Wu lectures on well posedness and singularities of water waves

sharp-crestsThe second set of lectures at the summer school were given by Sijue Wu on well-posedness of the initial value problem.  The first part of the lectures was an introduction to local and global existence for 2D waves.  The moving domain was mapped to the lower half plane using a Riemann mapping.  Tools included the theory of holomorphic functions, the Hilbert transform on the real line, commutator estimates, product formulae for singular integrals, and harmonic analysis.  A key step was finding the right formulation to which analysis in Sobolev spaces could be applied directly.  The extension to 3D involved generalizing complex analysis to Clifford analysis, introducing an analogue of the Hilbert transform, and key estimates for singular integrals.  Key background work includes Wu (1997), Wu (1999), Wu (2009), and Wu (2011). Slides from the lectures can be downloaded here.  Lecture notes will be forthcoming.

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