- Main
Low Regularity Solutions of Korteweg-de Vries and Chern-Simons-Schr\"{o}dinger Equations
- Liu, Baoping
- Advisor(s): Tataru, Daniel
Abstract
The aim of this thesis is to understand the locall wellposedness
theory for some nonlinear dispersive equations at low regularity.
The Korteweg-de Vries equation has sharp wellposedness at
$H^{-\frac{3}{4}}$ if we are concerned about the Lipschitz
dependence of solutions on the initial data. For lower regularity,
one might still have a weaker form of wellposedness only with
continuous dependence on data. Here we prove that the smooth
solutions satisfy a-priori local in time $H^s$ bound in terms of
the $H^s$ size of the initial data for $s\geq -4/5$. Together with
the bounds we obtained on the nonlinearity, the result here
ensures that the equation is satisfied in the sense of
distributions even for weak limits.
The Chern-Simons-Schr\"{o}dinger equation is a planar gauged
Schr\"{o}dinger equation which has some similarity to the
derivative formulation of the Schr\"{o}dinger map problem. We work
on to prove local wellposedness in the full subcritical range
$H^s(\mathbb{R}^2), s>0$.
One important idea in working on these problems is to find a
suitable space to characterize the solution. We use $X^{s,b}$
spaces introduced by Bourgain, and $U^2$, $V^2$ spaces introduced by Koch and Tataru. For the Chern-Simons-Schr\"{o}dinger equation, we also need to fix a suitable gauge to make the problem well-posed. The heat gauge is a variation of Coulomb gauge, and it serves as a good candidate for this problem.
Main Content
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