer R<sup>2</sup> results from [#YM_on_R^3 the R<sup>3</sup> results]. Thus, for instance, one has LWP for s > 3/4 in the temporal gauge and GWP in the temporal gauge for <math>s\geq 1</math>. These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure). |

Line 82: | Line 82: | ||

* Scaling is s_c = 1/2. | * Scaling is s_c = 1/2. | ||

− | * LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small [[Ta-p3]]. The main tools are bilinear estimates involving both X^{s,\theta} spaces and product Sobolev spaces. | + | * LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small [[Ta-p3]]. The main tools are bilinear estimates involving both <math>X^{s,\theta}</math> spaces and product Sobolev spaces. |

** Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge. | ** Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge. | ||

− | ** For s \geq 1 in the Temporal or Coulomb gauges LWP for large data was shown in [[KlMa1995]]. | + | ** For <math>s \geq 1 </math>in the Temporal or Coulomb gauges LWP for large data was shown in [[KlMa1995]]. |

** For s > 1 LWP for the Temporal, Coulomb, or Lorentz gauges follows from Strichartz estimates [[PoSi1993]]. | ** For s > 1 LWP for the Temporal, Coulomb, or Lorentz gauges follows from Strichartz estimates [[PoSi1993]]. | ||

** For s > 3/2 LWP for the Temporal, Coulomb, or Lorentz gauges follows from energy estimates [[EaMc1982]]. | ** For s > 3/2 LWP for the Temporal, Coulomb, or Lorentz gauges follows from energy estimates [[EaMc1982]]. | ||

** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge. | ** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge. | ||

− | ** For the model equation LWP fails for s < 3/4 [MaStz-p] | + | ** For the model equation LWP fails for s < 3/4 [[MaStz-p]] |

− | ** The endpoint s=1/2 looks extremely difficult, even for Besov space variants. | + | ** The endpoint s = 1/2 looks extremely difficult, even for Besov space variants. |

− | * GWP is known for data with finite Hamiltonian (morally, this is for s \geq 1) in the Coloumb or Temporal gauges [[KlMa1995]]. | + | * GWP is known for data with finite Hamiltonian (morally, this is for <math> s \geq 1 </math>) in the Coloumb or Temporal gauges [[KlMa1995]]. |

** For smooth data this was proven in [[EaMc1982]]. | ** For smooth data this was proven in [[EaMc1982]]. | ||

*** This result was extended to curved space in [[CcSa1997]] | *** This result was extended to curved space in [[CcSa1997]] | ||

Line 101: | Line 101: | ||

* Scaling is s_c = 1. | * Scaling is s_c = 1. | ||

− | * For the MKG equations in the Coulomb gauge, LWP is known for s > 1 [Sb-p5]. This is still not known for Yang-Mills. | + | * For the MKG equations in the Coulomb gauge, LWP is known for s > 1 [[Sb-p5]]. This is still not known for Yang-Mills. |

** For the model equations this is in [[KlTt1999]] | ** For the model equations this is in [[KlTt1999]] | ||

*** For general quadratic DNLW this is only known for s > 5/4 (e.g. by the estimates in [[FcKl2000]]). Strichartz estimates need s > 3/2 [[PoSi1993]], while energy estimates need s > 2. | *** For general quadratic DNLW this is only known for s > 5/4 (e.g. by the estimates in [[FcKl2000]]). Strichartz estimates need s > 3/2 [[PoSi1993]], while energy estimates need s > 2. | ||

Line 107: | Line 107: | ||

* It is conjectured that one has global well-posedness results for small energy, but this is open. | * It is conjectured that one has global well-posedness results for small energy, but this is open. | ||

** For small smooth compactly supported data, one can obtain global existence from the [#gwp_qnlw general theory of quasi-linear equations]. | ** For small smooth compactly supported data, one can obtain global existence from the [#gwp_qnlw general theory of quasi-linear equations]. | ||

− | ** For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time [[BizTb2001]], [Biz-p]. | + | ** For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time [[BizTb2001]], [[Biz-p]]. |

