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

Terry Moschandreou

doi:10.34257/GJSFRF256061

Successive Matrix Operations on the Viscous Term

1. Navier–Stokes viscous term

We begin with the viscous term in the three-dimensional Cartesian Navier–Stokes equations:

2. Definition of the first derived vector field

Let

Define the vector field

3. First row-operation matrix

Define the matrix operator

Applying the matrix to the viscous term:

Using the Laplacian product rule for scalar fields:

we obtain componentwise:

Define

Thus:

4. Second derived vector field

Define

5. Second row-operation matrix

Define

Apply the second matrix to the result of the first transformation:

Focus first on the Laplacian term:

6. Laplacian product rule for vector fields

For a scalar field $f$ and vector field $\vec g$ :

Let

Then:

Rearranging:

Multiplying by $\nu$ :

7. Successive matrix transformation

Applying both matrices to the original viscous term gives:

where

and

Final structural identity

Successive Matrix Transformations of the Pressure Gradient Term

1. Pressure gradient term in the Navier–Stokes equations

The pressure force term in vector form is

2. Definition of the velocity vector

Let

3. Definition of the first derived vector field

Define

4. Definition of the second derived vector field

Define

5. First row-operation matrix

Define

Apply the matrix to the pressure gradient:

Compute componentwise.

First component

Apply the product rule:

Therefore

Thus

Define

Therefore

6. Second row-operation matrix

Define

Apply the second matrix to the result of the first transformation:

Focus on the gradient term:

7. Product rule for gradient of scalar-vector product

For scalar $f$ and vector $\vec g$ :

Let

Then

Rearranging:

Therefore

8. Successive matrix transformation

Applying both matrices to the original pressure gradient term gives

where

Final structural identity

Successive Matrix Transformations of the Force Field and Final Navier–Stokes Form

1. Original Navier–Stokes equations

where

2. Derived vector fields

Define

3. Matrix operators

4. Force Field Transformation

Original force term

5. First transformation

Apply $M(\vec u)$ :

Componentwise:

Define

Thus

6. Second transformation

Apply $M(\vec b)$ to the result:

Componentwise:

Define

Therefore

Summary of all transformed terms

Time derivative

Inertial term

Pressure term

where

Viscous term

where

Force term

Final transformed Navier–Stokes equation

Applying both operators to the original system gives:

Final hierarchical Navier–Stokes structure

Meaning of Production = Diffusion

In the constructed hierarchy, the statement

has a precise mathematical meaning. It is an exact algebraic balance between two classes of terms arising after applying the successive operators $M(\vec u)$ and $M(\vec b)$

1) Starting identity

We obtained the exact decomposition

There are three structurally different pieces:

Divergence (transport)
Production
Diffusion

2) Definition of production

The term

contains products of the form

These terms:

increase gradient magnitude locally,
are nonlinear,
contain no Laplacian.

Thus they generate new gradient energy inside the domain.

Formally,

3) Definition of diffusion

The term

contains the Laplacian operator.

The Laplacian:

spreads gradients spatially,
smooths oscillations,
reduces local curvature.

This is the mathematical definition of diffusion.

Formally,

4) Exact meaning of Production = Diffusion

The statement means the pointwise equality

This equality is:

not approximate,
not statistical,
not integral,
but an exact algebraic identity.

5) Structural consequence

Substituting the equality into the identity for $\mathbf{G}_2$ gives

Thus the entire field becomes a pure divergence field.

6) Consequence on the periodic torus

On the periodic torus $T^3$

Therefore,

7) Physical interpretation

Production equals diffusion means:

nonlinear steepening creates gradients,
diffusion smooths gradients,
the two effects cancel exactly everywhere.

Mathematically,

Only transport remains.

8) Analogy with classical Navier–Stokes energy balance

In the standard kinetic energy equation,

balance occurs when

The present hierarchy represents a higher-order analogue of this structure.

