Turbulent flows are characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. In the 3D viscous incompressible flow the kinetic energy is transferred from large scale eddies to small scale eddies, in particular, if you inject energy at a wavenumber $k_F$ kinetic energy will be transferred to the wavenumbers k s.t. $k>k_F$. In 2D viscous incompressible flow the kinetic energy is transferred from small to large eddies; in particular, if you inject energy at wavenumber $k_F$, then your energy will be transferred both to the larger and smaller wavenumbers; this fact can be flagged as an inverse cascade. This double cascade is due to the presence of two inviscid quadratic invariants: energy and enstrophy. Experimental, numerical and theoretical works have shown that many turbulent configurations deviate from the ideal three and two dimensional homogeneous and isotropic cases characterized by the presence of a strictly direct and inverse energy cascade [1] . It can be assumed that the flow is confined in a periodic box of size L and that the external forcing is acting on a band with a limited range of scales centered around ${\ell }_{in}$. It is important to discuss the case when the horizontal size, L of the domain is finite and no drag force is present so that a condensate forms in a split cascade regime. With layers of fluid of finite height, there is a path that saturates the inverse cascade. In square boxes, the condensate takes the form of a vortex dipole. In [2] , mild solutions to the Cauchy problem for the forced Navier Stokes equations have been sought by applying the Duhamel principle and developing the corresponding (to NS equation) integral equation. ( $u\left(t\right)=S\left(t\right)u_0-B\left(u\mathrm{,}u\right)\left(t\right)+F\left(t\right)$) with the heat semigroup:

$S\left(t\right)u_0\left(x\right)={\displaystyle {\int }_{{ℝ}^3}}\frac{1}{{\left(4\pi t\right)}^{3/2}}\text{exp}\left(-\frac{{\left|x-y\right|}^2}{4t}\right)u_0\left(y\right)\text{d}y$ where the bilinear form $B\left(u\mathrm{,}v\right)\left(t\right)\equiv {\displaystyle \int \mathbb{P}\nabla S\left(t-\tau \right)\cdot \left(u\left(\tau \right)\otimes v\left(\tau \right)\right)\text{d}\tau }$. With external related force term: ( $F\left(t\right)\equiv {\displaystyle {\int }_0^{t}S\left(t-\tau \right)\mathbb{P}f\left(\tau \right)\text{d}\tau }$), where $\mathbb{P}$ is the Leray projector. See the

development in [2] , where it shows clearly how it is defined in terms of the Fourier multiplier. Using Fourier transforms and introducing general tempered distributions living in a unique functional space, first a result on the existence of small solutions to the integral equation associated with Navier Stokes equations in this functional space setting is referred to and then in (Theorem 2.3 ( [2] proven on page 17 shows that there is not an imposing of any smallness assumption neither on initial data nor external forces). There, it is concluded that there is a finite bound in norm of the expression: ${\Vert u\left(t\right)-S\left(t\right)u_0-F\left(t\right)\Vert }_q$ for each $q\in \left[\mathrm{2,3}\right)$, $u\left(t\right)-S\left(t\right)u_0-F\left(t\right)\in L^q\left({ℝ}^3\right)$ for all $t>0$. See [3] for asymptotic results using heat kernel associated with Heat equation. Considering the scaling $\lambda =\frac{1}{\delta }$ where $\delta >0$ as defined in Section 2, Planchon [3] has shown that for scale invariance on a critical space that studying the asymptotic behaviour of the velocity of NS equation for large time t is equivalent to the study of large $\lambda$ with fixed time. It is shown there that as t approaches infinity that the natural space scale is $\sqrt{t}$ as in the Heat equation. Replacing x by $x/\sqrt{t}$ in the solution of the other velocity term involved in the scaling and letting t approach infinity, we obtain the same result as if we let $\lambda$ approach infinity in $u_{\lambda }=u_{1/\delta }$ ( $\delta$ approaching zero here). It is proposed that this limit should also be a solution of the N.S. equations. For two-dimensional problems, Hasan et al. [4] have introduced a new approach to solving 2D unsteady incompressible Navier-Stokes equations. Now on the topic of well-posedness of solutions to Navier Stokes equations, Wang et al. [5] have examined globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional 3D incompressible Navier Stokes and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates has been obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. In the present paper, proposition B (hence A) is proposed to be true in the Millennium problem for the PNS Cauchy initial value system of equations. The blowup points for Hölder functions in higher derivatives can be shifted procedurally to arbitrarily large time t. However, at infinity it remains to show that there is no blowup there. Here the case of the asymptotic limit as $\delta \to 0$ is proposed to exist [3] . So the claim is that there is no finite time blowup on $\left[\mathrm{0,}\infty \right]$. Taking $\delta \ne 0$ establishes non smooth or singular solutions at a finite time. First it is singular solutions that we seek in $L^2$ space and subsequently claim that there exists a sequence of such functions that can converge to a $C^{\infty }$ function.

