Long Time Dynamics for Defocusing Cubic Nonlinear Schrödinger Equations on Three Dimensional Product Space
Zehua Zhao, Jiqiang Zheng
Abstract
In this article, we study long time dynamics for defocusing cubic nonlinear Schrödinger equations (NLS) on three dimensional product space. First, we apply the decoupling method in Bourgain and Demeter [Ann. of Math. (2), 182 (2015), pp. 351--389] to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on a three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on the I-method and the bilinear estimate. At last, we discuss the growth of the higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.
Topics & Concepts
MathematicsBilinear interpolationNonlinear systemSobolev spaceDecoupling (probability)Product (mathematics)Mathematical analysisNorm (philosophy)Space (punctuation)Dynamics (music)SpacetimeGeometryPhysicsQuantum mechanicsLawLinguisticsControl engineeringEngineeringAcousticsPolitical sciencePhilosophyStatisticsAdvanced Mathematical Physics ProblemsNonlinear Waves and SolitonsBlack Holes and Theoretical Physics