View source for Wave maps on R2

====Wave maps on <math>R^2</math>====

* Scaling is s_c = 1 (energy-critical).
* LWP in H^1 [[Tt-p2]]
** For B^{1,1}_2 this is in [[Bibliography#Tt2001b|Tt2001b]].
** LWP in H^s, s>1 was shown in [[Bibliography#KlSb1997|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 [[Bibliography#Tt2001b|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 [[Bibliography#SaSw1993 |SaSw1993]], [[Sw-p2]]; see also [[Bibliography#SaTv1992|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 [[Bibliography#BizCjTb2001|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 [[Bibliography#AcIb2000|AcIb2000]].
** Regularity is also known for large smooth radial data [[Bibliography#CdTv1993|CdTv1993]] assuming a convexity condition on the target manifold. This convexity condition was relaxed in [[Bibliography#Sw2002|Sw2002]], and then removed completely in [[Bibliography#Sw2003|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 [[Bibliography#CdTv1993b|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 [[Bibliography#MuSw1996|MuSw1996]], [[Bibliography#FrMuSw1998|FrMuSw1998]], but as far as is known these solutions might develop singularities or become "ghost" solutions.
* When the domain and target are S<sup>2</sup>, stationary-rotating solutions exist and are stable with respect to corotational perturbations [[Bibliography#SaTv1997|SaTv1997]]
* BMO-type estimates on distance functions were obtained in [[Gl1998]]

[[Category:Wave]]
[[Category:Equations]]