Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing

Abstract

<p style='text-indent:20px;'>We establish a smoothing result for the generalized KdV (gKdV) on the torus with polynomial non-linearity, damping, and forcing that matches the smoothing level for the gKdV at <inline-formula><tex-math id="M1">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>. As a consequence, we establish the existence of a global attractor for this equation as well as its compactness in <inline-formula><tex-math id="M2">\begin{document}$ H^s(\mathbb{T}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ s\in (1, 2). $\end{document}</tex-math></inline-formula></p>