A solution concept for some still open questions
Disclaimer: Only a solution concept is proposed to find answers to some still open fundamental questions about the existence & smoothness of "classical" solutions of the Navier-Stokes equations.
If the proposed solution concept enables people to develop a solution, this is appreciated and the primarily intension of this internet page.
For background information we refer to
C. L. Fefferman, "Existence & Smoothness of the Navier-Stokes Equation":
Baseline (the "what"?)
Quote (C. Fefferman): "Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas"
Rational (the "why"?)
The NSE are formulated as initial value problem for the unknown functions (u, p), where there is no initial value given/required for the pressure p. In a PDE lecture, just by voting of the present students, this would be classified as a not well-posed problem. The basic idea is, that there is a chance for an appropriate initial value of the pressure p in a less regular Hilbert space environment, which enables a well-posed NSE formulation. From a Sobolev-Hilbert scale perspective this goes in line with a closure of the Serrin gap.
If the below proposed approach ("how") would become successfull, then the non-well-posedness of the continuous case should become a straightforward task with standard functional analysis arguments.
Tools (the "where with/ with what"?)
The proposed approach are based on footprints of A. M. Arthurs, J. Plemelj and J.A. Nitsche:
- J. Plemelj: Alternative normal derivative definition and Stieltjes integral, requiring less regularity assumptions than standard definition; the "achieved" "regularity reduction" is in the same size as a reduction from a C(1) to a C(0) regularity
- J. A. Nitsche: Finite element approximations and non-linear parabolic differential equation with non-regular initial value function
- A. M. Arthurs, "Complementary variational principles", Clarendon Press, Oxford, 1970.
Approach (the "how"?)
For the space dimension n=3 the"natural" energy norm H(1) is being replaced by an appropriate alternative with reduced regularity requirements. The concept of J. Plemelj still keeps the physical model relevance and, at the same time, strengths the capabilities of a potential. For the non-stationary case this norm is modified by timely-weighted (integral-dt) norms. For such a Hilbert space there is a quasi-optimal shift theorem for the NSE (to be proven,based on footprints of J. A. Nitsche). The initial value function for the pressure is basically of Dirac function type. We note, that the embedding properties of the Dirac function into the continuous function space is depending from the space dimension. The same is valid for the Serrin gap.
The regularity of the Dirac function depends from its space-domain dimension. In case of n=3 this is expected to lead to a requirement of a non-integer energy Hilbert space. As proper "transformation tool" the Hilbert transform (resp. its n-dimensional transformation analog, the Riesz transforms) is proposed. The mathematical variational framework is provided by "Arthurs' " complementary variational theory.The correspondingly required reduced regularity assumptions to the normal derivative is "delivered" by the concept of J. Plemelj. The Oxygen-Diffusion-problem in the context of the one-dimensional Stefan problem leads for standard (free boundary) inital value data to transformed (non-linear parabolic) initial value data in form of a Dirac function (see J. A. Nitsche, "A finite element method for parbolic free boundary problems"). "Quasi-optimal" error estimates of corresponding FE approximations are still missing for such inital value problems, basically due to not adequate existing shift theorems resp. not appropriately defined adequate Hilbert or Banach space providing the "right" energy norm. A first step for a generalization of the one-dimensional Stefan problem to the two-dimensional analog of the NSE has been made by J. A. Nitsche built on a Hölder space framework ("Free boundary problems for Stokes flows and finite element methods").
Result (the "where"?)
In an appropriately defined Hilbert space framework an uniquely defined well posed weak solution is equivalent to the corresponding strong solution.Then, by standard functional analysis methods the Serrin gap enables the building of a counter example of a weak solution, which cannot have a corresponding strong solution in case of the "standard-energy-H(1)-Hilbert space".
A similar situation is given in case of the most simple non-linear parabolic approximation problem, which is the Stefan problem. A proof of quasi-optimal finite element approximations to the weak solution in case of non-regular initial value functions is still missing. A "nearly" quasi-optimal convergence has been proven by J. A. Nitsche, using defined timely-integral weighted norms. We provide a solution concept, to prove the quasi-optimal convergence. It is suggested to be seen as a warming-up exercise before starting to go for the main run.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." – John von Neumann
start date: 01.06.2013