Several general inequalities are used to arrive at these claims. In particular the Hölder inequality is used and because of the compactness of the 3-Torus the inequality involving the $L^2$ and $L^{\infty }$ norms is used through the finite measure of the space. Also Jensen’s inequality is used in a few places as well as the Preköpa-Leindler and Gagliardo-Nirenberg Inequalities. The main idea is that bounds are established for each compact torus and then in a limiting approach as a sequence of compact tori volumes approach infinity the Preköpa-Leindler and Gagliardo-Nirenberg Inequalities can be used to calculate the tensor product term in the expression given as $\mathcal{G}{\left(\eta \right)}_{{\delta }_3}$ in this paper. Leray’s criterion for Navier-Stokes singularity states that if a solution to the Navier-Stokes equations becomes singular at $t=T_{\mathrm{<mo>\; *\; </mo>}}$, then it is necessary that the velocity norm ${\Vert u\Vert }_{L^s}\left({ℝ}^3\right)$ for $s>3$, grows unbounded in the manner ${\Vert u\Vert }_{L^s}\left({ℝ}^3\right)\ge C_s/{\left(T_{\mathrm{<mo>\; *\; </mo>}}-t\right)}^{\frac{s-3}{2s}}$ as $t\to T_{\mathrm{<mo>\; *\; </mo>}}$. Here $C_s$ may depend on s [6] .

The aim of the author of the present work is to use the fact that solutions of the Navier-Stokes equations that become singular must do so in an isotropic manner. It appears in recent works [7] [8] that the perfect candidate for this manner of singularity is the solution shown to depend on the Lambert W function as shown in those works and in the present work as well. Future work will confirm that when each of the 3 velocities ${\nu }_i,i=1\cdots 3$ are set as ${\nu }_i=f\left({\nu }_j\times H_i\right)$ where $i\ne j$ and $H_i=H_i\left(x,y,z,t\right)$, then there exists an auxiliary PDE which gives a solution dependent on the W function as in this paper.

For the parameters of $\nu =\mu /\rho$, where $\mu =\frac{1}{C{C}_3^3\kappa \rho }$ and $\rho =1$, where optimal constants C and C₃ exist with eigenvalue $\kappa$ the existence of dipole-dipole interactions are proven to occur. These interactions are essentially collisions of evolving 4-vortices which are known in the literature to not be integrable [9] . Upon collision the dipoles can overlap. An integrable dipole-dipole collision corresponds to a configuration where two dipoles propagate along the same geometric axis in the form of a parallelogram. In this work for small times in the PNS flow evolution multi-dipoles are shown to result from the Navier Stokes equations that are arranged in a sequence of symmetric Riemann surface pancakes that do not overlap. The Riemann surface plots in this paper show the velocity $u_z$ versus the real and imaginary parts of small and large data $\zeta$ respectively. Finite time blowup occurs in the first derivative and higher wrt to t at the center $\zeta =0$ and $\zeta ={\zeta }_L$. The arbitrariness of small and large data addresses the problem posed in [10] . Essentially there is a flip-flop of two possible values for $\zeta$, either small $\zeta =0$ or $\zeta ={\zeta }_L$. For the dynamic viscosity $\mu$ there exists a function $C\left(\mu \right)$ such that as C decreases t decreases and as C increases, t increases. An equality is set to unity that involves the velocity $u_z$ and parameters $\rho$, $\mu$, $\kappa$ and $C_3$ associated with the z component of velocity for $u_z$ expressed in terms of the LambertW function. There is a deformation or non parallelization in the dipole space configuration along the imaginary axis $y=Im\left(\zeta \right)$. Eventually there is an optimal time where two dipoles of size d are situated away from the origin by distance L₁ and L₂ respectively and orientated with respect to each other by an angle $\psi$ around the origin [11] . As time increases a return to symmetry is obtained as $t\to \infty$. Thus, symmetry breaking for the PNS equations is proven to exist and there is a strong correlation between the optimal time of finite time blowup and the maximal symmetry breaking time. Strong symmetry breaking is also mentioned in [12] . For a variational formulation of Rayleigh Plesset cavitation dynamics with reference to the incompressible Navier-Stokes equations see Moschandreou et al. [13] . Finally high infinite spatially antisymmetric oscillations occur at the origin for $\zeta$ and at $\zeta ={\zeta }_L$.

