Uniqueness and Stability for an Ideal Fluid

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This project addresses mathematical problems related to the motion of an ideal fluid. The question of uniqueness will be studied for the system of equations describing fluid motion with special emphasis on "rough" flows. In this situation, the trajectories of fluid particles may not be uniquely defined at some points. The vanishing viscosity limit is deeply connected with uniqueness and will be investigated for such classes of flows. Qualitative properties of a possible blow-up set will also be considered. Another line of research focuses on the linear stability of an ideal fluid. The main question to be addressed in this area is how the growth rate of a small perturbation is related to the location of the spectrum of small oscillations. The interrelation between linear instability and nonlinear instability for flows of an ideal incompressible fluid will be studied.Mathematical study of the basic models describing fluid motion forms a foundation for a number of applications, such as meteorology, geophysics, astrophysics, engineering, and the theory of turbulence. This research explores one of the two commonly used models describing the motion of fluids, and investigates whether the initial distribution of velocity in a fluid flow determines the future velocity distribution. Stability of general classes of fluid flows will be studied, and in particular, the question of whether stability can be determined by characteristics of small oscillations. Potential applications of this research include modeling of large and small scale structures in turbulence, as well as stability of vortices in the atmosphere and ocean.