Final precise definition

and under periodic boundary conditions,

A crucial note here is that the nonlinear inertial (advective) term is introducing the blowup mechanism as seen in the following section. If production is not equal to diffusion then the calculations show that if,

where this will in fact be shown later to be a solution and $b_3=u_x u_y$ then $\nu \nabla b_3$ will approach zero when $\nu=\epsilon_\mathit{3}$ . Taking the limit of this term approaches zero for $\nu \ne 0$ In fact the second derivative of $b_3$ wrt to $z$ for example is,\

at the branch point of the LambertW function where $z=z*=t+\pi/2$ . See appendix C. The only danger is at the branch point not elsewhere. But there we have $\nu D^2_zb_3 \rightarrow 0$ for $\nu \ne 0$ For the $\epsilon_3$ to cancel $(\epsilon_3/\epsilon_3)$ in the product of expression in $G_2$ , $\epsilon_3$ must be non-zero. So for periodic domain $T^3$ the definition of Production equals Diffusion or not equal leads to only the singularity coming from the nonlinear inertial terms of the transport equation(Eq.(6) in . Since there I added the expression $\vec{a} \nabla{\vec{a}}$ I have to check when the integral of a divergence is zero. It happens that it is true when Production either equals or not equals the diffusion term in the transformed Navier Stokes equations. Moreover we will see that $M(\vec b)\mathbf{G}_1=0$ so that both terms in $G_2$ are independently vanishing at the branch point of LambertW function.(shown further below and in Appendix C) Finally we note that in the real-valued setting,

\

The existence of a LambertW solution of the governing equations of Fluid Mechanics

It is claimed that Lambert $W$ profiles are invariant manifolds of finite codimension in the Navier–Stokes flow and there is closure under all interaction operators $\mathcal{I}_{ij}$ . To formalize this, define the interaction operators

The nonlinear structure of the Navier-Stokes equations may then be viewed as the superposition of the actions of $\mathcal I_{ij}$ over all component pairs. This viewpoint naturally leads to the consideration of velocity profiles that are closed under all interaction channels, meaning that no new singular structures are generated when any $\mathcal I_{ij}$ acts on the profile.

We therefore consider the set

whose elements are velocity fields invariant, in a structural sense, under the full nonlinear interaction geometry. Being in $\mathcal S$ imposes severe rigidity: such profiles must be stable under multiplication, differentiation, and implicit inversion, while remaining compatible with Navier–Stokes scaling and incompressibility.

A main theorem which leads to the LambertW - $\wp$ closure solutions is as follows,\

Assuming the Poisson equation,

and the following PDE holds,

where $u_i=V_z+C$ , $C$ is a shift related to $\kappa:(\mathbb{R}^3,R^+)\rightarrow \mathbb{R}$ where $V_z$ is at least in $C^0(\mathbf{x},t)$ with the pressure $P$ in $C^2(\mathbf{x},t)$ , then,\

where $C_3 \in \mathbb{R}^+$ and $\wp$ is the WeierstrassP function, $\wp^{-1}$ is it’s inverse and $W$ is the LambertW function defined on an affine spatio-temporal space $\Xi=x - y + 2 z - t+C_6$ , with $C_6=O(\delta)$ . The expression for $\delta$ is connected to $\kappa$ as $\delta=-\frac{1}{\sqrt{\kappa1}}$ , where $\kappa1$ is the derivative of $\kappa$ in a preferred direction, say $z$ in this case. It can be proven that at the branch point of the LambertW function, $\kappa1>0$ . In principle both $\kappa$ and $\delta$ are functions in general, where the derivative of $\kappa$ wrt to $z$ ( $\kappa1$ ) approaches $0^-$ when we look at large values in the i’th direction(say $z$ direction), in particular for $V_z$ in the $z$ direction. (Recall $\kappa=\mathcal{R}_n$ from Eq.(1) from the main Introduction; This approaches zero for the LambertW solution given by Eq.([eq:PDE]) as $-z-t \rightarrow -\infty$ . As shown further in this paper $z$ will be chosen arbitrarily large in the positive direction for large positive initial data(similar approach for large negative initial data where $-z-t \rightarrow +\infty$ ) Then the solution problem can be defined by Eqs.([MAIN]-[MAIN3]) which can be shown to reduce to Eq.([eq:PDE]). Moreover the solutions given by $u_i$ or $V_z$ are periodic in both space and time by means of the Lambert to Weierstrass mapping.

