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.