Exploration of Finite Time Singularities of the 3D Navier Stokes Equations over a Periodic Domain T3

Terry Moschandreou

doi:10.34257/GJSFRF256061

General Transport Structure of $b_3$ , $u_y$ inputs:How PDE re-enters a square term
General Finite time blowup after gradient blowup
Transport Structure and Regularity Properties of the $b_3$ Solution
Listing of Matlab Code for the PDE with $b^2$ term
1. Characteristic transport formulation
2. Explicit Lambert $W$ representation
3. Branch singularity structure
4. First-order regularity
5. Behavior at $z=t+\pi/2$
6. Behavior at $z=t-\pi/2$
7. Failure of second-order regularity
8. Final conclusion
Defining $F(x,y,\xi(z))$ uniquely to obtain the branch point of the LambertW function at $z=z^*=t+\frac{\pi}{2}$
1. Finalized expression for $b_3$
Mathematical Description
Mathematical Context
Foliations
Step 1 — Compute the Gradient
Step 2 — Determine the level Sets
Step 3 — Show the Sets Partition the Space
Step 4 — Verify the Local Coordinate condition (Definition of Foliation)
Geometric Meaning
Conclusion
Useful General Principle
Proof that $b_1 \frac{\partial u_z}{\partial x}+b_2\frac{\partial u_z}{\partial y}=0$ through foliations $y=x+C$ where $C \in \mathbb{R}$
1. Substitute the foliation
2. Impose the condition $f_2 = - f_1$
3. Add $u_x + u_y$
4. Conclusion
Geometric compatibility analysis for $f_1=\alpha f_2^3$
Compatibility PDE when $f_1=\alpha f_2^3$ and $\alpha=\alpha(x,y)$
1. Express $f_2$ in terms of $f_1,\alpha$
2. New geometric decomposition
3. Because $\alpha=\alpha(x,y)$
4. Explicit consequence for blow-up
1. The Irrotational “Shock” Realization
Proof that $\frac{\partial u_z}{\partial y}+\frac{\partial u_y}{\partial z}=\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}$ along a plane foliation
Collection of terms with Pressure $P$ and $u_z$ defining $(f_a)_z$ (force in $z$ direction)
Proof of Periodicity of $u_x$ provided $\epsilon_3 \ne 0$
Explicit Formula for $u_x$
Periodicity in $z$
Periodicity of the Lambert $W$ Composition
Membership in the One-Dimensional Torus
Analysis of the Modified Denominator
Critical Geometry
Behavior of the Lambert Argument
Regularity Consequences
7. Final Statement
Analysis of $\epsilon_3 \nabla u_x \cdot \nabla u_y$ for the Modified Structure $1-\sin(t-z+\epsilon)+\epsilon_3$
1. Definitions
2. Identification of the Singular Point
3. Gradient of $u_y$
4. Behavior of the Lambert- $W$ Derivative
5. Gradient of $u_x$
6. Dot Product Scaling
7. Multiply by $\epsilon_3$
8. Integral behavior
Geometric difference from the Previous case
Final Conclusion
Condition for Vanishing
Viscosity Scaling and Vanishing of the Gradient Product Term
1. Asymptotic scaling of the gradient product
2. Integral on the torus
3. Localized collapse under vanishing support
4. Conclusion
A smooth extension of the Navier Stokes equations
Regularized Transport Equation for the Transformed Variable
1. Interpretation
2. Energy Identity

General Transport Structure of $b_3$ , $u_y$ inputs:How PDE re-enters a square term

Consider a sufficiently smooth function:

and

Define

Then

Also define

Let

so that

We now compute

and analyze the nonlinear transport condition

The goal is to determine a nontrivial surface relation among

which preserves the identity without forcing terms to vanish individually.

Since

we obtain

Using the product rule:

Now

Hence

Treating $x,y,z$ as independent coordinates,

Thus

Therefore

Hence

Again using the product rule:

Now

Thus

Hence

Therefore

So

Recall

Observe carefully that

contains no $z$ -dependence.

