The Geometry and Analysis of Non-newtonian Fluids and Vortex Methods

This paper is devoted to the geometric analysis of some initial-boundary value problems for differential-type fluids that commonly arise in non-Newtonian hydrodynamics. We shall mainly be concerned with second-grade fluids, because of their prominent role in modeling polymer flow, and because of the surprising fact that the mathematical equations which describe their motion not only coincide with the averaged Euler equations (averaged Navier-Stokes equations when viscosity is present) but are exactly Chorin’s vortex blob method for integrating the Euler equations when a certain type of blob or cut-off function is used. Remarkably, this system of equations is governed by the geodesic flow of a new right invariant Riemannian metric on any one of three new subgroups of the volume-preserving diffeomorphism group of a compact n dimensional smooth Riemannian manifold with boundary. We are able to obtain local well-posedness results for classical solutions to these PDEs on arbitrary compact n dimensional Riemannian manifolds by transferring the problem from the usual and difficult Eulerian setting to the Lagrangian setting of flow on certain subgroups of the diffeomorphism group of the fluid container, thus extending the program initiated by Arnold [4] and Ebin-Marsden [18] some thiry years ago. We prove the existence of new C∞ subgroups of the Hilbert Hs-class diffeomorphism and volume-preserving diffeomorphism groups which are in one-to-one correspondence with the Neumann, Dirichlet, and mixed boundary value problems of second-order elliptic equations, and prove that flows on these subgroups, governed by vector fields which are zeroth-order differential operators, give solutions to the above PDEs. We also establish the existence of a new C∞ projection map, which we call the Stokes projector (and which generalizes the L Hodge-Helmholtz projector), which maps vector fields onto the divergence-free vector fields that satisfy either the Neumann, Dirichlet, or mixed boundary conditions. We prove existence of unique classical solutions which are C∞ functions of time, and have C∞ dependence on initial data; as Kato proved, this cannot, in general, be expected of hyperbolic systems in Eulerian coordinates, where only C smoothness can be found. For the case of two-dimensional Riemannian manifolds, we prove global well-posedness. Owing to the geometric properties of the solutions to this non-Newtonian system, their linearization about a particular solution is given by the Jacobi equation of the new right invariant metric which we shall introduce. We are thus able to prove wellposedness of the linearized system, by showing that the weak infinite-dimensional Date: May 29, 1999; current version September 28, 1999. 1991 Mathematics Subject Classification. 35Q35, 35Q53, 58B20, 58D05. 1