Now to establish the grounds for these assertions made in the main theorem we are required to review the work in the proofs of the connection of the form of the Navier Stokes equations to the PDEs in and and references therein. Since the work in these references were valid in the $L^p$ spaces for $p\in [2,3)$ a note here is required. The Poisson equation was used to relate the velocities to the pressure terms. In order to remain in these spaces for $p$ (in particular it was found that $p>3/2$ is necessary) the following PDE was required in the definition of the Poisson equation,

Here $u_i$ are the components of the Navier Stokes flow with $P$ representing the pressure and $S$ some arbitrary function used in the solution approach. To be specific the solution in the previous references made use of this PDE(Poisson equation). In , and a geometric calculus approach was used to rewrite the Navier Stokes equations in a general $u_i$ direction for $i=1 \dots 3$ . The PDEs defining the $u_i$ were possible to develop mainly due to the Poisson equation. Thus we use the $\frac{\partial P}{\partial x_i}$ term in the governing PDE developed by Geometric Calculus approach. The transition to the PDE in Eqs.([MAIN]-[MAIN3]) is a result of adding to the original Navier Stokes equations after applying row operations , and a pivot function $\mathbf{a} \nabla \cdot \mathbf{a}$ which is used as a place holder in obtaining the solution of the Navier Stokes equations. See Eq.(6) in where this pivot function has been added. In the subsequent parts of the paper there the pivot function is used in the following sense: If we have two operators $L_1=L_2$ and $L_1=L_3$ then necessarily $L_2=L_3$ so that the appearance of $L_1$ is not seen anymore. It is in this way that the PDE problem for the Navier Stokes equations was set up. As mentioned the following PDE captures the dynamics of the original Navier Stokes equations, which now are written for the $u_i$ direction.
The 3D incompressible unsteady Navier-Stokes Equations (NSEs) in Cartesian coordinates may be listed for the velocity field,

where $\rho$ is constant density, $\mu$ is dynamic viscosity , $\textbf{F}^*=F^{*i}\bm{\textbf{e}}_i$ are the body forces on the fluid. In some cases, it may be elected to reparametrize the components of the velocity vector, and pressure to $\textbf{u}=(u)^i\bm{\textbf{e}}_i$ , $\textbf{P}=(P)^i\bm{\textbf{e}}_i$ , coordinates $\textbf{x}_i$ and time $t$ according to the following form utilizing the non-dimensional quantity $\delta$ ( $\delta \le 0$ ) :

The Navier-Stokes equations above in $*$ variables are proven to be equivalent to the following PDE and and in non-star variables for the $u_i$ component and $x_i$ , $t$ variables, (Here $u_i=V_z+\sqrt{\kappa1}$ (for example $\kappa1$ is a derivative of $\kappa$ in a preferred direction in space or time):

where $S=(V_z+\sqrt{\kappa1})^2 \ \nabla^2 P$ ,

where $S$ is defined in Eq(1). Note the Laplacian for the pressure is written as an integral over an epsilon ball along each of the infinitely many branches of LambertW function appearing in the WeierstrassP function.(The real branch is of interest here) There are precisely three Laplacians in Eqs.([MAIN]-[MAIN3]), one is for the pressure and the other two are in terms of the velocity $V_z$ . The work of Rumer and Fet was used in ( and ) to write the Laplacians as integrals over epsilon balls. In there, $x$ , $y$ components $V_x,V_y$ vanish on a suitable manifold as shown in also (where the space $J_{y_i}$ was defined with a calculation showing that an operator involving all three velocity components and their derivatives, $\mathcal{X}$ , is precisely zero on this space and we see it to be true on the boundary of an embedded ball in $\mathbb{T}^3=[0,1]^3$ . It is important to define $\delta$ which is used in two ways in this paper. The first is that we assume alignment of two vector fields in general and then separately non-alignment.The expression for $\delta$ is that it is the negative reciprocal of $\kappa1(x,y,z,t)$ ( $\kappa1$ is the derivative $\kappa'$ ) and the following eigen-type problem holds true:\