Therefore

Thus the right-hand side becomes

Hence the equation reduces to

We now prove this reduction explicitly.

Substituting the previously computed formulas gives

Multiply the equation by $S_1^3$ :

Group the cosine terms:

Factor $G$ :

This proves that the original nonlinear PDE reduces to the nonlinear conservation-type equation above. Setting $\mathcal{J}(y)=x+z-t$ defines a surface where the PDE is in variables $y,z,t$ .

General Finite time blowup after gradient blowup

and consider

We prove that this equation admits:

finite-time gradient blowup,
followed by finite-time amplitude blowup.

The mechanism is a nonlinear Riccati-type compression generated by the transport term.

Divide by $S_1^2\neq0$ :

Define

Then

This is a quasilinear transport equation with nonlinear forcing.

Define characteristics by

Along characteristics,

Thus

This is a Riccati equation.

Suppose

Since

this means

Then

Hence along characteristics:

For sufficiently large positive $G$ , the quadratic term dominates:

Therefore

for some $c>0$ .

Integrating:

Hence

Therefore

Thus

at finite time

Hence finite-time amplitude blowup occurs.

Now differentiate the PDE with respect to $z$ .

Let

Differentiate:

Thus

The crucial term is

Since

the differentiated equation contains a Riccati compression mechanism.

Along characteristics:

For sufficiently negative initial slope,

the quadratic term dominates:

Integrating:

Hence

Thus

in finite time.

Therefore

before the solution amplitude necessarily diverges.

This is classical gradient catastrophe.

The characteristic equation is

Neighboring characteristics satisfy

Hence

If

then

Thus characteristics intersect.

At intersection, the classical solution loses regularity first through slope blowup.

After characteristics compress, the Riccati amplitude equation

forces amplitude divergence.

Therefore:

followed by

The instability occurs on regions satisfying

Equivalently,

On these oscillatory compression surfaces:

transport compresses characteristics,
gradients diverge,
Riccati forcing amplifies amplitudes,
finite-time singularities form.

Transport Structure and Regularity Properties of the $b_3$ Solution

From Eq(24) we have that the nonlinear transport equation is:

where

We prove that the transport solution admits the explicit Lambert $W$ representation

and that the obtained solution satisfies

while

due to the Lambert $W$ branch singularity.
For the transport equation

the characteristic system is

The first singularity mechanism is characteristic crossing.

Differentiating gives the Riccati-type equation:

The term

forces finite-time blowup of the gradient whenever compression occurs.

Thus:

This is the classical shock-formation mechanism.

As

the forcing coefficient (coefficient of $b^2$ term) becomes singular:

Near the branch point ( $b_3$ and it’s derivatives are infinite whereas at $t-z=-\frac{\pi}{2}$ , $b_3$ is finite and it’s derivatives are infinite.) At:

we have

Thus:

Also:

Therefore the forcing behaves like

The maximal magnitude occurs when

giving

Hence along characteristics:

Now solve the asymptotic Riccati equation:

Separating variables:

Integrating:

Thus:

Therefore the characteristic blowup time is

Now take the limit:

Then:

Thus the amplification timescale collapses to zero.

Equivalently:

Calculations (and simulations) show that finite time blowup occurs at a time $t2>t1$ significantly strictly greater than catastrophic higher gradient blow up which occurs first at time $t1$ at the branch point of the LambertW function solution. These two times are distinct times for gradient blowups and finite time amplitude blowup. (in $b_3$ ) For the higher gradient blowups $b_3$ is finite whereas for finite time blowup $b_3$ is infinite for $T*<\infty$ . In the simulations conducted I started off with smooth sine functions as they evolved from $t=0$ to these sequential blowups above (for countable $z$ values, distorting and breaking up the smoothness of the function. See the following Matlab code for the simulation of catastrophic gradient blowup followed by finite time blowup(source term $\sin(t-z+\epsilon)^2$ term added from Eq(15)): Calculations and simulation show that at time $t2>t1$ where $t2=z+\pi/2+\epsilon$ and $t1=z-\pi/2+\epsilon$ there is finite time amplitude blowup at $t2$ whereas at time $t1$ there is a catastrophic higher gradient blowup at the branch point of the LambertW function solution. The interested reader can run the matlab code provided.