The 3-D incompressible unsteady Navier-Stokes equations (NS) in Cartesian coordinates for the velocity $u^{\mathrm{<mo>\; *\; </mo>}}=u^{\mathrm{<mo>\; *\; </mo>}i}{e}_i$, $u^{\mathrm{<mo>\; *\; </mo>}i}=\left\{u_x^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{u}_y^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{u}_z^{\mathrm{<mo>\; *\; </mo>}}\right\}$:

$\rho \left(\frac{\partial }{\partial t^{\mathrm{<mo>\; *\; </mo>}}}+u^{\mathrm{<mo>\; *\; </mo>}j}{\nabla }_{\mathrm{<mo>\; *\; </mo>}j}\right)u_i^{\mathrm{<mo>\; *\; </mo>}}-\mu {\nabla }_{\mathrm{<mo>\; *\; </mo>}}^2{u}_i^{\mathrm{<mo>\; *\; </mo>}}+{\nabla }_{\mathrm{<mo>\; *\; </mo>}i}{P}^{\mathrm{<mo>\; *\; </mo>}}=\rho F_i^{\mathrm{<mo>\; *\; </mo>}}$

where $\rho$ is constant density, $\mu$ is dynamic viscosity, $F^{\mathrm{<mo>\; *\; </mo>}}=F^{\mathrm{<mo>\; *\; </mo>}i}{e}_i$ are the body forces on the fluid. Expressing the components of the velocity vector and pressure to $u={\left(u\right)}^i{e}_i$, $P={\left(P\right)}^i{e}_i$, coordinates $x_i$ and time t according to the following form utilizing the non-dimensional quantity $\delta <0$:

$u_i^{*}=\frac{1}{\delta }{u}_i,P_i^{*}=\frac{1}{{\delta }^2}{P}_i,x_i^{*}=\delta x_i,t^{*}={\delta }^{2}t$

The continuity equation in Cartesian coordinates, is

${\nabla }^i{u}_i=0$

Using the previous transformations from $\ast$ to non- $\ast$ variables, multiplying the first three components of NS equation by unit vectors $i=\left(\mathrm{1,0,0}\right)$, $j=\left(\mathrm{0,1,0}\right)$ and $k=\left(\mathrm{0,0,1}\right)$ respectively and adding modified equations within the set given by the 3 parts of Navier Stokes equations (components 1, 2 and 3) give the following equations for the resulting composite vector

$b=\frac{1}{\delta }{u}_{x}i+\frac{1}{\delta }{u}_{y}j+\frac{1}{\delta }{u}_{z}k$,

$\begin{array}{l}\frac{1}{{\delta }^2}\frac{\partial b}{\partial t}+\frac{u_x}{{\delta }^2}\frac{\partial b}{\partial x}+\frac{u_y}{{\delta }^2}\frac{\partial b}{\partial y}+\frac{u_z}{{\delta }^2}\frac{\partial b}{\partial z}-\frac{\mu }{\rho {\delta }^2}{\nabla }^{2}b\\ +\frac{1}{{\delta }^3\rho }\frac{\partial P}{\partial x}i+\frac{1}{{\delta }^3\rho }\frac{\partial P}{\partial y}j+\frac{1}{{\delta }^3\rho }\frac{\partial P}{\partial z}k=F_{T_1}^{\mathrm{<mo>\; *\; </mo>}}i+F_{T_2}^{\mathrm{<mo>\; *\; </mo>}}j+F_{sz}^{\mathrm{<mo>\; *\; </mo>}}k\end{array}$

Multiplying the previous equation by ${\delta }^3$ and by $u_z^{\mathrm{<mo>\; *\; </mo>}}$ and the $z^{\mathrm{<mo>\; *\; </mo>}}$ component of the Navier Stokes equations(in $\ast$ variables) by ${\delta }^3$ and by $b$, (using transformations in $\delta$), the addition of the resulting equations recalling the product rule and finally setting the forcing to be equal to $F_{sz}^{*}=\frac{1}{{\delta }^3\rho }\frac{\partial P}{\partial z}$, (also equal to pressure gradient force if $\delta <0$) I obtain:

$\begin{array}{l}\frac{\partial a}{\partial t}+b\cdot \nabla a-\frac{\mu }{\rho }{u}_z{\nabla }^{2}b+\frac{1}{\delta \rho }{u}_z{\nabla }_{xy}P+b\frac{1}{\rho }\frac{\partial P}{\partial z}\\ +{\delta }^2{u}_z{F}_T^{\mathrm{<mo>\; *\; </mo>}}+{\delta }^3{F}_{sz}^{\mathrm{<mo>\; *\; </mo>}}b-\frac{\mu }{\rho }b{\nabla }^2{u}_z=0\end{array}$

Note that here the forcing term $F_{sz}^{\mathrm{<mo>\; *\; </mo>}}$ does not counterbalance the non zero pressure gradient which causes flow to occur. It is equal to it, hence for negative ( $\delta <0$), setting $F_{sz}^{\mathrm{<mo>\; *\; </mo>}}=\frac{1}{{\delta }^3}{F}_{sz}$, and equating the two $F_{sz}^{\mathrm{<mo>\; *\; </mo>}}$ expressions then $F_{sz}-\frac{1}{\rho }\frac{\partial P}{\partial z}=0$. This is the conditional equation that needs to be satisfied as well as the PDE’s associated with the problem. It turns out as the paper shows that this equation is fulfilled as each term in it is zero for the final solution in $u_z$ (similar equations in $u_y$ and $u_x$ follow.) The reason as will become apparent is that for a sequence converging to the z solution $u_z=C+{\Psi }_3\left(x,y\right)$ will ultimately be shown for all real t and z, $C\in ℝ$thereby showing by Newton’s second law that $F_{sz}=0$. Also for pressure function $P\in L^2$ defined as a Hölder continuous function in z, a sequence converges to the solution for P that is in the form of $P=C+\Theta \left(x,y\right)$ for all real t and z, $C\in ℝ$. Hence the pressure gradient in z, $\frac{\partial P}{\partial z}=0$. Finally the relationship between $a$ and $b$ is:

$a=u_{z}b$

The nonlinear inertial term when added to $b\nabla u_z\cdot b$ and factoring out $b$ gives: $b\cdot \nabla a$. Multiplying the previous PDE in (a and b) by $u_z$ and adding to it $u_{z}b\nabla \cdot a$ gives a new PDE (call it (I*)) which has been solved in reference ( [14] : Equation (6) there). This extra term when integrating the resultant PDE and using Ostrogradsky’s theorem(divergence theorem) has a zero contribution since $b=\omega \left|r\right|$. (Recall: velocity is the radius multiplied by angular velocity. It is shown in this paper that the $L^2$ norm of this term approaches zero.)

It is stressed here that the same analysis is possible where we can rearrange the terms in such a way as to add the first and third equations and get a PDE with $a=u_{y}b$ and $a=u_{x}b$. So there will be three PDE in these forms and they are all coupled. This is identical to solving the original NS equations where the forcing terms are not counterbalancing the pressure gradient terms.

Taking the geometric or dot product of the PDE (I*) defined above with respect to $f=a\cdot \nabla a$, it can be shown that a scalar and vector field PDE is produced. (See Reference [14] on page 388 to page 389 there.)

It has also been shown there that there exists a non linear operator $\mathcal{G}$ as shown in this paper which can be expressed in the form given in reference [14] on page 389 by Equation (19) there.(note that there in Equation (17) the left side of 3D (generalized from 2D to 3D in this section) Navier Stokes equation is set to zero which simplifies to the tensor product term. The analysis continues from Equation (19) in the present work. (see Equations (6)-(10) in Section (4))

The following theorem as proven in [2] is the foundational theorem for the existence of finite energy solutions in the space $L^q\left({ℝ}^3\right)$ for each $q\in \left[\mathrm{2,3}\right)$ and all $t>0$. Mild solutions to the Cauchy problem for the forced Navier Stokes equations has been sought for by applying the Duhamel principle and developing the corresponding (to problem (2) in Section 4) integral equation.

