Continuity and Analyticity for the Generalized Benjamin–Ono Equation

Abstract

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s>3/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in Hr-topology for all 0≤r<s with exponent α depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.