The solution of this PDE gives a general form in terms of the WeierstrassP function. It is:

In the PDE given by Eqs.([MAIN]-[MAIN3]) the expression $\left(\frac {\partial ^{2} V_z}{\partial z\partial t}\right)$ appears at a few places. Substituting in the Eqs.([MAIN]-[MAIN3]) introduces $\frac{1}{\kappa}$ at these places and then multiplying by $\kappa$ throughout aligns $\delta$ with $\kappa$ so that we have the PDE which gives the LambertW solution. It is a straightforward bookkeeping approach to see that this results as in Eq.([eq:PDE]) in the later section where we have defined the representative governing equations. Attention must be given to the $P_1$ operator expression in Eq.(10). Here the first term and sixth term expressions(the sixth part contains also $\Phi(t)$ which is set to zero due to expanding Tori) in the sum of 6 parts in Eq.(10), call the sum of the two parts of the six, the operator $Q_s$ which vanishes due to the existence of a $C^{\infty}$ pressure Laplacian. The Laplacian of the pressure is in a reciprocal relationship to the velocity $u^2_z$ . What remains in Eqs.([MAIN]-[MAIN3]) will lead to Eq.[eq:PDE] which solves as a LambertW solution .

A unique representation for the PNS system

Consider the function representing the $y_3$ component of the Navier Stokes equations ( $v_3=u_z=u^*_z\delta$ , $x=\delta y_1$ , $y=\delta y_2$ , $z=\delta y_3$ , $t=\delta^2 s$ ), $\delta \in \mathbb{R}(<0)$

and the PDE

where $\rho$ is the density and $\mu$ is the dynamic viscosity of the fluid. There was a relabeling of $V_z$ to $v_3$ . The function $\delta$ is defined to be $-1/\sqrt{\kappa1(y_1,y_2,y_3,s)}$ and is related to the WeierstrassP function as described previously in this work . Also the pressure part of the PDE is resolved in this reference where it is claimed that $P$ at the branch point of the LambertW function is of the order the reciprocal of $u^2_z$ . (See Section: 2.1.4. "Exact Solution of the Extended PDE via the Weierstrass Ansatz for General Spatio-Temporal Pressure" in . It can be shown that the base solution( $n=1$ ) is (use Maple 2026): $u_z=LambertW(-exp(-G(x,y,z,t)-1))+1$ .

Difference between
$u_z=\text{LambertW}(-\text{exp}(-G(x,y,z,t)-1))$
and constant $k(n)$ multipliers

To determine the difference, we need to look at how a constant multiplier $k = (1 + 2^n)$ changes the logarithmic derivative (or the "sensitivity") of the base of the chain.

If you replace the base $x = -e^{\xi}$ with $x_{alt} = -(1 + 2^n)e^{\xi}$ , you are essentially changing the initial condition of the recursive derivative chain.

1. The Derivative at the First Level

In the current derivation, we have:

This leads to the elegant cancellation in $\partial_z W_1 = \frac{W_1}{x(1+W_1)} \frac{\partial x}{\partial z} = -\omega_2 \frac{W_1}{1+W_1}$ .

If we use $x_{alt} = -k e^{\xi}$ (where $k = 1 + 2^n$ ):

Crucially, the ratio $\frac{1}{x} \frac{\partial x}{\partial z}$ remains exactly $-\omega_2$ .

Because the LambertW derivative $\frac{W}{x(1+W)}$ always contains that $1/x$ factor, the constant $k$ in the numerator of the partial derivative and the constant $k$ in the denominator ( $x$ ) cancel out completely in the first step.

2. The Chain Effect

Since the first derivative $\partial_z W_1$ remains functionally the same (expressed in terms of $W_1$ ), the recursive steps for $W_j$ follow the same algebraic path. The final "Compact Product" formula would look identical in structure:

3. The Functional Difference

While the formula for the derivative remains the same, the numerical values and singularities shift significantly:

The Branch Point Shift: In the original case, the branch point (singularity) occurs where $-e^{\xi} = -1/e$ , or $\xi = -1$ .