*** Further numerics suggests that the radius of the instanton in fact decays like C t / sqrt(log t) [[BizOvSi-p]]. | *** Further numerics suggests that the radius of the instanton in fact decays like C t / sqrt(log t) [[BizOvSi-p]]. | ||

− | ** GWP for small B^{1,1} data (with an additional angular derivative of regularity) in the Lorentz gauge is in [Stz-p2]. | + | ** GWP for small <math>B^{1,1}</math> data (with an additional angular derivative of regularity) in the Lorentz gauge is in [[Stz-p2]]. |

Line 118: | Line 118: | ||

* Scaling is s_c = n/2 - 1. | * Scaling is s_c = n/2 - 1. | ||

− | * LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in [Sb-p5]. The corresponding question for Yang-Mills is still open. | + | * LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in [[Sb-p5]]. The corresponding question for Yang-Mills is still open. |

** For the model equations one can probably achieve this by adapting the results in [[Tt1999]] | ** For the model equations one can probably achieve this by adapting the results in [[Tt1999]] | ||

− | * For dimensions n\geq 6, GWP for small H^{n/2} data in MKG-CG is in [RoTa-p]. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorentz gauge) from [Stz-p3]. | + | * For dimensions <math>n\geq 6</math>, GWP for small H^{n/2} data in MKG-CG is in [[RoTa-p]]. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorentz gauge) from [[Stz-p3]]. |

Line 128: | Line 128: | ||

====Yang-Mills-Higgs on R<sup>3</sup>==== | ====Yang-Mills-Higgs on R<sup>3</sup>==== | ||

− | * Suppose the potential energy V( f ) behaves like | f <nowiki>|^{p+1} (i.e. defocussing p^th power non-linearity). When p\ | + | * Suppose the potential energy V( f ) behaves like | f <nowiki>|^{p+1} (i.e. defocussing p^th power non-linearity). When p\leq 3, the Higgs term is negligible, and the theory mimics that of the ordinary Yang-Mills equation. The most interesting case is p=5, since the Higgs component is then H^1-critical.</nowiki> |

* There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is s_c = max(1/2, 3/2 - 2/(p-1)) | * There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is s_c = max(1/2, 3/2 - 2/(p-1)) | ||

* In the sub-critical case p<5 one has GWP for smooth data [[EaMc1982]], [[GiVl1982b]]. This can be pushed to H^1 by the results in [[Ke1997]]. The local theory might be pushed even further. | * In the sub-critical case p<5 one has GWP for smooth data [[EaMc1982]], [[GiVl1982b]]. This can be pushed to H^1 by the results in [[Ke1997]]. The local theory might be pushed even further. | ||

− | * In the critical case p=5 one has GWP for s \geq 1 [[Ke1997]]. | + | * In the critical case p=5 one has GWP for <math>s \geq 1</math> [[Ke1997]]. |

− | * In the supercritical case p>5 one probably has LWP for s \geq s_c (because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results. | + | * In the supercritical case p>5 one probably has LWP for <math>s \geq s_c </math>(because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results. |

---- [[Category:Equations]] [[Category:Geometry]] [[Category:Wave]] | ---- [[Category:Equations]] [[Category:Geometry]] [[Category:Wave]] |

## Revision as of 01:01, 3 February 2007

## Contents

#### The Yang-Mills equation

Let be a connection on which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the *Yang-Mills equation* if it is a critical point for the Lagrangian functional

where is the curvature of the connection . The Euler-Lagrange equations for this functional have the schematic form

where is the spacetime divergence of . A more succinct (but less tractable) formulation of this equation is

It is often convenient to split into temporal and spatial components as .

As written, the Yang-Mills equation is under-determined because of the gauge invariance

in the equation, where U is an arbitrary function taking values in . In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:

There are also several other useful gauges, such as the Cronstrom gauge Cs1980 centered around a point in spacetime.

The Lorentz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression acquires a null structure KlMa1995 which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see KlMa1995. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. Ta-p3.

In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic* portion of the gauge ( in the Coulomb gauge, or the curl-free portion of in the Temporal gauge). The resulting model equation is*

where is some null form such as

The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.

