Well-posedness of problems in fluid dynamics (a fluid-dynamical point of view)

The proofs of the existence, uniqueness, smoothness, and stability of solutions of problems in fluid dynamics are needed to give meaning to the equations and corresponding initial and boundary conditions that govern these problems.
For any arbitrary reasonable choice of a class of admissible initial data, a problem in fluid dynamics must be well posed (in the Hadamard sense [1]). This means that (a) the problem has a solution for any initial data in this class, (b) this solution is unique for any initial conditions, (c) the solution depends continuously on the initial data. In this paper we give a survey of some aspects of problems on well-posedness from the point of view of fluid dynamics itself; these problems form a very difficult and at the same time important part of fluid mechanics.