Difference between revisions of "YangMills equations"
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m (YangMills Equations moved to YangMills equations: Consistency with other equation pages) 
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Revision as of 15:44, 5 May 2007
Description  

Equation  
Fields  
Data class  
Basic characteristics  
Structure  Hamiltonian 
Nonlinearity  semilinear with derivatives 
Linear component  wave 
Critical regularity  
Criticality  energy critical for d=4 
Covariance  Lorentzian, gauge 
Theoretical results  
LWP  varies 
GWP  varies 
Related equations  
Parent class  DNLW 
Special cases  YangMills on R^2, R^3, R^4 
Other related  MKG, Cubic NLW, YangMillsHiggs 
Contents
The YangMills 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 YangMills equation if it is a critical point for the Lagrangian functional
where is the curvature of the connection . The EulerLagrange 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 YangMills equation is underdetermined 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 YangMills equation is not wellbehaved in this gauge for rough data. (For smooth data one can obtain local wellposedness 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 divergencefree and curlfree components). See e.g. Ta2003.
In the Coulomb or Temporal gauges, one can create a model equation for the YangMills system by ignoring cubic terms and any contribution from the "elliptic" portion of the gauge ( in the Coulomb gauge, or the curlfree 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 YangMills or MaxwellKleinGordon equations.
The YangMills 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 curlfree part of ). This causes some technical difficulties in the global wellposedness theory.
The YangMills equations can also be coupled with a gvalued scalar field , with the Lagrangian functional of the form
where are covariant derivatives and is some potential function (e.g. . The corresponding EulerLagrange equations have the schematic form
and are generally known as the YangMillsHiggs system of equations. This system may be thought of as a YangMills equation coupled with a semilinear wave equation. The MaxwellKleinGordon system is a special case of YangMillsHiggs.
The theory of YangMills 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 three and four dimensional cases; the onedimensional 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 Bizp). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension BizTb2001, Bizp.
The YangMills equations can also be coupled with a spinor field. In the case this becomes the MaxwellDirac equation.
The YangMills equations in dimension n have many formal similarities with the wave maps equation at dimension d2 (see e.g. CaSaTv1998 for a discussion).
YangMills on
 Scaling is .
 One can use the method of descent and finite speed of propagation to infer R^{2} results from 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 nonoptimal, 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).
YangMills on R^{3}
 Scaling is s_c = 1/2.
 LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small Ta2003. 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 nonlinearity 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 illposedness in the energy class for the Lorentz gauge.
 For the model equation LWP fails for s < 3/4 MaStzp
 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.
 For smooth data this was proven in EaMc1982.
 This result was extended to curved space in CcSa1997
 It seems likely that one can improve this to something like s>7/8, in analogy with the theory for the MaxwellKleinGordon equation.
 For smooth data this was proven in EaMc1982.
MKG and YangMills in R^4
 Scaling is s_c = 1.
 For the MKG equations in the Coulomb gauge, LWP is known for s > 1 Sbp5. This is still not known for YangMills.
 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 wellposedness results for small energy, but this is open.
 For small smooth compactly supported data, one can obtain global existence from the general theory of quasilinear equations.
 For large data YangMills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time BizTb2001, Bizp.
 Further numerics suggests that the radius of the instanton in fact decays like BizOvSip.
 GWP for small data (with an additional angular derivative of regularity) in the Lorentz gauge is in Stzp2.
MKG and YangMills in R^d, d>4
 Scaling is s_c = d/2  1.
 LWP is almost certainly true for MKGCG for s > s_c by adapting the results in Sbp5. The corresponding question for YangMills is still open.
 For the model equations one can probably achieve this by adapting the results in Tt1999
 For dimensions , GWP for small H^{d/2} data in MKGCG is in RoTap. The corresponding question for YangMills is still open, but a Besov result follows (in the Lorentz gauge) from Stzp3.
YangMillsHiggs on R^{3}
 Suppose the potential energy V( f ) behaves like (i.e. defocussing p^th power nonlinearity). When , the Higgs term is negligible, and the theory mimics that of the ordinary YangMills equation. The most interesting case is p=5, since the Higgs component is then H^1critical.
 There is no perfect scaleinvariance to this equation (unless p=3); the critical regularity is .
 In the subcritical 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 YangMills 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 smalldata GWP results.