An Improved Global Well-Posedness Result for the Modified Zakharov Equations in 1-D
Commun. Korean Math. Soc. Published online January 20, 2022
Agus Leonardi Soenjaya
Merlion School
Abstract : The global well-posedness for the fourth-order modified Zakharov equations in 1-D, which is a system of PDE in two variables describing interactions between quantum Langmuir and quantum ion-acoustic waves is studied. In this paper, it is proven that the system is globally well-posed in $(u,n)\in L^2\times L^2$ by making use of Bourgain restriction norm method and $L^2$ conservation law in $u$, and controlling the growth of $n$ via appropriate estimates in the local theory. In particular, this improves on the well-posedness results for this system in [Guo-Zhang-Guo, Z. Angew. Math. Phys. 64 (2013), 52-68] to lower regularity.
Keywords : global well-posedness, low regularity, modified Zakharov equations
