On the Uniqueness of Weak Solutions of Navier-Stokes Equations: Remarks on a Clay Institute Prize Problem Uniqueness of Weak Solutions of Navier-Stokes Equations

We consider the Clay Institute Prize Problem asking for a mathematical analytical proof of existence, smoothness and uniqueness (or a converse) of solutions to the incompressible Navier-Stokes equations. We argue that the present formulation of the Prize Problem asking for a strong solution is not reasonable in the case of turbulent flow always occuring for higher Reynolds numbers, and we propose to focus instead on weak solutions. Since weak solutions are known to exist by a basic result by J. Leray from 1934, only the uniqueness of weak solutions remains as an open problem. To seek to give some answer we propose to reformulate this problem in computational form as follows: For a given flow what quantity of interest can be computed to what tolerance to what cost? We give computational evidence that quantities of interest (or output quantitites) such as the mean value in time of the drag force of a bluff body subject to a turbulent high Reynolds number flow, is computable on a PC up to a tolerance of a few percent. We also give evidence that the drag force at a specific point in time is uncomputable even on a very high performance computer. We couple this evidence to the question of uniqueness of weak solutions to the Navier-stokes equations, and thus give computational evidence of both uniqueness and non-uniqueness in outputs of weak solutions. The basic tool of investigation is a representation of the output error in terms of the residual of a computed solution and the solution of an associated linear dual problem acting as a weight. By computing the dual solution coupled to a certain output and measuring the energy-norm of the dual velocity, we get quantitiative information of computability of different outputs, and thus information on output uniqueness of weak solutions.