In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig-Ponce-Vega from the Quasi-Linear Schrödinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.2000 Mathematics Subject Classification. Primary: 35Q53, 35G20.
An initial value problem for a very general linear equation of KdVtype is considered. Assuming non-degeneracy of the third derivative coefficient, this problem is shown to be well-posed under a certain simple condition, which is an adaptation of the Mizohata-type condition from the Schrödinger equation to the context of KdV. When this condition is violated, ill-posedness is shown by an explicit construction. These results justify formal heuristics associated with dispersive problems and have applications to non-linear problems of KdVtype.
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.
An initial value problem for a very general linear equation of KdV-type is considered. Assuming non-degeneracy of the third derivative coefficient this problem is shown to be wellposed under a certain simple condition, which is an adaptation of Mizohata-type condition from the Schrödinger equation to the context of KdV. When this condition is violated ill-posedness is shown by an explicit construction. These results justify formal heuristics associated with dispersive problems and have applications to non-linear problems of KdV-type.
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