# Difference between revisions of "Wave maps on R2"

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* Scaling is s_c = 1 (energy-critical). | * Scaling is s_c = 1 (energy-critical). | ||

− | * LWP in H^1 [[Tt-p2]] | + | * LWP in H^1 [[Bibliography#Tt-p2|Tt-p2]] |

** For B^{1,1}_2 this is in [[Bibliography#Tt2001b|Tt2001b]]. | ** For B^{1,1}_2 this is in [[Bibliography#Tt2001b|Tt2001b]]. | ||

** LWP in H^s, s>1 was shown in [[Bibliography#KlSb1997|KlSb1997]]. | ** 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>7/4 this can be shown by Strichartz methods. | ||

** For s>2 this can be shown by energy estimates. | ** 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]] | + | * GWP and regularity is known for small energy when the target manifold is boundedly parallelizable [[Bibliography#Tt-p2|Tt-p2]] |

− | ** When the target manifold is H^2, regularity was obtained by Krieger [[Kri-p]] | + | ** When the target manifold is H^2, regularity was obtained by Krieger [[Bibliography#Kri-p|Kri-p]] |

− | ** When the target manifold is a sphere, regularity was obtained in [[Ta-p6]] | + | ** When the target manifold is a sphere, regularity was obtained in [[Bibliography#Ta-p6|Ta-p6]] |

** For small B^{1,1}_2 data GWP is in [[Bibliography#Tt2001b|Tt2001b]] | ** 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]]. | + | ** 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]], [[Bibliography#Sw-p2|Sw-p2]]; see also [[Bibliography#SaTv1992|SaTv1992]], [[Bibliography#Gl-p|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. [[Bibliography#Sw-p2|Sw-p2]], or the numerics in [[Bibliography#BizCjTb2001|BizCjTb2001]], [[Bibliography#IbLie-p|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]] | ** 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. | ** 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. | ||

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** 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. | ** 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]] | * 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]] | + | * BMO-type estimates on distance functions were obtained in [[Bibliography#Gl1998|Gl1998]] |

[[Category:Wave]] | [[Category:Wave]] | ||

[[Category:Equations]] | [[Category:Equations]] |

## Revision as of 16:08, 31 July 2006

#### Wave maps on

- Scaling is s_c = 1 (energy-critical).
- LWP in H^1 Tt-p2
- 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 S
^{2}, stationary-rotating solutions exist and are stable with respect to corotational perturbations SaTv1997 - BMO-type estimates on distance functions were obtained in Gl1998