Listing of Matlab Code for the PDE with $b^2$ term

clear;
clc;
close all;

epsilon = 0.091;

L = 2*pi;

Nz = 800;

Nt = 1800;

Tfinal = 3.3;

dz = 2*L/(Nz - 1);

dt = Tfinal/Nt;

% Spatial grid
zgrid = linspace(-L,L,Nz);

% Initial condition
b_old = (1/3)*sin(zgrid);

b_new = zeros(1,Nz);

% Create figure
figure;

% Frame storage
Frames = struct('cdata',{},'colormap',{});
frameCount = 0;

for n = 1:Nt

    t = n*dt;

    % Interior points
    for i = 2:(Nz-1)

        z = zgrid(i);

        % Upwind derivative
        if b_old(i) >= 0

            bz = (b_old(i)-b_old(i-1))/dz;

        else

            bz = (b_old(i+1)-b_old(i))/dz;

        end

        % RHS forcing term
        RHS = ...
            b_old(i)^2*cos(t-z) ...
            /(1 - sin(t-z) + epsilon) ...
            + 0.75^2*sin(t-z)^2;

        % Explicit update
        b_new(i) = ...
            b_old(i) ...
            - dt*b_old(i)*bz ...
            + dt*RHS;

    end

    % Periodic boundary conditions
    b_new(1)  = b_new(Nz-1);
    b_new(Nz) = b_new(2);

    % Update solution
    b_old = b_new;

    % Save a frame every 20 steps
    if mod(n,20)==0

        plot(zgrid,b_old,...
             'b',...
             'LineWidth',2);

        xlabel('z');
        ylabel('b_3(z,t)');

        title(sprintf( ...
            'Smooth Sine Evolving Toward Kink: t = %.3f', ...
            t));

        axis([-L L -2 8]);

        grid on;

        drawnow;

        frameCount = frameCount + 1;
        Frames(frameCount) = getframe(gcf);

    end

end

% Infinite replay
while ishandle(gcf)

    movie(gcf,Frames,1,15);

end

Characteristic transport formulation

Rewrite the PDE as

Introduce the characteristic parameter $s$ . Then

Define the invariant variable

Then

Hence

Therefore

Integrating,

The left-hand side gives

while the right-hand side yields

Thus

Equivalently,

Explicit Lambert $W$ representation

Define

Then

Exponentiating and solving using the Lambert function gives

Branch singularity structure

The Lambert function satisfies

and possesses the branch point

Hence singularity formation occurs when

Equivalently,

At this point,

and therefore

Thus the function value itself remains finite.

First-order regularity

We now prove that all first-order derivatives remain finite.

Let

Then

Differentiate:

where $\alpha\in\{x,y,z,t\}$ .

Using

we obtain

Near the branch point,

Simultaneously,

because $F$ is smooth and the denominator

vanishes at most quadratically at the critical geometry.

Therefore

which remains finite.

Hence

Thus

Behavior at $z=t+\pi/2$

Let

Then

so

Hence

Therefore the denominator is strictly positive and smooth.

Moreover,

contain factors of $\cos(t-z+\epsilon)$ , which vanish at this point.

Consequently,

Behavior at $z=t-\pi/2$

Now let

Then

so

Hence

Since

the denominator remains strictly nonzero.

Again all first derivatives contain only bounded factors and vanish proportionally to $\cos(t-z+\epsilon)$ .

Therefore

Thus all spatial and temporal derivatives up to order one remain regular at both critical geometries.

Failure of second-order regularity

We now differentiate again.

Since

we obtain

Near the branch point,

Hence

Therefore second derivatives behave like

which diverges at the branch surface.

Consequently,

Hence

More generally,

Final conclusion

The transport equation admits the explicit Lambert $W$ solution

The solution satisfies

including at

where all first-order spatial and temporal derivatives remain finite.

