Linear and bilinear wave equation estimates, and well-posedness of NLWs for rough data

This field can be roughly divided into two halves, with results from the former half being used to tackle problems from the latter half:

• Wave equation estimates.  The problem here is to establish good estimates (linear or bilinear) in various spaces (Lebesgue, elliptic Sobolev, wave-equation Sobolev, Fourier-Lebesgue spaces, etc.) of solutions to the homogeneous or inhomogeneous wave equation or related operators (e.g. the Bochner-Riesz operator for the wave equation, wave equation square functions, etc.) in terms of the initial data (or inhomogeneous data).

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• Well-posedness of NLWs.  Consider the Cauchy problem for various non-linear wave equations (NLWs) such as the semi-linear, quasi-linear, wave map,Yang-Mills, Maxwell-Klein-Gordon, and Einstein equations.  Given rough initial data (e.g. finite energy but no better), can one determine local and global well-posedness of these equations (i.e. existence, uniqueness, and continuous dependence on the initial data)?  If there are global solutions, what is the long-term behaviour of these solutions (e.g. does it scatter to a free solution? do the Sobolev norms stay bounded? do solitons appear?)  If there is only local existence, how long does the solution exist, how do singularities develop, and what is the nature of these singularities (e.g. which norms must blow up)?
The problem of establishing wave equation estimates is closely related to the Bochner-Riesz and restriction theory for the light cone in R^{n+1}.  Optimal fixed-time estimates have been known for quite some time, but they do not give the best spacetime estimates because there is a local smoothing effect arising from averaging in time.  This local smoothing effect can be captured by the L^2-based Strichartz estimates or by the L^p estimates predicted by the Sogge's local smoothing conjecture.  This latter conjecture is very difficult, and is known to imply the Bochner-Riesz, restriction and Kakeya estimates for the sphere; these arguments are based on the method of descent, and use the fact that the sphere is a section of the light cone.  In particular, failure of the Kakeya conjecture would imply that certain conjectured wave equation estimates are impossible.  (This can be shown directly, by using the hypothetical counterexample to Kakeya to construct a collection of wave trains which strongly interfere with each other and do not exhibit much local smoothing).

The estimates mentioned above are linear, however in the past ten years it has been realized (by Klainerman, Machedon, Bourgain, and others) that bilinear estimates for wave equations may be more fundamental and yield better results for applications.  In particular, the problem of estimating null forms such as

Q(\phi,\psi) = \phi_t \psi_t - \nabla \phi . \nabla \psi

where \phi, \psi solve the homogeneous or inhomogeneous wave equation, are key to the low-regularity theory of non-linear wave equations.  These estimates have traditionally been proven using Plancherel's theorem and were thus restricted to L^2-based spaces, but recent advances in bilinear restriction theory have opened the way to L^p null form estimates which may soon prove useful in applications.  Null forms are also sometimes connected to Strichartz estimatesThe bilinear theory for the cone parallels that of the sphere, and has many applications beyond non-linear wave equations.

Once one can obtain good control on solutions to the linear equation, one can often obtain local well-posedness for small perturbations of this equation, even if the data is somewhat rough.  There is a standard technique to do this, which goes by various names including perturbation theory, Picard iteration, the method of power series, and the bootstrap argument.  The effectiveness of this technique depends on whether the regularity of the initial data is sub-critical, critical, or super-critical.  In the sub-critical regime these methods work fairly well.  When the regularity is critical then the argument becomes much more delicate, and perturbation theory techniques can sometimes fail completely.  Indeed, some equations are ill-posed at the critical regularity.  In the super-critical regime it is expected that ill-posedness occurs for virtually all equations.

For simple equations such as the semi-linear wave equation, a satisfactory local well-posedness theory can be obtained by the Strichartz estimates, which give L^p control on the solution, although in the case of very low regularity one must use more intricate norms.  However, for more interesting equations the non-linearity is more complicated and tends to involve the null forms mentioned earlier.  In this case one seems forced to use null form estimates to get close to the critical regularity, and it seems one must use even more precise techniques (multilinear estimates?) to push these equations to the critical level.

Once local well-posedness has been obtained, the next step is to look for global well-posedness, and possibly scattering to a linear solution.  For very smooth, rapidly decaying data this can often be achieved by conformal compactification or by showing that the solution decays and becomes more regular in certain directions as time progresses; without enough decay, these equations can blow up.  For small data one can sometimes use the critical theory (which is somewhat insensitive to the local/global distinction).  If the data has finite energy, then one can also exploit energy conservation to get global well posedness (but not necessarily scattering); this works best when the energy is positive definite, and the energy norm is sub-critical.  When the energy norm is critical then this strategy can be threatened by the possible phenomenon of energy concentration; however, many equations have conservation laws or monotonicity formulae which prevent this concentration from occuring.  Recently, Bourgain has demonstrated a general argument which shows that global existence can also be obtained slightly below the energy norm if there is an additional smoothing effect in the equation, although it seems that one can sometimes do without this smoothing.

I have worked on endpoint Strichartz estimates and their failure in low dimension for both the wave and Schrodinger equation.  I've also investigated blowup for the Klein-Gordon and 1D wave-map equations under certain circumstances, and global existence below the energy norm for 1D wave maps and Maxwell-Klein-Gordon under different circumstances.  This work is mostly in collaboration with Mark Keel.

This area is only a small part of the field of non-linear wave equations, which in turn is a sub-field of non-linear PDE, and yet there are an incredible number of people who are currently working in this area.  There are also many other people working on analogues of these results in curved space, with obstacles, or on Lie groups, and on the closely related problems for Schrodinger and KdV.

• Sergiu Klainerman, On the regularity of classical field theories in Minkowski space-time R^{3+1}. Nonlinear partial differential

• equations in geometry and physics (Knoxville, TN, 1995), 29--69, Progr. Nonlinear Differential Equations Appl., 29, Birkhäuser, Basel, 1997
Sergiu is a strong advocate of the philosophy of using the latest developments in the Fourier analysis of waves to control the solutions to non-linear wave equations.  This survey article details this viewpoint.

• Michael Struwe, Wave maps. Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995), 113--153, Progr.

• Nonlinear Differential Equations Appl., 29, Birkhäuser, Basel, 1997.
This is a detailed survey on the wave map equation, which is a beautifully geometric non-linear wave equation whose global regularity lies tantalizingly just out of reach of perturbative harmonic analysis techniques.  Presumably one would have to blend these techniques with some deep differential geometry in order to clinch the problem.

• Lars Hormander, Lectures on nonlinear hyperbolic differential equations, Mathematiques & Applications, 26. Springer-Verlag, 1997

• A little light on Strichartz and null-form estimates, but otherwise an extensive modern introduction to the field.

• Sigmund Selberg, Multilinear spacetime estimates and applications to local existence theory for nonlinear wave equations, Princeton University Thesis 1999

• This thesis has a very nice exposition of H^{s,\delta} spaces, the estimates they satisfy, and their role in the local well-posedness theory for nonlinear wave equations.

• Chris Sogge, Lectures on nonlinear wave equations, Cambridge University Press, 1993

• This is a good reference for the material up to the late '80s, although some of the later arguments and methods in the book have since been simplified.

• Mark Keel, Terence Tao, Endpoint Strichartz Estimates, Amer. J. Math., 120 (1998), 955-980.

• If you want to learn about Strichartz estimates and how they are proved, this is the place.
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