At the moment I am attending a research programme called ‘Quantitative studies of nonlinear wave phenomena’ at the Erwin Schrödinger Institute in Vienna. This has stimulated me to think again about a topic in the study of nonlinear wave equations which has seen remarkable developments just recently. This concerns critical wave maps. Wave maps are nonlinear generalizations of the wave equation associated to a Riemannian manifold called the target manifold. These days wave maps are one of the most important model problems in the study of nonlinear wave equations. If is a pseudo-Riemannian manifold and a mapping from to define a Lagrange function by the norm squared of the derivative of with the norm being determined by and . In local coordinates . Solutions of the corresponding Euler-Lagrange equations are called harmonic maps when is Riemannian and wave maps when is Lorentzian. Harmonic maps are elliptic while wave maps are hyperbolic. It is common to classify nonlinear evolution equations by the scaling of the energy into subcritical, critical and supercritical. The standard point of view is then that subcritical problems are relatively straightforward, supercritical problems are impossible at present and critical problems are on the borderline. It is therefore natural that critical problems are a hive of activity at the moment. For more information about this classification and its significance see this post of Terence Tao. Actually there are a couple of
results around on problems which could be called marginally supercritical. One, due to Tao, concerns the nonlinear wave equation in three space dimensions (arXiv:math/0606145). Note that the equation with nonlinearity is critical in three space dimensions. Another, due to Michael Struwe, concerns the equation in two space dimensions, where every power law nonlinearity is subcritical. Here criticality is decided by the value of the energy and Struwe’s result applies to the supercritical case, under the assumption of rotational symmetry. He gave a talk on this here on Monday.
Getting back to wave maps the criticality is decided by the dimension of the manifold . If its dimension is two (i.e. space dimension ) then the problem is subcritical. The case is critical while the problem is supercritical for . From now on I concentrate on the critical case. In this context there is a general guide for distinguishing between global smoothness of solutions and the development of singularities which is the curvature of the target manifold. Negative curvature tends to be good for regularity while positive curvature tends to encourage singularities. This is related to an analogous feature of harmonic maps which has been known for a long time. A good model problem for the negatively curved case is that where is the hyperbolic plane. The thesis of Shadi Tahvildar-Zadeh played an important role in making this problem known in the mathematics community. In this work the manifold is three-dimensional Minkowski space. It was done at the Courant Institute under the supervision of Jalal Shatah and Demetrios Christodoulou. To make the problem more tractable some symmetry assumptions were made. Consider the action of on by rotations about a point. There is also a natural action of the rotation group on the hyperbolic plane. One symmetry assumption is invariance – applying a rotation to a point leaves invariant. Another symmetry assumption is equivariance – applying a rotation by an angle to leads to a rotation of by an angle where is a positive integer. It can be seen that there are different types of equivariance depending on the parameter . The invariant case is related to the Einstein equations of general relativity. Cylindrically symmetric solutions of the vacuum Einstein equations lead to exactly this problem. On the other hand one of the key techniques used in the proofs originates in a paper of Christodoulou where he showed a stability result for an exterior region in a black hole spacetime in the presence of a scalar field. The results on the invariant case were published by Christodoulou and Tahvildar-Zadeh in 1993. They made no mention of the connection to the vacuum Einstein equations. The results on the equivariant case were published by Shatah and Tahvildar-Zadeh at about the same time. Interestingly the two papers use rather different terminology. While the work on the equivarant case uses the language of multipliers, traditional in PDE theory, the work on the invariant case uses the language of vector fields coming from the direction of general relativity.
Without the symmetry the problem becomes much harder and was the subject of a major project of Tao (project heatwave). For more information see this post of Tao and the comments on it. By the time the project was finished there were already alternative approaches to solving the same problem by Sterbenz and Tataru and Krieger and Schlag. Now this problem can be considered solved. The different papers use different methods and the detailed conclusions are different. Nevertheless they all include global regularity for wave maps from three-dimensional Minkowski space to the hyperbolic plane without any symmetry assumptions. The proof of Sterbenz and Tataru goes through an intermediate statement that if there were a singularity there would be a finite energy harmonic map into the given target. They actually concentrate on the case of a compact target manifold but the case of the hyperbolic plane can be handled by passing to the universal cover. There also give a more general statement for the case that there is a non-trivial harmonic map around. Then regularity is obtained as long as the energy is less than that of the harmonic map. These ideas are also related to results on the positive curvature case which show what some blow-up solutions look like when they occur. Consider the case that the target manifold is the two-sphere. Here it is possible, as in the case of the hyperbolic plane, to define invariant and equivariant wave maps. In recent work Raphaël and Rodnianski are able to describe the precise nature of the blow-up for an open set of initial data and for all values of the integer describing the type of equivariance. The case is exceptional. They have also proved similar results for an analogous critical problem, the Yang-Mills equations in dimensions. These results provide a remarkably satisfactory confirmation of conjectures made by Piotr Bizoń and collaborators on the basis of numerical and heuristic considerations which were an important stimulus for the field.