Wave maps on R2

Wave maps

Description
Equation	$(\phi ^{*}\nabla )^{\alpha }\partial _{\alpha }\phi =0$
Fields	$\phi :\mathbb {R} ^{1+2}\to {\mathfrak {g}}$
Data class	$\phi [0]\in H^{s}\times H^{s-1}(\mathbb {R} ^{2}\to TM)$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear with derivatives
Linear component	wave
Critical regularity	${\dot {H}}^{1}(\mathbb {R} ^{d})$
Criticality	energy critical
Covariance	Lorentzian, diffeomorphism of target
Theoretical results
LWP	$s\geq 1$
GWP	$s\geq 1$ for small energy, radial, or equivariant data
Related equations
Parent class	Wave maps
Special cases	-
Other related	Einstein equations

The local theory for wave maps on $R^{1+2}$ is as follows.

• Scaling is s_c = 1 (energy-critical).
• LWP in H^1 (Tt-p2)
  • For $B_{2}^{1,1}$ this is in Tt2001b.
  • LWP in H^s, s>1 was shown in KlSb1997.
  • For s>7/4 this can be shown by Strichartz methods.
  • For s>2 this can be shown by energy estimates.

The global theory is as follows.

• GWP and regularity is known for small energy when the target manifold is boundedly parallelizable (Tt-p2)
  • When the target manifold is H^2, regularity was obtained by Krieger (Kri-p)
  • When the target manifold is a sphere, regularity was obtained in Ta2001d
  • For small $B_{2}^{1,1}$ data GWP is in Tt2001b
  • GWP and regularity for small H^1 data was known for corotational wave maps, and can be extended to large H^1 data when the target is geodesically convex (SaSw1993, Sw-p2); see also SaTv1992, Gl-p. In the later papers the result is obtained for quite general rotationally symmetric manifolds, such as non-compact manifolds, although one generically expects blow-up for certain manifolds such as the sphere (see e.g. Sw-p2, or the numerics in BizCjTb2001, IbLie-p). The question of large H^1 GWP and regularity is equivalent to the non-existence of non-constant harmonic maps on the target (Sw-p2). The corotational results have been extended to wave maps with torsion in AcIb2000.
  • Regularity is also known for large smooth radial data (CdTv1993) assuming a convexity condition on the target manifold. This convexity condition was relaxed in Sw2002, and then removed completely in Sw2003. One also has a pointwise bound on the diameter of the range of the wave map in the radial case under similar conditions on the manifold (CdTv1993b).
  • Global weak solutions are known for large energy data (MuSw1996, FrMuSw1998) but as far as is known these solutions might develop singularities or become "ghost" solutions.
• When the domain and target are S², stationary-rotating solutions exist and are stable with respect to corotational perturbations (SaTv1997).
• BMO-type estimates on distance functions were obtained in Gl1998.

Global regularity problem

It is an important open problem whether one has regularity for all large smooth data, at least in the negative curvature case. A slightly harder problem would be to obtain GWP in the critical space H^1.

• When the target manifold is a sphere, large data blowup for equivariant data was established in RoStz-p.
  • Numerical evidence for this blowup was obtained earlier in (IbLie-p).
  • In the equivariant case, examples of blowup in $H^{1+\epsilon }$ on domains $|x|^{\alpha }<t$ can be constructed if one adds a forcing term on the right-hand side (GeIv-p).

In the case of negative curvature targets, a heat flow renormalization for large energy wave maps was proposed in Ta2004c.