Wave maps on R2

Wave maps on $R^{2}$

• Scaling is s_c = 1 (energy-critical).
• LWP in H^1 Tt-p2
  • For B^{1,1}_2 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.
• 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 Ta-p6
  • For small B^{1,1}_2 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
  • 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, numerical evidence seems to suggest energy concentration and singularity formulation for large equivariant data [IbLie-p].In the equivariant case, examples of blowup in H^{1+eps} on domains |x|^alpha < t can be constructed if one adds a forcing term on the right-hand side [GeIv-p]
  • 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