concerning the Navier-Stokes equations
The Navier-Stokes Equations (NSE) describe a flow of incompressible, viscous fluid. The three central questions of every PDE is about existence, uniqueness and smooth dependency on initial data can develop singularities in finite time, and what these might mean. For the NSE satisfactory answers to those questions are available in two dimensions, i.e. 2D-NSE with smooth initial data possesses unique solutions which stay smooth forever. In three dimensions, those questions are still open. Only local existence and uniqueness results are known. Global existence of strong solutions has been proven only, when initial and external forces data are sufficiently smooth. Uniqueness and regularity of non-local Hopf solutions are still open problems. The question of global existence of smooth solutions vs. finite time blow up (ODE problem) is one of the Clay Institute millennium problems.
We provide a global unique weak H(-1/2) solution of the Navier-Stokes initial value problem. The complete content of this homepage is w/o any review and aknowledgement of academic institution.
The Navier-Stokes equations describe the motion of fluids. The Navier–Stokes existence and smoothness problem for the three-dimensional NSE, given some initial conditions, is to prove that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.
The d’Alembert “paradox” is not about a real paradox but it is about the failure of the Euler equation (the model of an ideal incompressible fluid) as a model for fluid-solid interaction. The difficulty with ideal fluids and the source of the d’Alembert paradox is that in incompressible fluids there are no frictional forces. Two neighboring portions of an ideal fluid can move at different velocities without rubbing on each other, provided they are separated by streamline. It is clear that such a phenomenon can never occur in a real fluid, and the question is how frictional forces can be introduced into a model of a fluid.
With respect to the below we mention that there are basically two kinds of required "energy" measures as part of the energy inequality which are defined by the H(1/2)- and the L(2)(H(1/2)) norms. The first one could be interpreted as the quantum energy measure while the second one could be interpreted as the (average) energy of this quantum energy on the "classical "phenomenon PDE level.
Some further details building on the paper / book of M. Cannone (CaM) / M. Shinbrot (ShM), whereby all not itallic marked text is cited from those references:
Given a smooth datum at time zero, will the solution of the NSE continue to be smooth and unique for all time?
There is no uniqueness proof except for over small time intervals:
the existence of weak solutions can be provided, essentially by the energy inequality. If solutions would be classical ones, it is possible to prove their uniqueness. On the other side for existing weak solutions it is not clear that the derivatives appearing in the inequalities have any meaning.
It has been questioned whether the NSE really describes general flows.
The difficulty with ideal fluids, and the source of the d'Alembert paradox, is that in such fluids there are no frictional forces. Two neighboring portions of an ideal fluid can move at different velocities without rubbering on each other, provided they are separated by a streamline. It is clear that such a phenomenon can never occur in a real fluid, and the question is how frictional forces can be introduced into a model of a fluid.
The uniqueness question is among the most important unsolved problems in fluid mechanics: “instant fame awaits the person who answers it. (Especially if the answer is negative!)” uniqueness of the solutions of the equations of motion is the cornerstone of classical determinism.
Moreover, as for the solutions of the Euler equations of ideal fluids, or the Boltzmann equation of rarefied gases, or the Enskog equation of dense gases either.
The intention of the sections related to the work of J. Plemelj is to motivate an alternative mathematical framework, which does not require “ideal” boundary layer assumptions. The mathematical requirements to define boundary layers and corresponding potentials are very much depending from the definition and regularity requirements of the normal derivative. It is perpendicular to the boundary itself and therefore requires regularity assumptions, affecting "points" outside of the domain w/o any physical meaning.
There is a similar challenge related to Einstein's field equations, when trying to describe a ground state energy in a vacuum: tensor analysis, manifolds and the affine connexion concept require differentiable manifolds, while for the physical model it would be sufficient to require continuous manifolds only:
The question intimately related to the uniqueness problem is the regularity of the solution. Do the solutions to the NSE blow-up in finite time? The solution is initially regular and unique, but at the instant T when it ceases to be unique (if such an instant exists), the regularity could also be lost.
In the twentieth century, instead of explicit formulas in particular cases, the problems were studied in all their generality. This leads to the concept of weak solutions. The prize to pay is that only the existence of the solutions can be ensured. In fact the construction of weak solutions as the limit of subsequence of approximations leaves open the possibility that there is more than one distinct limit, even for the same sequence of approximations.
The intention of the sections related to the footprints of J. A. Nitsche is to motivate approximation theory in Hilbert scale (which included approximation theory of negative Hilbert scale and semi-group generated even more weak norms). This is, when
- Navier meets Fourier enabled by the stationary linear NSE case, the self-adjoint, bounded Stokes operator and its fractional powers, enabling the definition of fractional, negative Hilbert space inner product and norm to reduce the power of 3 of the square of the Sobolev norm NSE solution down to 2, ending up in a Riccati-type ordinary differential equations solution problem
- Ritz-Galerkin and semi-group approximation methods meet (regular and non-regular) non-linear parabolic equations (n=1, Stefan problems), whereby appropriate time-weight norms govern the balance of the linear (motion) and the non-linear (vortex) terms.
For the stationary NSE the existence of a solution has been proven for all space dimensions. The uniqueness has been proven for n<5 under certain conditions to the data. The underlying functions space is a reflexive, separable Banach space X, which is compact embedded into its dual space X(*).
For the full non-linear NSE case (non-stationary, non-linear) the classical solution definition are provided in (CaM). Basically the existence of solutions is proven only for “large” Banach spaces. The uniqueness is proven only in “small” Banach spaces.
M. Shinbrot, Lectures on fluid mechanics, Dover Publications, New York, 2012
"In fact, to date, 3D regular flows are known to exist either for all times but for data of "small size", or for data of "arbitrary size" but for a finite interval of time only, .... it is not known whether, in the 3D case, the associated initial-boundary value problem is well-posed in the sense of Hadamard, ...
"The prescription of the pressure p as a solution of the Neumann problem at the boundary walls or at the initial time is independently of the velocity v and, therefore, could be incompatible with the initial-boundary value NSE problem, which could render the problem ill-posed."
G. P. Galdi, The NSE: A Mathematical Analysis