Wave maps

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Wave maps

Wave maps are maps f from R^{d+1} to a Riemannian manifold M which are critical points of the Lagrangian

\int f _ a . f ^ a dx dt.

When M is flat, wave maps just obey the wave equation (if viewed in flat co-ordinates). More generally, they obey the equation

Box f = G ( f ) Q_0( f , f )

where G ( f ) is the second fundamental form and Q_0 is the null form [#ddnlw mentioned earlier]. When the target manifold is a unit sphere, this simplifies to

Box f = - f Q_0( f , f )

where f is viewed in Cartesian co-ordinates (and must therefore obey | f |=1 at all positions and times in order to stay on the sphere). The sphere case has special algebraic structure (beyond that of other symmetric spaces) while also staying compact, and so the sphere is usually considered the easiest case to study. Some additional simplifications arise if the target is a Riemann surface (because the connection group becomes U(1), which is abelian); thus S2 is a particularly simple case.

This equation is highly geometrical, and can be rewritten in many different ways. It is also related to the Einstein equations (if one assumes various symmetry assumptions on the metric; see e.g. BgCcMc1995).

The critical regularity is s_c = d/2. Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory s > d/2 is fairly well understood, but the s_c = d/2 theory is quite delicate. A big problem is that H^{d/2} does not control L^ ¥ , so one cannot localize to a small co-ordinate patch (or perform algebraic operations properly).

The positive and negative curvature cases are suspected to behave differently, especially at the critical regularity. Intuitively, the negative curvature space spreads the solution out more, thus giving a better chance for LWP and GWP.More recently, distinctions have arisen between the boundedly parallelizable case (where the exists an orthonormal frame whose structure constants and derivatives are bounded), and the isometrically embeddable case.For instance, hyperbolic space is in the former category but not in the latter; smooth compact manifolds such as the sphere are in both.

The general LWP/GWP theory (except for the special [#wm_on_R one-dimensional] and [#wm_on_R^2 two-dimensional] cases, which are covered in more detail below) is as follows.

  • For d\geq2 one has LWP in H^{n/2}, and GWP and regularity for small data, if the manifold can be isometrically embedded in Euclidean space [Tt-p2]
    • Earlier global regularity results in H^{n/2} are as follows.
      • For a sphere in d\geq5, see [Ta-p5]; for a sphere in d \geq 2, see [Ta-p6]..
      • The d \geq 5 has been generalized to arbitrary manifolds which are boundedly parallelizable [KlRo-p].
      • This has been extended to d=4 by SaSw2001 and NdStvUh2003b. In the constant curvature case one also has global well-posedness for small data in H^{n/2} NdStvUh2003b. This can be extended to manifolds with bounded second fundamental form SaSw2001.
      • This has been extended to d=3 when the target is a Riemann surface Kri2003, and to d=2 for hyperbolic space [Kri-p]
    • For the critical Besov space B^{d/2,1}_2 this is in Tt1998 when d \geq 4 and Tt2001b when d\geq2. (See also Na1999 in the case when the wave map lies on a geodesic). For small data one also has GWP and scattering.
    • In the sub-critical spaces H^s, s > d/2 this was shown in KlMa1995b for the d\geq4 case and in KlSb1997 for d\geq2.
      • For the model wave map equation this was shown for d\geq3 in KlMa1997b.
    • If one replaces the critical Besov space by H^{n/2} then one has failure of analytic or C^2 local well-posedness for d\geq3 [DanGe-p], and one has failure of continuous local well-posedness for d=1 Na1999, Ta2000
    • GWP is also known for smooth data close to a geodesic Si1989. For smooth data close to a point this was in Cq1987.
  • For d \geq 3 singularities can form from large data, even when the data is smooth and rotationally symmetric CaSaTv1998
    • For d=3 this was proven in Sa1988
    • For d\geq7 one can have singularities even when the target has negative curvature CaSaTv1998
    • For d=3, numerics suggest that there is a transition between global existence for small data and blowup for large data, with the self-similar blowup solution being an intermediate attractor [Lie-p]

For further references see Sw1997, SaSw1998, [KlSb-p].

====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].
====Wave maps on ====
  • 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 references#SaSw1993 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 references#Gl1998 Gl1998