Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms
In this chapter, we study the hydrodynamics of a viscous incompressible fluid. The Lagrangian hydrodynamical systems (LHSs) of a viscous incompressible fluid were introduced in  as generalizations of those of an ideal incompressible fluid. Namely, these LHSs were defined as systems on the group of volume-preserving diffeomorphisms with an additional right-invariant force field which depends on the velocity of the fluid. In the tangent space at id, the force field is v△ where △ is the Laplace-de Rham operator and v is the viscosity coefficient. Note, however, that the operator △ does not preserve the space of H8-smooth vector fields. In fact, △sends H8-smooth vector fields to fields which belong to a broader Sobolev class. As a consequence, the method relies heavily on the theory of partial differential equations, leading to the loss of many natural geometric properties of the LHSs of an ideal incompressible fluid (Chap. 8) in the passage to a viscous incompressible fluid.
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