Wave maps

Wave maps
Description
Equation ${\displaystyle (\phi ^{*}\nabla )^{\alpha }\partial _{\alpha }\phi =0}$
Fields ${\displaystyle \phi :\mathbb {R} ^{1+d}\to {\mathfrak {g}}}$
Data class ${\displaystyle \phi [0]\in H^{s}\times H^{s-1}(\mathbb {R} ^{d}\to TM)}$
Basic characteristics
Structure Hamiltonian (completely integrable when d=1)
Nonlinearity semilinear with derivatives
Linear component wave
Critical regularity ${\displaystyle {\dot {H}}^{d/2}(\mathbb {R} ^{d})}$
Criticality energy critical for d=2
Covariance Lorentzian, diffeomorphism of target
Theoretical results
LWP varies
GWP varies
Related equations
Parent class DDNLW
Special cases Wave maps on R, on R^2
Other related Einstein equations

Wave maps are maps ${\displaystyle \phi \,}$ from ${\displaystyle R^{d+1}}$ to a Riemannian manifold ${\displaystyle M}$ which are critical points of the Lagrangian

${\displaystyle \int \phi _{\alpha }\cdot \phi ^{\alpha }dxdt.}$

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

${\displaystyle \Box \phi =G(\phi )Q_{0}(\phi ,\phi )}$

where ${\displaystyle G(\phi )}$ is the second fundamental form and ${\displaystyle Q_{0}\,}$ is the standard null form. When the target manifold is a unit sphere, this simplifies to

${\displaystyle \Box \phi =-\phi Q_{0}(\phi ,\phi )}$

where ${\displaystyle \phi \,}$ is viewed in Cartesian co-ordinates (and must therefore obey ${\displaystyle |\phi |=1\,}$ 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 ${\displaystyle U(1)\,}$, which is abelian); thus ${\displaystyle S^{2}\,}$ 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 ${\displaystyle s_{c}=d/2\,.}$ Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory ${\displaystyle s>d/2\,}$ is fairly well understood, but the ${\displaystyle s_{c}=d/2\,}$ theory is quite delicate. A big problem is that ${\displaystyle H^{d/2}\,}$ does not control ${\displaystyle L^{\infty }\,}$, 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 one-dimensional and two-dimensional cases) is as follows.

• For ${\displaystyle d\geq 2}$ one has LWP in ${\displaystyle H^{n/2}\,}$, and GWP and regularity for small data, if the manifold can be isometrically embedded in Euclidean space Tt-p2
• Earlier global regularity results in ${\displaystyle H^{n/2}\,}$ are as follows.
• For a sphere in ${\displaystyle d\geq 5\,}$, see Ta2001c; for a sphere in ${\displaystyle d\geq 2\,}$, see Ta2001d.
• The ${\displaystyle d\geq 5\,}$ has been generalized to arbitrary manifolds which are boundedly parallelizable KlRo-p.
• This has been extended to ${\displaystyle d=4\,}$ by SaSw2001 and NdStvUh2003b. In the constant curvature case one also has global well-posedness for small data in ${\displaystyle H^{n/2}\,}$ NdStvUh2003b. This can be extended to manifolds with bounded second fundamental form SaSw2001.
• This has been extended to ${\displaystyle d=3\,}$ when the target is a Riemann surface Kri2003, and to ${\displaystyle d=2\,}$ for hyperbolic space Kri-p
• For the critical Besov space ${\displaystyle B_{2}^{d/2,1}\,}$ this is in Tt1998 when d \ge 4 and Tt2001b when ${\displaystyle d\geq 2\,}$. (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 ${\displaystyle H^{s},s>d/2\,}$ this was shown in KlMa1995b for the ${\displaystyle d\geq 4\,}$ case and in KlSb1997 for ${\displaystyle d\geq 2\,}$.
• For the model wave map equation this was shown for ${\displaystyle d\geq 3\,}$ in KlMa1997b.
• If one replaces the critical Besov space by ${\displaystyle H^{n/2}\,}$ then one has failure of analytic or ${\displaystyle C^{2}\,}$ local well-posedness for ${\displaystyle d\geq 3\,}$ [DanGe-p], and one has failure of continuous local well-posedness for ${\displaystyle d=1\,}$ 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 ${\displaystyle d\geq 3\,}$ singularities can form from large data, even when the data is smooth and rotationally symmetric CaSaTv1998
• For ${\displaystyle d=3\,}$ this was proven in Sa1988
• For ${\displaystyle d\geq 7\,}$ one can have singularities even when the target has negative curvature CaSaTv1998
• For ${\displaystyle d=3\,}$, 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