concerning classical Navier-Stokes solutions
Only a solution concept is given to build answers to some still open fundamental questions about the existence & smoothness of "classical" solutions of the Navier-Stokes Equations (NSE).
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"
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.
Based on the energy method for n>2 there is no global boundedness result possible from the corresponding norm estimates. This is caused by corresponding time-depending (!) norm estimates for the non-linear term. Every solution of the related ordinary differential equation with respect to the time variable t blows up, due to the power of "3" of the Sobolev norm term in the ODE. Therefore no global estimates for those terms are possible.
The "power 3" challenge is a given due to Sobolev embedding theorem and corresponding Serrin scale conditions. A reduced Hilbert scale factor of the (energy) Hilbert space is proposed to reduce the problematic "power of 3" term down to "power of 2" (still problematic) term, as one element of a solution concept. The consequences to the physical model parameters would be to define a pressure force based on a non-harmonic pressure function at the boundary layers.
From a Galerkin approximantion method perspective addressing the non-regular behavior of the N-S-E solution(s) at zero and blow-up time go in line with local convergence analysis for non-linear parabolic equations with reduced regularity of the intial value function and/or not fulfilled compatibility conditions. The transformation of the (one dimensional) free boundary Stefan problem to a (non-linear) fixed boundary value equation provides the simplest model problem for such situations, which still has open questions, as well.
From a functional analysis perspective the above goes in line with (hyper) singular integral operators, which correspond to single / double layer potential operator, the tangential derivative of the single layer operator and the normal derivative of the double layer operator.
If the below proposed approach ("how") would become successfull, then the non-well-posedness of the N-S-E concerning the "strong solution" question could get a purely functional analysis answer.
Objective : the "where?"
In an appropriately defined distributional Hilbert space framework the incompressible (!) N-S-E are well posed. The shift on the Hilbert scale to the left closes the Serrin gap and "time-weighted" norms "control" the singular velocity and pressure behavior for t --> 0. At the same time the Hilbert scale shift jeorpadies the application of the Sobolev embedding theorem to make conclusions about classical solutions. By standard functional analysis this should enable the building of a counter example (following the idea of J. Heywood) that existing weak solutions in "standard" scaled Sobolev spaces lead to "strong solution".
Just from a common sense feeling the same argument should not be valid in case for compressible "fluids". This would be in line with J. Plemelj's concept of a mass element, replacing an only mass density, which require less regularity assumptions to enable the Hilbert scale shift to the left, while keeping the option to apply the Sobolev embedding theorem. This would also be in line with
A. M. Arthurs, "Complementary variational principles", ([ArA]), where one example (§ 4.8) is about nonlinear variation method applied to compressible fluid flow (see also section "idea").
Remark: It's suggested to apply J. Plemelj's theory also to the still not solved "radiation problem" (R. Courant, D. Hilbert, "Methods of mathematical physics, II", 1937, VI, §5, section 6, VI §10, section 3).
Approach: the "how?"
The proposed approach is based on footprints of J. Plemelj and J.A. Nitsche:
J. Plemelj proposed an alternative definition of a normal derivative, based on Stieltjes integral. It requires 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. Plemelj's concept is proposed to be applied to ensure physical model requirements modeled by normal derivatives within a distributional Hilbert space framework.
The concept of Pseudo-Differential operators is about the study of the differential and integral operators within the same algebra of operators. The concept of Hilbert scale is about an appropriate Hilbert space with respect to the eigenpairs of self-adjoint, positive definite operators with corresponding order (respecitvely coercive operators in combination with the Garding inequality). The study of uniqueness and existence of solution of the Navier-Stokes equation is about appropriate weak variational representation of the NSE and related approximation solutions, which converge to the NSE solution(s).
For the quasi-optimal approximation estimates of Ritz-Galerkin method in Hilbert scales (which, of course,also includes collocation methods) we refer to
Braun K., "Interior Error Estimates of the Ritz methods for Pseudo-Differential Equations".
Defining the right Hilbert space framework goes along with defining the proper Pseudo-Differential operators with their related domains (!). In this context we refer to the model (singular integral) Pseudo-Differential operators in
Braun K., "An alternative quantization of H=xp"
Braun K., "A new ground state energy model"
Lifanov, I. K., Nenashev A. S., "Generalized functions on Hilbert spaces, singular integral equations, and problems of aerodynamics and electrodynamics".
For the linkage to the normal derivative concept of Plemelj (especially to the double layer potential, as well as its normal derivative, a hyper singular operator of Calderon-Zygmund type, which is essentially a once differentation operator) we refer to
Amini S., "On Boundary Integral Operators for the Laplace and the Helmholtz Equations and Their Discretization":
- the operator defined by the normal derivative of the single layer potential is the dual operator of the double layer potential
- the operator defined by the tangential derivative of the single layer potential is the Hilbert transform (S(0)). It is discontinuous on the surfaces with a jump according to the Plemelj formula
- the operator of the normal derivative of the double layer potential is a hypersingular operator, which behaves as the derivative of the Hilbert transform operator (S(1)). It is continuous (!) on the surface.
