2011/09/08 by Boris Khesin, Jonatan Lenells, Khesin, Boris +5
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1109.1816
Written for "Geometric and Algebraic Structures in Mathematics," a conference in celebration of Dennis Sullivan's 70th birthday
arxiv created 2011/09/09 · arxiv updated 2011/09/12
Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev H0 and H1 energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.