Well-posedness of fully nonlinear KdV-type evolution equations

Abstract

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, ut = f [u], where f may depend on up to the first three spatial derivatives of u.We make three primary assumptions about the form of f : a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion.Because the third derivative of u is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type.We prove the well-posedness of the initial value problem in the Sobolev space H 7 (R).The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.

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