However the Lambert $W$ branch geometry produces higher-order derivative singularities:

Thus the solution exhibits finite-value higher-gradient singularity formation without blowup of the function itself.

Defining $F(x,y,\xi(z))$ uniquely to obtain the branch point of the LambertW function at $z=z^*=t+\frac{\pi}{2}$

Finalized expression for $b_3$

To determine if the branch point is realized at $z_* = t + \frac{\pi}{2}$ , we need to evaluate the argument of the Lambert $W$ function at that point as $\varepsilon_3 \to 0$ and,\

Evaluate $F(x, y, \xi)$ at $z_*$ . The function $F$ is given as $F = \sin(t-z+\epsilon) - (\ln(2) - 2)$ . At $z = z_* = t + \frac{\pi}{2}$ :The argument of the sine is $t - (t + \frac{\pi}{2}) = -\frac{\pi}{2}$ and $\sin(-\frac{\pi}{2}) = -1$ .
Substituting this into $F$ :

2. Evaluate the Full Argument. The argument of the Lambert $W$ function is:

Substitute the values at $z_*$ :Numerator: $-\exp(-(1 - \ln(2))) = -\exp(-1 + \ln(2)) = -e^{-1} \cdot e^{\ln(2)} = -\frac{2}{e}$ .Denominator: $1 + \varepsilon_3 - \sin(-\frac{\pi}{2}) = 1 + \varepsilon_3 - (-1) = 2 + \varepsilon_3$ .As $\varepsilon_3 \to 0$ , the argument becomes:

3. Conclusion: The branch point of the Lambert $W$ function is indeed defined as the point where the argument equals $-1/e$. The choice of $F = \sin(t-z+\epsilon) - (\ln(2) - 2)$ perfectly balances the denominator’s value of $2$ at the point where the sine is $-1$.The branch point is realized exactly at $z_* = t + \frac{\pi}{2}$. Note on the Coordinate $z_*$, we use an expansion where $\sin(t-z+\epsilon) \to 1$ (near its maximum). In this current formulation, the singularity is placed at $\sin(t-z+\epsilon) = -1$ (its minimum).If the goal is to match the quadratic expansion $1 - \frac{\eta^2}{2}$, we would look at the neighborhood of $z = t - \frac{\pi}{2}$.At the current $z_* = t + \frac{\pi}{2}$, the sine function is at a minimum, so the local expansion is:

, this still provides the quadratic $\eta^2$ spatial dependence required for the $O(\varepsilon)$ cancellation, just with a reflected geometry. Here $\eta=z-z*$ . It is true that we gain two derivative regularity here. At $z=z*=t+\frac{\pi}{2}$ , at the branch point both spatial and time derivatives up to order 1, that is we have $C^0$ and $C^1$ regularity but not higher than $C^n$ for $n \ge 2$ as $\epsilon_3\rightarrow 0.$ \

Loss of regularity at T^*. The derivative (orange) diverges as it approaches the singular time, while the velocity amplitude (blue) remains bounded at a finite value.

Mathematical Description

In the context of the 3D Navier-Stokes equations, the criteria for blowup are often expressed through the Beale-Kato-Majda (BKM) condition. A solution remains smooth if:

where $\symbf{\omega} = \nabla \times \mathbf{u}$ is the vorticity.

The structure shown above depicts a scenario where:

Energy Conservation: The $L^2$ norm (energy) and $L^\infty$ norm of the velocity stay within a finite range.
Enstrophy Blowup: The $H^1$ norm (enstrophy) or higher Sobolev norms $\|\mathbf{u}\|_{H^s}$ diverge.
Physical Interpretation: This corresponds to the "shredding" of fluid elements where the velocity is not large, but the shear rates and local rotations become infinitely steep.

The following figure illustrates the difference between Type 1 and Type 2 blowup,

Comparison of Type I and Type II blowup profiles in 𝕋³. Type I follows the natural scaling of the Navier-Stokes equations, while Type II represents a "super-scaling" singularity.

