The author considers the low regularity of the initial value problem for the Kawahara equation \[ \partial_t u +\alpha \partial_x^5 u +\beta\partial_x^3u +\partial_x(u^2)=0, \quad (t,x)\in[0,T]\times \mathbb{R}, \]
\[ u(0,x)=u_0(x),\quad x\in\mathbb{R}. \] Note that the Kawahara equation is a fifth-order KdV equation; however, it has fewer symmetries than the KdV equation, in particular, it is not completely integrable.
The main result of the paper establishes a local well-posedness with initial data in the Sobolev space \(H^{s}(\mathbb{R})\) if \(s\geq-2\). Also, the author shows that some ill-posedness occurs if \(s<-2\). Note that the local well-posedness in \(H^{s}(\mathbb{R})\) with \(s>-7/4\) was established previously in [W. Chen et al., J. Anal. Math. 107, 221–238 (2009; Zbl 1181.35229)] and with \(s\geq -7/4\) in [W. Chen and Z. Guo, J. Anal. Math. 114, 121–156 (2011; Zbl 1238.35127)]. Moreover, in the last paper it was shown that the initial value problem is globally well-posed in \(H^s(\mathbb{R})\) with \(s\geq -7/4\).