The Multiplied Case: With $k = (1 + 2^n)$ , the singularity occurs when $-(1+2^n)e^{\xi} = -1/e$ . This moves the critical value of $\xi$ to:

Magnitude: Since $1 + 2^n$ is a large number, it forces $W_1$ deeper into the principal branch (approaching $0$ ) or further into the lower branches.

Summary

There is no difference in the derivation logic or the final formula. The structure of the LambertW derivative is "scale-invariant" with respect to a constant multiplier in the argument’s exponent.

However, in the context of my fluid dynamics research, this $k$ factor would act as a damping or scaling coefficient. It changes "where" and "how fast" the blowup occurs in coordinate space, but it does not change the recursive "Product Law" I derived for the gradients.

No Finite-Time Blowup of $b$ and Blowup of Higher Time Derivatives

The equations developed in this section analyze the evolution of the derived nonlinear variable $b_3$ under the dynamics of the three-dimensional incompressible Navier–Stokes system. In particular, Equation(6) in expresses the time derivative $\partial_t b_3$ as a rational combination of transport, production, and gradient-coupling terms involving the velocity component $u_z$ and the vector field $\vec b$ . Such evolution equations arise naturally when the nonlinear convective operator $(\vec u\cdot\nabla)\vec u$ is rewritten in terms of composite variables formed from products of velocity gradients. These reformulations are standard in the mathematical analysis of nonlinear partial differential equations and are often used to expose cancellation structures or boundedness mechanisms in transport systems; see, for example, Ladyzhenskaya , Constantin and Foias , and Temam .

A central feature of the present formulation is the appearance of the matrix operator $M(\vec b)$ acting on the gradient vector $\vec G_1$ , producing quadratic gradient interaction terms of the form

Terms of this type represent nonlinear production or stretching mechanisms in fluid dynamics, analogous to the gradient-amplification processes that appear in vorticity and strain evolution equations. The balance between these production terms and the transport terms is fundamental in determining whether a solution remains bounded or develops singular behavior. The mathematical study of such balances is a central theme in the regularity theory of the Navier–Stokes equations; see Doering and Gibbon and Evans .
The interesting approach by using the vector $\bm{a}$ where $\bm{a}=\bm{b u_z}$ is that it can be proven that $\bm{b}$ is regular in space and time, that is $\bm{b} \in C^1$ . To see the proof of this see Appendix D.
Also I have suppressed the term $\nabla u_z \cdot \nabla \bm{b}$ term given in $-2 \nu G_2$ expression by integrating by parts and using the periodicity of $\mathbb{T}^3$ to give the $D^2_z b_3$ term I found previously that vanishes by multiplying by a nonzero $\nu$ and taking the limit. See Appendix E for the calculation of $\nabla u_z \cdot \nabla \bm{b}$ . I have also shown in this paper that you do not need to integrate $\nabla u_z \cdot \nabla \bm{b}$ by parts necessarily. A direct approach is to construct(as done) a smooth $u_z$ function and prove (as done in this paper) $\bm{b} \in C^1$ . This way $\nabla u_z \cdot \nabla \bm{b} \in C^0.$
The velocity component $u_z$ is modeled using a reciprocal structure depending on a scalar function $U(z,t)$ and an auxiliary function $f_3(x,y,t;\epsilon)$ , leading to the representation

This inverse form introduces a controlled pole structure in the velocity field, regulated by the parameter $\epsilon>0$ , and is consistent with analytic constructions in which singularities are shifted or regularized through additive perturbations. Such reciprocal velocity representations frequently appear in similarity solutions, potential-flow constructions, and analytic continuation methods for nonlinear evolution equations.

More generally, the velocity components are expressed using functions related to the Weierstrass $\zeta$ function,

where $g_2$ and $g_3$ are the invariants defining the associated elliptic lattice. Elliptic-function representations provide a natural framework for describing periodic or quasi-periodic structures and controlled pole behavior in nonlinear differential equations. The analytic properties of the Weierstrass functions and their role in constructing periodic solutions to nonlinear systems are well documented in classical function theory; see Whittaker and Watson and modern treatments of elliptic functions and nonlinear dynamics.