Mathematical Context

In the study of 3D Navier-Stokes regularity, we distinguish singularities based on the rate of the $L^\infty$ norm growth:

Type I Singularity: Characterized by the Leray scaling. If the solution blows up at $T^*$ , it satisfies:

$sup_{t \in [0, T^{*})} (T^{*} - t)^{1 / 2} ‖ u (\cdot, t) ‖_{L^{\infty}} < \infty$

These are often associated with potential self-similar blowup solutions.
Type II Singularity: Any blowup that is not Type I. The norm grows faster than the natural scaling:

$\underset{t \to T^{*}}{lim sup} (T^{*} - t)^{1 / 2} ‖ u (\cdot, t) ‖_{L^{\infty}} = \infty$

Type II blowup is known to occur in other nonlinear evolution equations (like the energy-critical wave equation) and remains a subject of intense research for Navier-Stokes.

Foliations

Let

where:

$\alpha\in\mathbb{R}^n$ is a constant vector
$C\in\mathbb{R}$ is a constant
$\alpha\neq 0$

Then the level sets

form a codimension-1 foliation of $\mathbb{R}^n$ .

Step 1 — Compute the Gradient

Since $\alpha\neq 0$ ,

So the function is regular everywhere.

Step 2 — Determine the level Sets

Solve:

Rearrange:

Define:

So each level set is:

This is an affine hyperplane perpendicular to $\alpha$ .

Thus every leaf is:

smooth
connected
dimension $n-1$

Step 3 — Show the Sets Partition the Space

For any $x\in\mathbb{R}^n$ :

so $x$ belongs to exactly one level set.

Therefore:

and the sets are disjoint.

So we have a partition.

Step 4 — Verify the Local Coordinate condition (Definition of Foliation)

Define coordinates:

and choose coordinates:

along the hyperplane orthogonal to $\alpha$ .

Because:

the mapping

is a smooth local coordinate system (Implicit Function Theorem).

In these coordinates, the leaves are:

This exactly matches the definition of a foliation.

Geometric Meaning

Adding the constant $C$ does not change the foliation — it only shifts the labeling of the leaves.

Specifically:

and

generate the same family of parallel hyperplanes.

The constant just moves the origin of the coordinate.

Conclusion

Useful General Principle

More generally:

Any smooth function whose gradient is nonzero everywhere defines a foliation by its level sets.

So in PDE or transport settings, any affine phase function like

produces planar leaves in space-time — a very common geometric structure in linear transport and wave propagation.

Proof that $b_1 \frac{\partial u_z}{\partial x}+b_2\frac{\partial u_z}{\partial y}=0$ through foliations $y=x+C$ where $C \in \mathbb{R}$

We are given the velocity components

We also consider a foliation of the form

We want to prove that if

then

Substitute the foliation

Along a leaf $y = x + C$ , we have

and

Impose the condition $f_2 = - f_1$

We are given that along the foliation

Hence,

Add $u_x + u_y$

Conclusion

Therefore, on a foliation $y = x + C$ , if

then the sum of the velocities in the $x$ and $y$ directions vanishes:

Note that $u_z \frac{\partial u_z}{\partial x}$ factors out. This result holds pointwise along each leaf of the foliation. The crucial point is the matching of the arguments along the foliation.

Geometric compatibility analysis for $f_1=\alpha f_2^3$

We begin with the vector field

The velocity components are

Therefore

and

Hence

and

Using the structure of the original ansatz,

so

Compatibility PDE when $f_1=\alpha f_2^3$ and $\alpha=\alpha(x,y)$

We consider the foliation relation:

and assume

only, so

The compatibility condition still comes from

1. Express $f_2$ in terms of $f_1,\alpha$

From

we get

Hence

The $f_2$ -part of the vector field becomes

Now differentiate:

so

Similarly,

2. New geometric decomposition

After inserting these into

and collecting terms, the structure is

The power $m$ depends on the chosen normalization, but the zero condition is independent of it. Thus the compatibility equation is

3. Because $\alpha=\alpha(x,y)$

We have

Therefore

Hence the full compatibility PDE is:

This is the reduced two-dimensional $\alpha(x,y)$ compatibility system.

