No abstract

Abstract. In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces B s 2,r in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-tosolution map provided that the initial data belongs to B s 2,r . We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B s 2,r to C([−T, T ]; B s 2,r ). Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.

The Novikov equation (NE) has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the Camassa-Holm and Degasperis-Procesi equations, which have quadratic nonlinearities. NE is well-posed in Sobolev spaces Hs on both the line and the circle for s > 3/2, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. This work studies the continuity properties of NE further. For initial data in Hs, s > 3/2, it is shown that the solution map for NE is Hölder continuous in Hr-topology for all 0 ⩽ r < s with exponent α depending on s and r.

The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space H s (R 2 ) is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is continuous. We prove that the data-to-solution map on the plane is not uniformly continuous on any bounded subset of Sobolev class functions.