Cauchy problem for the Ostrovsky equation in spaces of low regularity (Q860752)

The paper aims to produce a rigorous proof of the local well-posedness (for finite times) of the initial-value problem for the equation derived by L. A. Ostrovskii in 1978 to describe long surface waves in the ocean in the rotating reference frame: \[ u_t - u_{xxx} \mp \partial_x^{-1}u + uu_x =0. \] The main result is that the local existence of the single solution to the equation with initial condition \(u_0(x)\) can be proved if the function \(u_0(x)\) belongs to the Sobolev space \(H^s\), with \(s>-1/2\), for the upper sign in the equation (in front of the nonlocal term), and to the Sobolev space with \(s>-3/4\), for the lower sign in the equation (the Sobolev space is defined as the space of functions with the finite measure \(\int_{-\infty}^{+\infty}k^{2s}| ^{u}(k)| ^2\,dk\), where \(^{u}(k)\) is the Fourier transform of the function). The proof uses special properties of the so-called Bourgain space for the solutions as functions of the two variables \(x\) and \(t\).