Wave maps on R

From DispersiveWiki
  • Scaling is s_c = 1/2.
  • LWP in H^s for s > 1/2 (KeTa1998b)
    • Proven for s \geq 1 in (Zh1999)
    • 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 using the I-method.
    • 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 (Pm1976), but differs slightly from the KdV, mKdV or 1D NLS in that 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 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.