Feng, Xueshang Well-posed solutions of the third order Benjamin-Ono equation in weighted Sobolev spaces. (English) Zbl 0917.35103 Bull. Belg. Math. Soc. - Simon Stevin 4, No. 4, 525-537 (1997). 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. Cited in 2 Documents 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 Keywords:initial value problem; global well-posedness PDFBibTeX XMLCite \textit{X. Feng}, Bull. Belg. Math. Soc. - Simon Stevin 4, No. 4, 525--537 (1997; Zbl 0917.35103) Full Text: EuDML