The terms "vortex density" and "rotation of a fluid element" are sometimes used synomymly. The rotation of a vector field v describes the vortext field of v. The corresponding vortext force at a point x is defined as the product of the constant normal vector of all (infinitesimal small) areas F containing x with the rotation(v). The vortex density in combination with the concept of "circulation" (L. Prandtl) is interpreted as the route cause of the "rotation" of a fluid element. The definition of "circulation" is integral part of the definition of "vortext force". We propose the use Plemelj's concept/definition of a mass element (alternatively to a mass density) to define a "vortex" of a fluid element, which would require less regularity assumptions to the vector field v than H(1), but being still consistent with the definitions of fluid density, vortex force and rotation in case of corresponding H(1)- regularity assumptions.
We note that
- moving from the "convection form" to the "rotation form" goes along with a movement from "kinematic pressure" to "Bernoulli pressure"
- the Euler equation is nonlocal, i.e. one cannot compute the time derivative of the solution u at (x,t) only from the knowledge of the function u in the neighborhood of x at the time t. This gets proven by taking the divergence of the NSE, which gives the Laplacian of the pressure p. Therefore the pressure p is determined by local Information, but not the gradient of p (P. Constantin, "On the Euler equations of incompressible fluids").
J. A. Nitsche
1. "L(infinitety) boundedness of the FE Galerkin operator for parabolic problems"
2. several papers in the context of "Finite element approximations and non-linear parabolic differential equation"
3. "Direct proofs of some unusual shift theorems".
ad 1: The paper gives an ("optimal", i.e. shift on Hilbert scale by factor "2") shift theorem for the heat equation based on Fourier transforms estimates. The "trick" (which is about changing the order of integration with respect to the time-variable) to prove an optimal Sobolev scale shift by scale factor "-2" (analogue to the elliptic Laplace equation) is proposed to be applied for appropriately defined time-depending norm estimates for the Oseen kernel in the context of harmonic analysis for solving the incompressible Navier-Stokes equations.
ad 2: In this paper appropriately defined time-weighted Sobolev norm are applied aare applied to "manage" singular time coefficients. Following this concept analogue norms are proposed, which is about "a finite time integral over an integrand, which is built by the product of a weak singular time factor (=t exp(1/2), resp (=t exp(-1/2) depending from the Hilbert scale value) multiplied with an appropriate (square of a) Sobolev space-norm").
The link of 1. & 2. to the N-S-E is given by the Leray-Hopf operator and its Oseen kernels. The Fourier transformations of the Oseen kernel are required to apply "heat equation trick", ad 1). for the latter ones we refer to
[LeM] N. Lerner, "A note on the Oseen kernels"
[CaM] M. Cannone, "Harmonic analysis tools for solving the incompressible Navier-Stokes equations" ... "when Navier meets Fourier".
ad 3: For the Stokes equation an unusual shift therorem (negative norm estimates) has been proven by solving auxiliary problems, building on the Cauchy-Riemann differential equations. The effect is, that the Stokes problem is decoupled into two elliptic problems. The defintion of the auxiliary functions (w,z) for given C-R- relationships for (u,v) can be interpreted as defining curl(u) and div(v) as representations of the C-R equations (n=2). For n=3 the curl operator (which is for n=2 one of the two C-R-equation) is linked to the Leray-Hopf projector P (and the related Riesz operators) by the following properties (M. Lerner):
- the commutator (P,curl) vanishes
- P(curl) = curl
- (curl(u),curl(v)) =(gradP(u),gradP(v)) = D(P(u),P(v)).
We note that curl(u) describes the mean rotation of a fluid and 1/2*rot(u) gives the mean value of the angular velocity.
Replacing the gradient operator by the Calderon-Zugmund operator enables the definition of a "curl"-operator with distributional Hilbert space domain. This follows the same idea concerning alternative "energy inner product in distributional Hilbert spaces" ([BrK1]) to overcome current issues with different domain regularity/scope of momentum and location operator.
The conjecture is, that with the above shift theorems for appropriately defined norms can be proven for both, velocity and pressure, anticipating the divergent behavior for t-->0.
We emphasis that the L(2)-norm of the (Oseen) tensor kernel of the solutions of the incompressible N-S-E can be estimated by t*exp(-a) with a=1/4 ([CaM] ), which is the same (space dimension n independent!) divergence behavior as for the one-dimensional Stefan problem with non-regular intial value function [NiJ4] .
As a consequence 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. 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 correspondingly required reduced regularity assumptions to the normal derivative is "delivered" by the concept of J. Plemelj ([PlJ]), respectively the corresponding Pseudo-Differential operator.
"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