The Dirichlet–Neumann Paradifferential Problem
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The preceding chapters of this book provided a proof of our main almost global existence result for capillary-gravity wave equations, assuming that the latter have been reduced to some paradifferential system, which admits energy estimates. The goal of this chapter, and of the following ones, is to perform such a reduction. A key step is to obtain a convenient paradifferential representation of the Dirichlet–Neumann operator with a homogeneous expansion in the free wave profile at any degree. This is what we achieve in Chap. 6, constructing a paradifferential version of the Boutet de Monvel parametrix of the Dirichlet problem.
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