( $u\left(t\right)=S\left(t\right)u_0-B\left(u\mathrm{,}u\right)\left(t\right)+F\left(t\right)$) with the heat semigroup: $S\left(t\right)u_0\left(x\right)={\displaystyle {\int }_{{ℝ}^3}}\frac{1}{{\left(4\pi t\right)}^{3/2}}\text{exp}\left(-\frac{{\left|x-y\right|}^2}{4t}\right)u_0\left(y\right)\text{d}y$ where the bilinear form $B\left(u\mathrm{,}v\right)\left(t\right)\equiv {\displaystyle \int \mathbb{P}\nabla S\left(t-\tau \right)\cdot \left(u\left(\tau \right)\otimes v\left(\tau \right)\right)\text{d}\tau }$. With external related force term: ( $F\left(t\right)\equiv {\displaystyle {\int }_0^{t}S\left(t-\tau \right)\mathbb{P}-f\left(\tau \right)\text{d}\tau }$), where $\mathbb{P}$ is the Leray projector. The Leray projector is defined as the Fourier multiplier with the matrix component ${\left(\hat{P}\left(\xi \right)\right)}_{j\mathrm{,}k}={\delta }_{j\mathrm{,}k}-{\xi }_j{\xi }_k/{\left|\xi \right|}^2$ satisfying $\left|{\left(\hat{P}\left(\xi \right)\right)}_{j\mathrm{,}k}\right|\le 1$ for all $\xi \in {ℝ}^3\backslash \left\{0\right\}$ and all $j\mathrm{,}k\in \left\{\mathrm{1,2,3}\right\}$. The tensor product for two vector fields $u\otimes v$ is the $3\times 3$ matrix whose $\left(m\mathrm{,}n\right)$ entry is $u_m{v}_n$. The Fourier transform of an integrable function is

$\hat{f}\left(\xi \right)={\left(2\pi \right)}^{\frac{3}{2}}{\displaystyle {\int }_{{ℝ}^3}{\text{e}}^{-ix\cdot \xi }f\left(x\right)\text{d}x}$

The following functional spaces are used in the theorem in the space of tempered distributions ( $S^{\prime }$),

$\begin{array}{l}\frac{\partial u}{\partial t}-\Delta u+u\cdot \nabla u=-\nabla p+f\\ \text{div}u=0\\ u_{t\mathrm{<mo>\; =\; </mo>0}}=u_0\mathrm{.}\end{array}$(2)

where $u=u\left(x\mathrm{,}y\mathrm{,}z\mathrm{,}t\right)$ is velocity, $p=p\left(x\mathrm{,}y\mathrm{,}z\mathrm{,}t\right)$ is pressure and $f=f\left(x\mathrm{,}y\mathrm{,}z\mathrm{,}t\right)$ is the forcing function. Here $u=\left(u_x\mathrm{,}{u}_y\mathrm{,}{u}_z\right)$, where $u_x$, $u_y$, and $u_z$ denote respectively the x, y and z components of velocity. Introducing Poisson’s equation (see [13] [14] and [15] ), the second derivative $P_{zz}$ is set equal to the second derivative obtained in the ${\mathcal{G}}_{{\delta }_1}$ expression further below, as part of $\mathcal{G}$, and

$\begin{array}{c}{P}_{zz}=-2u_z{\nabla }^2{u}_z-{\left(\frac{\partial u_z}{\partial z}\right)}^2+\frac{1}{\eta }\frac{\partial }{\partial z}\left(\frac{\partial u_z}{\partial x}+\frac{\partial u_z}{\partial y}\right)-\delta u_x\frac{{\partial }^2{u}_z}{\partial x\partial z}\\ -\delta u_y\frac{{\partial }^2{u}_z}{\partial z\partial y}+{\left(\frac{\partial u_x}{\partial x}\right)}^2+2\frac{\partial u_x}{\partial y}\frac{\partial u_y}{\partial x}+{\left(\frac{\partial u_y}{\partial y}\right)}^2\end{array}$(3)

where the last three terms on rhs of Equation (3) can be shown to be equal to $-\left(P_{xx}+P_{yy}\right)$. Along with equations below, the continuity equation in Cartesian co-ordinates is ${\nabla }^i{u}_i=0$. Now for $\delta <0$,

