Wave maps on R

Wave maps on R

• Scaling is s_c = 1/2.
• LWP in H^s for s > 1/2 KeTa1998b
  • Proven for s \geq 1 in Zh-p
  • Proven for s > 3/2 by energy methods
  • One also has LWP in the space L^1_1 KeTa1998b. Interpolants of this with the H^s results are probably possible.
  • One has ill-posedness for H^{1/2}, and similarly for Besov spaces near H^{1/2} such as B^{1/2,1}_2 Na1999, Ta2000. However, the ill-posedness is not an instance of blowup, only of a discontinuous solution map, and perhaps a weaker notion of solution still exists and is unique.
• GWP in H^s for s>3/4 KeTa1998b when the target manifold is a sphere
  • Was proven for s \geq 1 in Zh1999 for general manifolds
  • Was proven for s \geq 2 for general manifolds in Gu1980, LaSh1981, GiVl1982, Sa1988
  • One also has GWP and scattering in L^1_1. KeTa1998b One probably also has asymptotic completeness.
  • Scattering fails when the initial velocity is not conditionally integrable KeTa1998b.
  • It should be possible to improve the s>3/4 result by correction term methods, and perhaps to obtain interpolants with the L^{1,1} result. One should also be able to extend to general manifolds.
• Remark: The non-linear term has absolutely no smoothing properties, because of the double derivative in the non-linearity and the lack of dispersion in the one-dimensional case.
• Remark: The equation is completely integrable [[Bibliography#Pm1976|Pm1976]], but not in the same way as KdV, mKdV or 1D NLS. (The additional conserved quantities do not control H^s norms, but rather the pointwise distribution of the energy. Indeed, the energy density itself obeys the free wave equation!).When the target is a symmetric space, homoclinic periodic multisoliton solutions were constructed in TeUh-p2.
• Remark: When the target manifold is S², the wave map equation is related to the [#Sine-Gordon sine-Gordon equation] Pm1976.Homoclinic periodic breather solutions were constructed in SaSr1996.
• When the target is a Lorentzian manifold, local existence for smooth solutions was established in [Cq-p2].A criterion on the target manifold to guarantee global existence of smooth solutions is in [Woo-p]; however if the target manifold is the Lorentz sphere S^{1,n-1} then there is a large class of data which blows up Woo-p.