The local well-posedness, existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude

Abstract

This paper deals with the Cauchy problem for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime, which was proposed by Constantin and Lannes (2009) [20]. Applying the pseudoparabolic regularization technique, the local well-posedness of strong solutions in Sobolev space Hs(R) with s>3/2 is established via a limiting procedure. Moreover, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1<s≤3/2 is obtained.