4. Explicit consequence for blow-up

Solving the first two equations:

and

so

This is the cleanest form. It says:

where the integrand is determined entirely by the $f_1$ -curvature. Therefore, in the case

the compatibility law is:

In a blow-up regime of $f_1$ , the right side typically becomes singular, forcing either:

or a special alignment condition making

So this foliation allows $\alpha$ to act as the compensating field even though it has no $z,t$ dependence.

From $(A\cdot \nabla)A= \nabla(|A|^2)/2 - A \times(\nabla \times A)$ The following flux amplification condition is introduced:

Now from the compatibility equation if $A\times (\nabla \times A)=0$ , $u_z \nabla u_z=\nabla \frac{u^2_z}{2}$ then

Multiply Eq.(15) by $e_3$ to get this form of the expression. For a 3-D gradient to point only in the z-direction everywhere the scalar function it is derived from cannot have any horizontal( $x$ or $y$ ) dependence. Therefore, $|A|^2= M(z,t)$ for some function $M$ .
The remaining $z$ -component of the equation is:

The RHS of this equation must balance and derivatives in $x,y$ directions vanish. So $(A\cdot \nabla)A=0$ in horizontal $X,Y$ directions. The flux matching condition creates Ricatti Type coupling. Energy of $A$ feeds into gradients of $u_z$ and gradients of $u_z$ feedback into energy growth. The matching condition says that the energy of $A$ evolves exactly like scalar transport of $u^2_z$ along $A$ .
See the following feedback amplification growth mechanism:

The previous derivation maps onto the foundational themes explored by Evan Miller in his recent work on the Navier-Stokes strain equation and finite-time blow-up . Miller’s research directly tackles how the self-amplification of strain—rather than classic vortex stretching—can act as the definitive driver for fluid singularities and the turbulent energy cascade. By mapping out how the horizontal velocity components ( $A_x, A_y$ ) compress the fluid filaments and pump energy into a vertical sheet gradient ( $\frac{\partial \Phi}{\partial z}$ ), the loop and analysis of this paper mirrors Miller’s analytical framework. [t]0.48

The strain matrix $S$ for an incompressible fluid is trace-free ( $\text{Tr}(S) = 0$ ), meaning its eigenvalues must satisfy:

Sorted by size, $\lambda_1 \ge \lambda_2 \ge \lambda_3$ , meaning $\lambda_1$ is positive (stretching), $\lambda_3$ is negative (compressing), and the middle eigenvalue $\lambda_2$ dictates whether the local geometry is flattening or pulling.

Miller established scale-critical conditions showing that finite-time blow-up is severely restricted by the history and alignment of this middle eigenvalue $\lambda_2$ . How my Loop Maps onto the Matrix Geometry In the system, the horizontal components ( $A_x, A_y$ ) drive the Horizontal Advection Engine, which sharpens the horizontal shear boundaries. This acts mathematically as a profound planar compression ( $\lambda_3 \ll 0$ ). Because the total energy density $\Phi$ is forced to be uniform in $x$ and $y$ ( $\nabla_{x,y}\Phi = 0$ ), the fluid is geometrically forbidden from stretching or moving energy sideways. Here is how the above loop and the analysis in this paper aligns with Miller’s work. 1. Strain Self-Amplification Over Vortex Stretching In standard fluid mechanics, the focus is almost always on the vorticity equation:

where $S = \nabla^{\text{sym}}u$ is the symmetric strain tensor. Blow-up is traditionally thought to occur when the strain matrix $S$ stretches the vorticity vector $\omega$ into a runaway spiral.

However, Miller investigated what happens when you focus heavily on the evolution of the strain tensor itself ( $S$ ). The strain evolution equation contains a quadratic self-interaction term:

Miller demonstrated (particularly in his 2023 paper, “Finite-time blowup for a Navier–Stokes model equation for the self-amplification of strain”) that the quadratic $S^2$ term can drive a finite-time singularity via purely compressional strain self-amplification, even if one damps out or severely restricts the rotational vortex stretching mechanisms. The irrotational model in my work ( $\nabla \times A = 0$ ) does exactly this: it completely deletes the vorticity terms ( $\omega = 0$ ), yet the quadratic convective advection still forces a runaway compression of potential fields. Because the total energy density $\Phi$ is forced to be uniform in $x$ and $y$ ( $\nabla_{x,y}\Phi = 0$ ), the fluid is geometrically forbidden from stretching or moving energy sideways. This locks the fluid into a strictly 1D-stratified vertical alignment along the $e_3$ axis, effectively forcing $\lambda_2 \to 0$ or locking its behavior into a perfect geometric sandwich. The fluid elements are compressed flat horizontally, forcing a massive runaway gradient spike along the vertical $z$ -axis.

The Irrotational “Shock” Realization

The horizontal components act as the geometric compressor. The irrotational constraints ( $\frac{\partial A_x}{\partial z} = \frac{\partial u_z}{\partial x}$ ) substitute for vorticity. The spatial energy pump feeds back into the horizontal velocities. By removing the geometric “orthogonal drift” of rotation, the derived equations form a Burgers-type convective shock channel. The fluid endlessly concentrates its energy into an infinitely thin vertical layer. This establishes a highly restricted manifestation of pure strain-driven regular stratification collapsing into a geometric singularity.

Proof that $\frac{\partial u_z}{\partial y}+\frac{\partial u_y}{\partial z}=\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}$ along a plane foliation

We are given

Assume a plane foliation

Express derivatives along the plane.

Along the plane, any function $f(x,y,z,t)$ can be written as $f(x, x+C, z, t)$ . Therefore,

Hence,

Equality of $V_1$ and $V_2$ .
Similarly, along the plane,

Thus, along the plane $y = x + C$ ,

Collection of terms with Pressure $P$ and $u_z$ defining $(f_a)_z$ (force in $z$ direction)

We have a left over term in the Navier Stokes transformed equation given as:\

However from the section on matrix transformations we examined for the forcing term in the Navier Stokes equations we calculated it as:\

where here we have calculated the third component to align with Equation (6) in . Since we are considering foliations $y=x+C$ , $C\in \mathbb{R}$ , then this will reduce to:\

since $f_x b_1 u_z+f_y b_2 u_z =0$ and $b_1 u_x=u_x u_yu_z$ and $b_2 u_y=u_x u_y u_z$ . So $(f_a)_3= b_1u_x f_z$ ,
Thus we have:\

since at the branch point of the LambertW function, $P\, b_3\, \partial_j u_z +P\, u_z\, \partial_j b_3=0$
when $j=3$ . Thus we have

But $b_1 u_x=u_z b_3$ and one arrives at the fundamental fluid dynamics equation for force $f_z$ in the $z$ direction of flow :\

Here the dimension of the right side are in terms of force per unit time. Here $P= F/L^2$ .\

Proof of Periodicity of $u_x$ provided $\epsilon_3 \ne 0$

We are given the modified definition

and

Suppose

Therefore

Explicit Formula for $u_x$

Since

we obtain

Using

this becomes

Periodicity in $z$

Define

Then

Since sine is $2\pi$ -periodic,

hence

Therefore

Thus every trigonometric component of $u_x$ is periodic with period $2\pi$ .

Periodicity of the Lambert $W$ Composition

Define the Lambert argument

Because

we obtain

Therefore, provided the same branch of the Lambert $W$ function is chosen,

Consequently,

Hence $u_x$ is periodic in the spatial variable $z$ .

Membership in the One-Dimensional Torus

Therefore

Hence

More precisely,

for every finite $p$ , provided the denominator remains nonzero.

Analysis of the Modified Denominator

The key structural difference from the previous case is the replacement

We now analyze the implications.

