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A para-differential renormalization technique for nonlinear dispersive equations. (English) Zbl 1214.35089

The authors study initial value problems for the generalized Benjamin-Ono equation. They define a pseudodifferential operator \(D^{\alpha}(x,t)\) by \((D^{\alpha} \hat u) (\xi ,t)=|\xi|^{\alpha} \hat u (\xi ,t)\), and they consider the following initial value problem:
\[ \partial_t u+D^{\alpha}\partial_x u +\partial_x (u^2 /2)=0,\quad u(0,)=\varphi (x)\;\;\;\text{on }{\mathbb R}_x \times {\mathbb R}_t. \]
If \(\alpha =1\), then this is the usual Benjamin-Ono equation, which is a nonlinear pseudodifferential equation describing one-dimensional internal waves in deep water. If \(\alpha =2\), then this is the KDV equation. The well-posedness of the initial value problems for these special cases \(\alpha =1,2\) is already known, and the authors try to give a new result unifying these special cases (considering general values of \(\alpha\)). They show that the above initial value problem is well-posed in Sobolev spaces, and also that the \(L^2\) energy is preserved. For the proof, it is sufficient to show an a priori estimate for the solutions, and for this purpose the authors decompose the solution into small frequency bands of width \(\sqrt{\lambda}\) at frequency \(\lambda\), and study the effect of a gauge transformation. In particular, they use a frequency dependent renormalization method to control the strong low-high frequency interactions.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
35Q53 KdV equations (Korteweg-de Vries equations)
35B45 A priori estimates in context of PDEs
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