$\begin{array}{l}{u}_x=\frac{u_x^{*}}{\delta };u_y=\frac{u_y^{*}}{\delta };u_z=\frac{u_z^{*}}{\delta };P=\frac{P^{*}}{{\delta }^2}\\ x=x^{*}\cdot \delta ;y=y^{*}\cdot \delta ;z=z^{*}\cdot \delta ;t=t^{*}\cdot {\delta }^2,\\ \frac{\partial }{\partial x}={\delta }^{-1}\frac{\partial }{\partial x^{*}};\frac{\partial }{\partial y}={\delta }^{-1}\frac{\partial }{\partial y^{*}};\frac{\partial }{\partial z}={\delta }^{-1}\frac{\partial }{\partial z^{*}};\frac{\partial }{\partial t}={\delta }^{-2}\frac{\partial }{\partial t^{*}}\end{array}$(4)

Next the right hand side of the group of transformations Equation (4) are mapped to $\eta$ variable terms. Here $\eta$ is unity,

$u_i^{*}=\eta v_i;P^{*}={\eta }^{2}Q;x_i^{*}={\eta }^{-1}{y}_i;t^{*}={\eta }^{-2}s,i=1,2,3.$(5)

The double transformation here is used for notational clarity. For a deeper connection see [16] where two scalings can be used for the Euler equations and the ratio there shows a hidden scale symmetry. Note that the form of the rescaled velocities is related to the definition of the LambertW function W, such that

${\text{e}}^{-W\left(\xi \right)}=\frac{W\left(\xi \right)}{\xi }$, where $\xi$ depends on t and proportional to $\delta$ approaching zero for infinite time t. It is true that the ratio $\frac{W\left(\xi \right)}{\xi }$ is equal to 1 and not zero or infinity so that it is well defined making u finite in the limit. Note that the original Navier-Stokes equations are preserved and rearranged in the following form,

$\mathcal{G}\left(\eta \right)=\mathcal{G}{\left(\eta \right)}_{{\delta }_1}+\mathcal{G}{\left(\eta \right)}_{{\delta }_2}+\mathcal{G}{\left(\eta \right)}_{{\delta }_3}+\mathcal{G}{\left(\eta \right)}_{{\delta }_4}=0$(6)

$\begin{array}{c}\mathcal{G}{\left(\eta \right)}_{{\delta }_1}=\frac{1}{{\eta }^6}[\left({\delta }^{-1}-1\right){\left(\frac{\partial v_3}{\partial s}\right)}^2+\frac{\mu \left(\frac{\partial v_3}{\partial s}\right)\left(\frac{{\partial }^2{v}_3}{\partial y_1^2}+\frac{{\partial }^2{v}_3}{\partial y_2^2}+\frac{{\partial }^2{v}_3}{\partial y_3^2}\right)}{\rho }\left(1-{\delta }^{-1}\right)\\ +\frac{\left({\delta }^{-1}-1\right)\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial Q}{\partial y_3}}{\rho }]\end{array}$(7)

$\begin{array}{c}\mathcal{G}{\left(\eta \right)}_{{\delta }_2}=\frac{v_3}{{\eta }^6}\left(\frac{\partial v_3}{\partial y_3}\right)\frac{\partial v_3}{\partial s}+\frac{{\left(v_3\right)}^2}{{\eta }^6}\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\\ +\frac{2\left(\frac{\partial v_1}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_1}+2\left(\frac{\partial v_2}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_2}+2\left(\frac{\partial v_3}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_3}}{\delta {\eta }^6}\end{array}$(8)

$\mathcal{G}{\left(\eta \right)}_{{\delta }_3}=\frac{1}{ϵ{\eta }^3}\times \left[{\displaystyle {\iint }_S}\left(\frac{1}{\delta \rho }{v}_3^2{\nabla }_{y_1{y}_2}Q+\frac{1}{{\delta }^2}\stackrel{\to }{v}\frac{1}{\rho }{v}_3\frac{\partial Q}{\partial y_3}\right)\cdot \stackrel{\to }{n}\text{d}S-{\displaystyle {\int }_{\Omega }}\frac{\Vert \frac{\partial v_3}{\partial t}\stackrel{\to }{b}\cdot \left(\stackrel{\to }{b}\otimes \nabla v_3\right)\Vert }{\Vert \stackrel{\to }{b}\Vert }\text{d}V\right]$(9)

