Wellposedness of fully nonlinear KdVtype evolution equations
Abstract
We study the wellposedness of the initial value problem for fully nonlinear evolution equations, [ image ] where f may depend on up to the first three spatial derivatives of u. We make three primary assumptions about the form of [ image ] 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 righthand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdVtype. We prove the wellposedness of the initial value problem in the Sobolev space [ image ], which is close to optimal for the energy estimates we make. The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data. There is evidence that the backward diffusion condition we express is optimal.
 Publication:

Nonlinearity
 Pub Date:
 August 2019
 DOI:
 10.1088/13616544/ab1bb3
 arXiv:
 arXiv:1810.05117
 Bibcode:
 2019Nonli..32.2914A
 Keywords:

 wellposedness;
 dispersive;
 gauged energy method;
 antidiffusion;
 Mathematics  Analysis of PDEs
 EPrint:
 doi:10.1088/13616544/ab1bb3