Eulerian and Lagrangian stability of fluid motions

In this thesis, we study stability of inviscid fluids in the Lagrangian sense. That is, we look at how paths of particles diverge from each other under a small perturbation of the initial velocity field. The more usual question of stability is in the Eulerian sense, when one looks at how velocities at each point diverge from a steady state under a small perturbation. We demonstrate, for certain restricted types of flows as well as for some very particular examples, the connection between these two forms of stability. In particular, we observe that in many cases, Eulerian stability is incompatible with exponential Lagrangian instability, although it is compatible with polynomial Lagrangian instability. Several of the most important features of fluid stability, especially in the incompressible case, are actually general properties of Lie groups with one-sided invariant metrics. In finite dimensions, we can perform many computations on such groups explicitly. For example, the geodesic equation linearized about a steady solution is just a differential equation with constant coefficients, whose solution may always be found explicitly. In addition, the Jacobi equation in this case can always be written as two decoupled first-order equations, and this makes them even easier to study. There is a simple criterion for the Eulerian stability of a steady flow on a Lie group; we verify that it captures nearly all instances of Eulerian stability on a three-dimensional group. On the other hand, we compile some of the known criteria for Lagrangian stability and demonstrate that they are much less broadly applicable. Since these results all involve constant-coefficient differential equations, we are able to avoid much technical machinery and prove most results quite directly. We finally list many examples of Jacobi fields and curvature tensors which behave unlike what one might intuitively expect. For example, we have Jacobi fields which grow exponentially despite having a strictly positive curvature tensor (the Rauch comparison theorem only bounds the Jacobi field up to the first conjugate point). We have geodesics along which the curvatures are both positive and negative, and such that their Jacobi fields can oscillate, grow exponentially, or even grow polynomially. (Surprisingly, directions of negative