$\mathcal{G}{\left(\eta \right)}_{{\delta }_4}=\frac{1}{ϵ{\eta }^3}\left[{\delta }^2{F}_T^{\mathrm{<mo>\; *\; </mo>}}\cdot {\nabla }_{y_1{y}_2}{v}_3^2-{\delta }^3{v}_3\frac{\partial v_3}{\partial y_3}{F}_{sz}^{\mathrm{<mo>\; *\; </mo>}}+{\delta }^2\stackrel{\to }{v}\cdot \nabla \left(v_3{F}_{sz}^{\mathrm{<mo>\; *\; </mo>}}\right)\right]$(10)

where $\mathcal{G}{\left(\eta \right)}_{{\delta }_i}\mathrm{,}i=1,2,3,4$ are given in [7] and [8] and it has been shown there that this decomposition is valid and that on a volume of an arbitrarily small sphere embedded in each cell of the lattice centered at the central point of each cell of the 3-torus, $\mathcal{G}{\left(\eta \right)}_{{\delta }_3}$ is negligible for the case of no viscosity (Euler equation) and for viscosity $\nu =1$ the existence of a dipole occurs with the centre of the dipole occurring shifted away from the centre of the given cell. From this equation we can solve for $\frac{\partial Q}{\partial y_3}$ algebraically and differentiating wrt to $y_3$ and using Poisson’s

equation by setting the representation of each of the two partial derivatives wrt to $y_3$ equal to each other, we can obtain,

$L=0$(11)

$L=L_1+L_2+L_3$(12)

where each $L_i,i=1,2,3$ is given as,

$\begin{array}{c}{L}_1={\left(\frac{\partial v_3}{\partial s}\right)}^2\mu \left(\delta -1\right)\frac{{\partial }^3{v}_3}{\partial y_3\partial y_1^2}+{\left(\frac{\partial v_3}{\partial s}\right)}^2\mu \left(\delta -1\right)\frac{{\partial }^3{v}_3}{\partial y_3\partial y_2^2}\\ +{\left(\frac{\partial v_3}{\partial s}\right)}^2\mu \left(\delta -1\right)\frac{{\partial }^3{v}_3}{\partial y_3^3}+\left(\frac{\partial v_3}{\partial s}\right){\left(v_3\right)}^2\left(\frac{{\partial }^3{v}_3}{\partial y_3^2\partial s}\right)\delta \rho \end{array}$

$\begin{array}{c}-{\left(v_3\right)}^2{\left(\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\right)}^2\delta \rho -2\rho (\left(\delta /2-1/2\right){\left(\frac{\partial v_3}{\partial s}\right)}^2-v_3\left(\frac{\partial v_3}{\partial s}\right)\left(\frac{\partial v_3}{\partial y_3}\right)\delta \\ +(v_3\left(F_{T_1}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_1}+v_3\left(F_{T_2}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_2}{\partial s}\right)\frac{\partial v_3}{\partial y_2}\\ \begin{array}{c}\\ \end{array}+1/2\Lambda \left(y_1,y_2,y_3,s\right)+1/2\Phi \left(s\right))\delta )\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\end{array}$

$\begin{array}{c}{L}_2=[(\left(\delta -1\right)\left(\delta v_1\left(y_1,y_2,y_3,s\right)-1\right)\frac{\partial v_3}{\partial s}\\ +2v_3\rho \delta \left(F_{T_1}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_1}{\partial s}\right))\times \frac{{\partial }^2{v}_3}{\partial y_3\partial y_1}\\ +(\left(\delta -1\right)\left(v_2\left(y_1,y_2,y_3,s\right)\delta -1\right)\frac{\partial v_3}{\partial s}\\ +2v_3\rho \delta \left(F_{T_2}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_2}{\partial s}\right))\times \frac{{\partial }^2{v}_3}{\partial y_3\partial y_2}\\ +3v_3\left(-2/3+\left(\rho +2/3\right)\delta \right)\left(\frac{\partial v_3}{\partial s}\right)\frac{{\partial }^2{v}_3}{\partial y_3^2}+\cdots \end{array}$

