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Well-posed solutions of the third order Benjamin-Ono equation in weighted Sobolev spaces. (English) Zbl 0917.35103

Summary: Here, we continue the study of the initial value problem for the third-order Benjamin-Ono equation \[ \partial_tu= -\partial_x(u^3+ 3u Hu_x+ 3H(uu_x)- 4\partial^2_x u),\quad t>0,\quad x\in\mathbb{R} \]
\[ u(x, 0)= \phi(x) \] (\(H\) denotes the Hilbert transform \((Hf)(x)= \text{P.V. }\int {f(y)\over \pi(y- x)}dy\)) in the weighted Sobolev spaces \(H^s_\gamma= H^s\cap L^2_\gamma\), where \(s>3\), \(\gamma\geq 0\). The result is the proof of well-posedness of the afore mentioned problem in \(H^s_\gamma\), \(s>3\), \(\gamma\in [0,1]\). The proof involves the use of parabolic regularization, the Riesz-Thorin interpolation theorem and the construction technique of auxiliary functions.

MSC:

35Q35 PDEs in connection with fluid mechanics
35G25 Initial value problems for nonlinear higher-order PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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