Wave maps

Wave maps

Description
Equation	$(\phi ^{*}\nabla )^{\alpha }\partial _{\alpha }\phi =0$
Fields	$\phi :\mathbb {R} ^{1+d}\to {\mathfrak {g}}$
Data class	$\phi [0]\in H^{s}\times H^{s-1}(\mathbb {R} ^{d}\to TM)$
Basic characteristics
Structure	Hamiltonian (completely integrable when d=1)
Nonlinearity	semilinear with derivatives
Linear component	wave
Critical regularity	${\dot {H}}^{d/2}(\mathbb {R} ^{d})$
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 $\phi \,$ from $R^{d+1}$ to a Riemannian manifold $M$ which are critical points of the Lagrangian

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int \phi_\alpha \cdot \phi^\alpha dx dt.}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box \phi = G ( \phi ) Q_0( \phi , \phi )}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G( \phi )} is the second fundamental form and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_0\,} is the standard null form. When the target manifold is a unit sphere, this simplifies to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box \phi = - \phi Q_0( \phi , \phi ) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi\,} is viewed in Cartesian co-ordinates (and must therefore obey Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(1)\,} , which is abelian); thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = d/2\,.} Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > d/2\,} is fairly well understood, but the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = d/2\,} theory is quite delicate. A big problem is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}\,} does not control Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\geq 2} one has LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}\,} are as follows.
    • For a sphere in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 5\,} , see Ta2001c; for a sphere in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \ge 2\,} , see Ta2001d.
    • The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \ge 5\,} has been generalized to arbitrary manifolds which are boundedly parallelizable KlRo-p.
    • This has been extended to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=4\,} by SaSw2001 and NdStvUh2003b. In the constant curvature case one also has global well-posedness for small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}\,} NdStvUh2003b. This can be extended to manifolds with bounded second fundamental form SaSw2001.
    • This has been extended to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=3\,} when the target is a Riemann surface Kri2003, and to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2\,} for hyperbolic space Kri-p
  • For the critical Besov space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B^{d/2,1}_2\,} this is in Tt1998 when d \ge 4 and Tt2001b when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s, s > d/2\,} this was shown in KlMa1995b for the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge4\,} case and in KlSb1997 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 2\,} .
    • For the model wave map equation this was shown for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 3\,} in KlMa1997b.
  • If one replaces the critical Besov space by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}\,} then one has failure of analytic or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^2\,} local well-posedness for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 3\,} [DanGe-p], and one has failure of continuous local well-posedness for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \ge 3\,} singularities can form from large data, even when the data is smooth and rotationally symmetric CaSaTv1998
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=3\,} this was proven in Sa1988
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\ge 7\,} one can have singularities even when the target has negative curvature CaSaTv1998
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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

Special cases

• one-dimensional wave maps
• two-dimensional wave maps

Further reading

Surveys of wave maps can be found in Sw1997, SaSw1998, KlSb-p.