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The goal of this paper is two-fold. Firstly, by using the Fourier restriction norm method and the fixed point theorem, we prove that the Cauchy problem for a generalized Ostrovsky equation $$ partial_{x} biggl(u_{t}-betapartial_{x}^{3}u +frac{1}{3}partial_{x}bigl(u^{3}bigr) biggr) - gamma u=0,quadbetau003e0,gammau003e0, $$is locally well-posed in (H^{s}(mathbf{R})) with (sgeqfrac{1}{4}). Secondly, we prove that the Cauchy problem for a generalized Ostrovsky equation is not well-posed in (H^{s}(mathbf{R})) with (su003cfrac{1}{4}) in the sense that the solution map is (C^{3}).

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion