Continuity properties of the solution map for the generalized reduced Ostrovsky equation

It is shown that the data-to-solution map for the generalized reduced Ostrovsky (gRO) equation is not uniformly continuous on bounded sets in Sobolev spaces on the circle with exponent s>3/2. Considering that for this range of exponents the gRO equation is well posed with continuous dependence on initial data, this result makes the continuity of the solution map an optimal property. However, if a weaker Hr-topology is used then it is shown that the solution map becomes Hölder continuous in Hs.