Lagrangian Navier-Stokes Diffusions and Trudinger-Moser Inequalities on non compact Riemannian manifolds

Abstract

We give the eigenvalue estimate , a variational analysis , multiple solutions with stability and instability of Einstien-scalar field Lichnerowicz equations on compact Riemannian manifolds. We construct a minimization problem arising from prescribing scalar curvature functions , a variational principle for the Navier-Stokes equation, the estimate of the first eigenvalue of the Laplacian on a compact Riemannian manifolds and of a closed manifold with positive Ricci curvature. We characterize the Navier-Stokes flow, equations and diffusions on Riemannian manifolds and on the group of homomorphisim of the torus. We also characterize the Lagrangian Navier-Stokes diffusions on manifolds. We discuss the sharp form of the trace Moser-Trudinger inequality on compact and complete Riemannian surface with boundary and complete non-compact Riemannian manifolds.