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Local and global well-posedness for the Ostrovsky equation. (English) Zbl 1089.35061
Summary: We consider the initial value problem for $\partial_tu- \beta\partial_x^3u- \gamma\partial_x^{-1}u+ uu_x=0, \quad x,t\in\mathbb R,$ where $$u$$ is a real valued function, $$\beta$$ and $$\gamma$$ are real numbers such that $$\beta\cdot\gamma\neq 0$$ and $$\partial_x^{-1}f= ((i\xi)^{-1} \widehat{f}(\xi))^\vee$$.
This equation differs from the Korteweg-de Vries equation in a nonlocal term. Nevertheless, we obtain local well-posedness in $$X_s= \{f\in H^s(\mathbb R): \partial_x^{-1}f\in L^2(\mathbb R)\}$$, $$s>\frac34$$, using techniques developed by C. E. Kenig, G. Ponce and L. Vega [J. Am. Math. Soc. 4, No. 2, 323–347 (1991; Zbl 0737.35102)]. For the case $$\beta\cdot\gamma>0$$, we also obtain a global result in $$X_1$$, using appropriate conservation laws.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 86A05 Hydrology, hydrography, oceanography
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