Time: 16:00 to 17:00
Place: 216 McBryde
Speaker: Bing-yu Zhang of University of Cincinnati
Title: On Conditional and Unconditional Well-posedness for the Cauchy Problem of Nonlinear Evolution Equations
| du/dt + Lu = N(t, u), u(0) = f. | (1) |
Xs
if r > s. For small values of s, some equations of the form in
(1) are well-posed in a conditional sense that the uniqueness
aspect depends upon the imposition of auxiliary conditions.
In the latter context, it is natural to inquire whether or not
the auxiliary condition are essential to securing uniqueness.
In this talk we will show that for a conditionally well-posed Cauchy problem (1), the auxiliary specification is removable if and only if a certain persistence of regularity holds. As a consequence, it will transpire that conditionally well posed problem (1) is (unconditionally) well posed if and only if the aforementioned persistence property holds.
These results will be applied to study several recent conditional well-posedness results of the KdV equation, the nonlinear Schrödinger equations and the nonlinear wave equations. It will be demonstrated that those auxiliary conditions used to secure the uniqueness are all removable and the corresponding Cauchy problems are, in fact, unconditionally well-posed. In addition, the well-posedness of the initial-boundary-value problem of the generalized KdV equation posed in a quarter plane will be also considered. An affirmative answer will be provided for a uniqueness problem left open by Colliander and Kenig in their recent work: The generalized KdV equation on the half-line.
The results presented in this talk are joint work with Jerry L. Bona and Shu Ming Sun.
Return to