Abstract
We study a Gross-Pitaevskii (GP) system associated with a Bose-Einstein condensate (BEC). The two-component system is highly related to fingering instabilities. The latter arises in many fields ranging from nuclear and supernova explosions to magnetic fluids. Recently, some numerical results have been obtained in Caries (Portugal Math NS 65:191-209,2008), after the authors were able to reduce (GP) to a lower dimension (2D), instead of (3D). This reduction eased things numerically, but have resulted in a new type of (2D) dipolar interaction that presents some mathematical challenges. Moreover, the new system is (2D) mass-critical. In this paper, we study the local and global energy well-posedness of the system. Using variational methods, we obtain a sharp threshold for blowup, which seems to agree with the physical experiments of Caries (Portugal Math NS 65:191-209, 2008).