Remarks on the uniqueness of weak solutions of the incompressible Navier-Stokes equations (original) (raw)
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At first we identify the main error in the formulation of the concept of the weak solution to Navier-Stokes (NS) equations which is the completely insufficient treatment of the incompressibility condition on the fluid (expressed in the standard way by div u = 0). The repair requires the complete reformulation of the NS problem. The basic concept must be the generalized motion (i.e. the generalized flow) which replaces the standard velocity field. Here we define the generalized flow on the bases of Geometric measure theory extended to the theory of Cartesian currents and weak diffeomorphisms (see [1], [2]). Then the key concept of the complete weak solution to the NS problem is defined and the two conjectures (the existence and the regularity ones) concerning the complete weak solutions are formulated. In two appendices many technical details are described (concerning e.g. Cartesian currents, homology conditions, weak diffeomorphisms, etc.). Our approach is based on the unification of the standard analysis of NS equations with the methods of Geometric measure theory and of the theory of Cartesian currents.
ON THE UNIQUENESS OF BOUNDED WEAK SOLUTIONS TO THE NAVIER-STOKES CAUCHY PROBLEM
In this note we give a uniqueness theorem for solutions (u, π) to the Navier-Stokes Cauchy problem, assuming that u belongs to L ∞ ((0, T ) × R n ) and (1 + |x|) −n−1 π ∈ L 1 (0, T ; L 1 (R n )), n ≥ 2. The interest to our theorem is motivated by the fact that a possible pressure field π, belonging to L 1 (0, T ; BMO), satisfies in a suitable sense our assumption on the pressure, and by the fact that the proof is very simple.
2004
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.
Applied Mathematics, 2018
We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.