# Wave maps on R2

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

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

• Scaling is s_c = 1 (energy-critical).
• LWP in H^1 (Tt-p2)
• For ${\displaystyle B_{2}^{1,1}}$ 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 ${\displaystyle B_{2}^{1,1}}$ 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 ${\displaystyle H^{1+\epsilon }}$ on domains ${\displaystyle |x|^{\alpha } 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.