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Local well-posedness for third order Benjamin-Ono type equations on the torus. (English) Zbl 1437.35611

Summary: We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35G25 Initial value problems for nonlinear higher-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B45 A priori estimates in context of PDEs
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