Within this framework, the boundedness of the variable $b$ follows from the algebraic structure of the governing equation, provided the denominator $U(z,t)+f_i$ remains nonzero. However, higher time derivatives may still exhibit rapid growth or blowup due to repeated differentiation of reciprocal factors. This distinction between bounded primary variables and potentially unbounded higher derivatives is a familiar phenomenon in nonlinear evolution equations and transport systems, where derivative growth can occur even when the underlying field remains finite. Such behavior has been analyzed in the context of regularity and singularity formation in fluid mechanics and related nonlinear systems; see Constantin and Foias and Doering and Gibbon .

The mathematical development that follows therefore focuses on the interaction between reciprocal velocity structures, nonlinear gradient production terms, and transport dynamics, with particular attention to conditions ensuring boundedness of the variable $b$ while allowing the possibility of blowup in higher-order time derivatives. We write the following in terms of third component $b_3$ corresponding to the $z$ component of velocity direction in the Navier Stokes equations of Eq(6) in (the appearance of $b^2_3$ term exists), in $\frac{\partial \bm{b}}{\partial t}$ ,

Note we have that:\

Solution of $u_z$ PDE and definition of $f_1$ , $f_2$ and $f_3$

Now in reference $u_z$ is:\

where $g_2,g_3$ are the invariants of the Weierstrass $\zeta$ function and we take them to be zero.
Here $f_3$ for $\epsilon_3>0$ is defined as,\

In the same fashion, one can obtain through the Geometric Calculus approach as in the approach used to obtain $u_z$ velocity, the components $u_x$ and $u_y$ . These in the same light will have the general form,\

and

Let us formalize the assumptions carefully.

Logic structure of singularities of NS Analysis of equation

The analysis presented in this section examines the logical structure of possible singularities in the three-dimensional incompressible Navier-Stokes equations by studying products of velocity components. In the classical formulation, the velocity field $\mathbf{u}=(u_x,u_y,u_z)$ evolves under nonlinear transport and viscous diffusion, and the formation of singularities is associated with the loss of boundedness of either the velocity field or its spatial derivatives. Determining which component of the velocity field is responsible for a divergence or a non-smoothness is a central problem in the mathematical theory of fluid dynamics and remains closely connected to the global regularity question for the Navier–Stokes equations; see Constantin and Foias , Temam , and Ladyzhenskaya .

The reasoning developed below relies on elementary but rigorous principles from real analysis: if a product of two quantities diverges or lacks smoothness while one factor remains bounded, then the divergence or non-smoothness must originate from the remaining factor. In the context of fluid mechanics, this principle allows singular behavior to be localized to a specific velocity component by examining the behavior of nonlinear interaction terms such as $u_x u_y$ and $u_x u_z.$ Such quadratic products appear naturally in the convective term $(\mathbf{u}\cdot\nabla)\mathbf{u},$ where they represent momentum transport between velocity components. The identification of singular growth through these nonlinear interactions is consistent with standard energy and regularity arguments used in the mathematical analysis of incompressible flows; see Doering and Gibbon and Evans .

Within this framework, the smoothness of selected velocity components imposes boundedness constraints that restrict the possible sources of divergence. If two distinct nonlinear products involving the same velocity component both become unbounded while the remaining factors stay finite, then the only mathematically consistent conclusion is that the shared component itself becomes singular. This logical structure provides a direct mechanism for identifying the origin of blowup in a multi-component system without requiring explicit solution formulas. Such component-wise reasoning is commonly used in the study of singularity formation, gradient amplification, and finite-time blowup criteria in nonlinear partial differential equations.

The argument developed in the following section therefore formalizes a simple but powerful diagnostic principle: the blowup of multiple quadratic interaction terms involving a common velocity component, combined with boundedness of the other components and their derivatives, implies that the shared component is the source of the singularity. In the Navier-Stokes setting, this conclusion corresponds to the loss of boundedness of the velocity gradient and signals the onset of finite-time singular behavior in the flow field. We are considering three components of velocity in the Navier-Stokes equations:

Suppose the following logical structure holds:

$u_z$ is smooth.
$u_x u_y$ blows up.
$u_y$ is smooth.
$u_x u_z$ blows up.

