Well-posedness of fully nonlinear KdV-type evolution equations

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ 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 $H^{7}(\mathbb{R}).$ The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.