Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation

Abstract

<p style='text-indent:20px;'>We consider the low regularity behavior of the fourth order cubic nonlinear Schrödinger equation (4NLS) <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \begin{cases} i\partial_tu+\partial_x^4u = \pm \vert u \vert^2u, \quad(t,x)\in \mathbb{R}\times \mathbb{R}\\ u(x,0) = u_0(x)\in H^s\left(\mathbb{R}\right). \end{cases} \end{align*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>In [<xref ref-type="bibr" rid="b29">29</xref>], the author showed that this equation is globally well-posed in <inline-formula><tex-math id="M1">\begin{document}$ H^s(\mathbb{R}), s\\\geq -\frac{1}{2} $\end{document}</tex-math></inline-formula> and mildly ill-posed in the sense that the solution map fails to be locally uniformly continuous for <inline-formula><tex-math id="M2">\begin{document}$ -\frac{15}{14}&lt;s&lt;-\frac{1}{2} $\end{document}</tex-math></inline-formula>. Therefore, <inline-formula><tex-math id="M3">\begin{document}$ s = -\frac{1}{2} $\end{document}</tex-math></inline-formula> is the lowest regularity that can be handled by the contraction argument. In spite of this mild ill-posedness result, we obtain an a priori bound below <inline-formula><tex-math id="M4">\begin{document}$ s&lt;-1/2 $\end{document}</tex-math></inline-formula>. This an a priori estimate guarantees the existence of a weak solution for <inline-formula><tex-math id="M5">\begin{document}$ -3/4&lt;s&lt;-1/2 $\end{document}</tex-math></inline-formula>. Our method is inspired by Koch-Tataru [<xref ref-type="bibr" rid="b17">17</xref>]. We use the <inline-formula><tex-math id="M6">\begin{document}$ U^p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ V^p $\end{document}</tex-math></inline-formula> based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.