The longitudinal analysis of the brain morphology in Alzheimer's disease(AD) is fundamental for understanding and quantifying the dynamics of the pathology. This study provides a new measure of the brain longitudinal changes based on the Helmholtz decomposition of deformation fields. We used the scalar pressure map associated to the irrotational component in order to identify a consistent group-wise set of areas of maximal volume change. The atrophy was then quantified in these areas for each subject by the probabilistic integration of the flux of the longitudinal deformations across the boundaries. The presented framework unifies voxel-based and regional approaches, and robustly describes the longitudinal atrophy at group level as a spatial process governed by consistently defined regions. Our experiments showed that the resulting regional flux analysis is able to detect the differential atrophy patterns across populations, and leads to precise and statistically powered quantifications of the longitudinal changes in AD, even in mild/premorbid cases.