Since

we obtain

Thus, if

then

for all $(z,t)$ .

Therefore the denominator never vanishes.

Critical Geometry

The minimum occurs when

This happens at

equivalently,

At this location,

Hence if

the denominator approaches zero precisely at

This differs from the previous $1+\sin(t-z+\epsilon)+\epsilon_3$ case, where the singular geometry occurred at

Behavior of the Lambert Argument

The Lambert argument is

At

we have

Therefore

Since

we obtain

As

we have

Thus the Lambert argument crosses the branch region and may leave the real-valued principal branch.

Regularity Consequences

When

the denominator remains strictly positive, so:

$u_y$ remains finite,
$u_x$ remains periodic,
the Lambert composition remains well-defined on a fixed branch,
all first-order derivatives remain finite away from the branch singularity.

However, as

the denominator degenerates at

and:

$u_y$ becomes unbounded,
the Lambert argument approaches the branch singular regime,
higher derivatives may fail to exist.

Thus periodicity is preserved independently of smoothness.

7. Final Statement

For the modified transport structure

the velocity component

satisfies

Hence

The modified singular geometry now occurs at

rather than

Therefore the sign change in the trigonometric structure shifts the location of possible loss of smoothness while preserving periodicity.

Analysis of $\epsilon_3 \nabla u_x \cdot \nabla u_y$ for the Modified Structure $1-\sin(t-z+\epsilon)+\epsilon_3$

We analyze whether

This holds only if the gradients do not blow up faster than $1/\epsilon_3$ .

1. Definitions

We define

and

Suppose

Hence

2. Identification of the Singular Point

The critical location is

At this point

Therefore

3. Gradient of $u_y$

Compute

Near the singular point,

4. Behavior of the Lambert- $W$ Derivative

Near the branch point,

In this construction,

Therefore

Thus derivatives of $b_3$ scale like

5. Gradient of $u_x$

Recall

Differentiate:

Near the singular point:

First term:

Second term:

Therefore

Thus

6. Dot Product Scaling

We combine the estimates

and

Therefore

7. Multiply by $\epsilon_3$

Hence

8. Integral behavior

Consider the integral near the singular point

Then

Thus even the integral typically diverges unless additional cancellation occurs.

Geometric difference from the Previous case

The only structural change from the previous analysis is the replacement

Consequently, the singular geometry shifts from

to

The asymptotic scaling structure remains identical.

Final Conclusion

For this modified structure, the dominant scaling again implies

Condition for Vanishing

To obtain

one would need

Viscosity Scaling and Vanishing of the Gradient Product Term

We now replace the parameter $\epsilon_3$ by a viscosity coefficient $\nu$ multiplying the nonlinear gradient interaction term:

Assume the scaling relation

1. Asymptotic scaling of the gradient product

From the previous analysis, near the singular point

we obtained

Therefore,

Hence

However, this is a pointwise scaling. To obtain vanishing behavior, we examine the integral structure.

2. Integral on the torus

Consider the integral over the torus $\mathbb{T}^1$ :

Using the localized singular structure near $z_*$ , we write

Thus,

3. Localized collapse under vanishing support

If the singular region shrinks with $\epsilon_3$ , i.e.

then the effective scaling becomes

Thus

4. Conclusion

With the viscosity scaling

and assuming localization of the singular structure near $z_*$ , we obtain

Thus the nonlinear gradient interaction is asymptotically suppressed in the weak (integrated) sense under this viscosity scaling.

A smooth extension of the Navier Stokes equations

Regularized Transport Equation for the Transformed Variable

I propose the following regularized evolution equation for the transformed variable $b(x,y,z,t)$ :

where

is the biharmonic operator, $\nu>0$ is the viscosity coefficient, and $\epsilon>0$ is a higher-order regularization parameter.

Interpretation

The additional term

introduces hyperviscosity, providing enhanced smoothing while preserving the nonlinear transport structure.

Energy Identity

Multiplying the equation by $b$ and integrating over the spatial domain $\Omega$ :

Hence,