The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic Lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is (2 + 1), the critical norm coincides with energy, the only manifestly conserved quantity in this (Lagrangian) theory. As a consequence, in this dimension the result is an important step in establishing global regularity at all energies, conjectured when the target manifold is negatively curved. The work advanced our understanding of the critical equations and already has been a catalyst for the new results for general target manifolds and other equations (Maxwell-Klein-Gordon, Yang-Mills).
|Original language||English (US)|
|State||Published - Aug 1 2007|
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