The Yang-Mills equations come with a positive definite conserved Hamiltonian

which mostly controls the norm of and the norm of . However, there are some portions of the norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is ; in the Temporal gauge, it is the norm of the curl-free part of ). This causes some technical difficulties in the global well-posedness theory.

The Yang-Mills equations can also be coupled with a g-valued scalar field , with the Lagrangian functional of the form

where are covariant derivatives and is some potential function (e.g. . The corresponding Euler-Lagrange equations have the schematic form

and are generally known as the *Yang-Mills-Higgs system of equations*. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [#mkg The Maxwell-Klein-Gordon system] is a special case of Yang-Mills-Higgs.

The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.

Attention has mostly focussed on the [#YM_on_R^3 three] and [#YM_on_R^4 four dimensional] cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to ). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data CaSaTv1998 (see also Biz-p). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension BizTb2001, Biz-p.

The Yang-Mills equations can also be coupled with a spinor field. In the case this becomes the Maxwell-Dirac equation.

The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension n-2 (see e.g. CaSaTv1998 for a discussion)

#### Yang-Mills on

- Scaling is .
- One can use the method of descent and finite speed of propagation to infer R
^{2}results from [#YM_on_R^3 the R^{3}results]. Thus, for instance, one has LWP for s > 3/4 in the temporal gauge and GWP in the temporal gauge for . These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure).

#### Yang-Mills on R^{3}

- Scaling is s_c = 1/2.
- LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small Ta-p3. The main tools are bilinear estimates involving both spaces and product Sobolev spaces.
- Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge.
- For in the Temporal or Coulomb gauges LWP for large data was shown in KlMa1995.
- For s > 1 LWP for the Temporal, Coulomb, or Lorentz gauges follows from Strichartz estimates PoSi1993.
- For s > 3/2 LWP for the Temporal, Coulomb, or Lorentz gauges follows from energy estimates EaMc1982.
- There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge.
- For the model equation LWP fails for s < 3/4 MaStz-p
- The endpoint s = 1/2 looks extremely difficult, even for Besov space variants.

- GWP is known for data with finite Hamiltonian (morally, this is for ) in the Coloumb or Temporal gauges KlMa1995.

#### MKG and Yang-Mills in R^4

- Scaling is s_c = 1.
- For the MKG equations in the Coulomb gauge, LWP is known for s > 1 Sb-p5. This is still not known for Yang-Mills.
- For the model equations this is in KlTt1999
- The latter two results (Strichartz and energy) easily extend to the actual MKG and YM equations in all three standard gauges.

- It is conjectured that one has global well-posedness results for small energy, but this is open.
- For small smooth compactly supported data, one can obtain global existence from the [#gwp_qnlw general theory of quasi-linear equations].
- For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time BizTb2001, Biz-p.
- Further numerics suggests that the radius of the instanton in fact decays like C t / sqrt(log t) BizOvSi-p.

- GWP for small data (with an additional angular derivative of regularity) in the Lorentz gauge is in Stz-p2.

#### MKG and Yang-Mills in R^n, n>4

- Scaling is s_c = n/2 - 1.
- LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in Sb-p5. The corresponding question for Yang-Mills is still open.
- For the model equations one can probably achieve this by adapting the results in Tt1999

- For dimensions , GWP for small H^{n/2} data in MKG-CG is in RoTa-p. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorentz gauge) from Stz-p3.

#### Yang-Mills-Higgs on R^{3}

- Suppose the potential energy V( f ) behaves like | f |^{p+1} (i.e. defocussing p^th power non-linearity). When p\leq 3, the Higgs term is negligible, and the theory mimics that of the ordinary Yang-Mills equation. The most interesting case is p=5, since the Higgs component is then H^1-critical.
- There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is s_c = max(1/2, 3/2 - 2/(p-1))
- In the sub-critical case p<5 one has GWP for smooth data EaMc1982, GiVl1982b. This can be pushed to H^1 by the results in Ke1997. The local theory might be pushed even further.
- In the critical case p=5 one has GWP for Ke1997.
- In the supercritical case p>5 one probably has LWP for (because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results.