We ask: what can we conclude?

1) First logical consequence from $u_x u_y$ blowup

Assume

That means (locally in space-time) there exists a finite bound

for some constant $M<\infty$ where D is a sequence of partial derivatives of $u_y$ with respect to space and time.

Now suppose the derivative of:

Then necessarily

or Reason: a bounded factor cannot create divergence in a product.

So from these two facts alone,

2) Use the second product

Assume also

So similarly,

for some finite constant $M'<\infty$ .

If

then again

This is consistent with the first deduction.

3) Combine both statements

We now have two independent implications:

and

Therefore the unavoidable conclusion is

Analysis of $1/(\sin(Z-t)+1-\epsilon)$ and $1/(\sin(Z-t)+1)$

We consider the functions

and

Denominator bounds

For the second function, the denominator is

Since $-1 \le \sin(x) \le 1$ , we have

Check for zeros

The denominator of $f_2$ can vanish if

This occurs at

Finite-time blowup

Since the denominator can vanish at finite $t$ , the function

has finite-time blowup at those points.

Comparison with $\epsilon>2$

For the function

the denominator satisfies

so it does not vanish for any finite $t$ .

Conclusion

$1/(\sin(Z-t)+1-\epsilon)$ with $\epsilon>2$ has no finite-time blowup.
$1/(\sin(Z-t)+1)$ has finite-time blowup whenever $\sin(Z-t)=-1$ , i.e., at

$Z - t = \frac{3 π}{2} + 2 k π, k \in Z .$

Analysis of the Function $\displaystyle f(Z,t)=\frac{1}{\eta+\sin(Z-t)+1-\epsilon}$

We analyze the function

We determine precisely when there is finite-time blowup and when there is none.

Blowup condition

Finite-time blowup occurs if the denominator equals zero:

Rearranging gives

Use the boundedness of sine

We use the fundamental bound

Therefore a real solution exists if and only if

This gives the inequality

Solve the inequality

Add $1$ to all parts:

This is the necessary and sufficient condition for finite-time blowup.

Final classification of finite time blowup

Finite-time blowup occurs if and only if

because then there exists a time $t$ such that

No finite-time blowup occurs if and only if

because then

so the denominator never vanishes.

Useful special cases

Case 1 Original blowup case

Then

Blowup occurs.

Case 2 Small positive regularization

If

then

and therefore

and the function is bounded:

Case 3 Large negative shift

If

then

which is impossible, so again:

Interpretation

The effective shift in the denominator is

If the net shift is positive enough $(\eta>\epsilon)$ , the denominator stays away from zero and the function remains smooth.
If the net shift falls in the interval

$0 \leq ϵ - η \leq 2,$

then the denominator can hit zero and finite-time blowup occurs.

This provides a complete and exact criterion for the form

From the general solution of each $u_i$ component of the 3D Navier Stokes equations for $i=1\dots3$ , for $i=3$ first:

where $u_z$ is derived from the WeierstrassZeta function with zero invariants and $\eta_3$ is in terms of $I_n$ (from section ). The logic followed here is that since $I_n \rightarrow 0$ as $n \rightarrow \infty$ then we choose a range of $\epsilon$ values such that $u_z$ is smooth in fact sinusoidal. Then we attempt to determine if $b_3=u_x u_y$ is smooth or has a finite time blowup. Now from the general solutions obtained for $u_x$ and $u_y$ based on the geometric calculus approach,(as was obtained for $u_z$ , the following occurs:\

On a foliation of planes $y=x+C$ we set $\mathit{f_1} \! \left(y , z , t\right)=\mathit{f_1} \! \left(x+C , z , t\right) =\mathit{f_2} \! \left(x, z , t\right)^3$ and $\eta_1 \rightarrow 0$ and $\eta_2 \rightarrow 0$ giving,\

Substituting into Equation 6 of reference we obtain,\

Here we let $\eta_3 \rightarrow 0$ and $\epsilon \rightarrow 1$ (blowup regime).\