Long time bounds for the periodic Benjamin-Ono-BBM equation (Q838117)

This paper deals with the existence, uniqueness and global well-posedness in time of the Cauchy problem associated with the Benjamin-Ono-BBM equation \[ u_t + u_x +\alpha u u_x +\beta H (u_{xt}) = 0,\tag{1} \] \[ u (x, 0) = f (x) ,\tag{2} \] where \(0 <\alpha ,\beta\leq 1\), \(\alpha\) is the ratio of the wave amplitude \(a\) and depth \(h\), \(\beta\) is the ratio of the depth \(h\) and wavelength \(\lambda\), and \(H\) is the Hilbert transform. Using the Brézis-Gallouët method [\textit{H. Brézis} and \textit{T. Gallouët}, Nonlinear Anal., Theory Methods Appl. 4, 677--681 (1980; Zbl 0451.35023)], and writing equation (1) in the normal form \[ v_t + L (v) = F (v) ,\tag{3} \] where \(L\) is the linear operator of the BO-BBM equation and \(F\) is the multilinear operator of order greater than 2, the author proves his main result in the form of the following Theorem: If \(0 <\alpha ,\beta\leq 1\) and \(s > \frac12\), then there exist \(0 < \varepsilon_0 <\beta^2 /\alpha\), \(C_1 > 0\) and \(C_2 > 0\) such that if \(f\in H_0^s (T )\) with \(\|f\|_s\leq\varepsilon\) for \(0 <\varepsilon <\varepsilon_0\), then the unique solution \(u\) of the periodic equation (1) with the initial data (2) satisfies \[ \|u (t)\|_s\leq C_1\varepsilon, \tag{4} \] for \(|t|\leq C_2(\frac{1}{\varepsilon}\frac{\beta}{\alpha})^2\). In other words, if the initial data of size \(\varepsilon\), the solution remains smaller than \(\varepsilon\) for a time scale of order \((\varepsilon^{-1}\beta/\alpha)^2 \), whereas the local well-posedness provides only a time scale of order \((\varepsilon^{-1}\beta/\alpha)^2 \). Finally, the author considers the Cauchy problem of the KP-BBM-II equation in the form \[ v_t + v_x +\alpha v v_x -\beta v_{xxt} +\gamma\partial_x^{-1} v_{yy}=0,\tag{5} \] where \(\alpha =\frac a h\), \(\beta =(\frac h\lambda)^2\), (\(\lambda\) = wavelength in the \(x\)-direction) and \(\gamma\) is the square of the ratio of the wavelengths in the two directions of the surface. Based on the local in time well-posedness of the Cauchy problem, there exist two constants \(C_1 > 0\) and \(C_2 > 0\) such that the initial data \(f\) satisfies \(\|f\|_s\leq\varepsilon\) for \(s > 1\), then for \(|t| < C_2\varepsilon^{-1} (\beta/\alpha)\), the solution \(v\) satises \(\|v\|_s < C_1\varepsilon\). This bound cannot be extended to a longer time.