$\begin{array}{c}{L}_3=2v_3\left(\frac{\partial v_3}{\partial s}\right)\left(\delta -1\right)\frac{{\partial }^2{v}_3}{\partial y_1^2}+2v_3\left(\frac{\partial v_3}{\partial s}\right)\left(\delta -1\right)\frac{{\partial }^2{v}_3}{\partial y_2^2}\\ +2\left(\frac{{\partial }^2{v}_1}{\partial y_3\partial s}\right)v_3\left(\frac{\partial v_3}{\partial y_1}\right)\rho \delta +2\left(\frac{{\partial }^2{v}_2}{\partial y_3\partial s}\right)v_3\left(\frac{\partial v_3}{\partial y_2}\right)\rho \delta \\ +(\left(-1+\left(3\rho +1\right)\delta \right){\left(\frac{\partial v_3}{\partial y_3}\right)}^2+\left(\delta -1\right)\\ \times \left({\left(\frac{\partial v_1}{\partial y_1}\right)}^2+2\left(\frac{\partial v_1}{\partial y_2}\right)\frac{\partial v_2}{\partial y_1}+{\left(\frac{\partial v_2}{\partial y_2}\right)}^2\right))\frac{\partial v_3}{\partial s}\\ +2\rho ((\left(F_{T_1}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_1}\\ +\left(\frac{\partial v_3}{\partial y_2}\right)\left(F_{T_2}^{*}\left(y_1,y_2,y_3,s\right)+\frac{\partial v_2}{\partial s}\right))\frac{\partial v_3}{\partial y_3}+v_3\left(\frac{\partial v_3}{\partial y_1}\right)\frac{\partial F_{T_1}^{*}}{\partial y_3}\\ +v_3\left(\frac{\partial v_3}{\partial y_2}\right)\frac{\partial F_{T_2}^{*}}{\partial y_3}+1/2\frac{\partial \Lambda (y_1,y_2,y_3,s)}{\partial y_3})\delta ]\frac{\partial v_3}{\partial s}\end{array}$

$\begin{array}{c}\Lambda \left(y_1,y_2,y_3,s\right)=2\frac{f_0\left(s\right)F\left(y_1,y_2,y_3\right)v_3\left(y_1,y_2,y_3,s\right)\frac{\partial v_3}{\partial y_1}}{\delta }\\ +2\frac{f_0\left(s\right)G\left(y_1,y_2,y_3\right)v_3\left(y_1,y_2,y_3,s\right)\frac{\partial v_3}{\partial y_2}}{\delta }\\ -{\delta }^3{v}_3\left(\frac{\partial v_3}{\partial y_3}\right)F_{sz}^{*}\left(y_1,y_2,y_3,s\right)\\ +{\delta }^2\left(\left(\frac{\partial v_3}{\partial y_3}\right)F_{sz}^{*}\left(y_1,y_2,y_3,s\right)+v_3\frac{\partial F_{sz}^{*}}{\partial y_3}\right)\end{array}$(13)

where $\stackrel{\to }{f}=\left(F_{T_1}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{F}_{T_2}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{F}_{sz}^{\mathrm{<mo>\; *\; </mo>}}\right)$ is the external forcing vector and $\stackrel{\to }{v}=\left(v_1\mathrm{,}{v}_2\mathrm{,}{v}_3\right)$ is the velocity in each cell of the 3-Torus.

The singular forcing is expressed in terms of WeierstrassP (P) and LambertW (W) functions in $y_i$ directions and defined as,

$F_{T_1}^{*}=F_{T_2}^{*}=F_{sz}^{*}=P{\left({\left(W\left(F\left(y_1,y_2,y_3,s\right)\right)+1\right)}^{-1},3m^2,m^3\right)}^{-3}$

for a full 3D dimensional plus time general function $F\left(y_1\mathrm{,}{y}_2\mathrm{,}{y}_3\mathrm{,}s\right)$. When $W\left(F\left(y_1\mathrm{,}{y}_2\mathrm{,}{y}_3\mathrm{,}s\right)\right)=-1$ the forcing is singular at $m=0$. An r-ball exists about any point in ${\mathbb{T}}^3$ where $\left|F+\frac{1}{\exp \left(1\right)}\right|\le r$. Here $F=-\eta +{\left(y_1-a_i\right)}^2+{\left(y_2-b_i\right)}^2+{\left(y_3-c_i\right)}^2$ where $\left(a_i\mathrm{,}{b}_i\mathrm{,}{c}_i\right)$ is the center of a given r-ball. The idea is that we can choose an arbitrary ball in ${\mathbb{T}}^3$. We set $F=-\frac{1}{e}+r$. At the center of ball forcing is singular as $r\to 0$. Forcing will be arbitrarily large in an r-ball for $\eta \to \frac{1}{e}$ with $\eta \ge \frac{1}{e}$. Considering circumscribed cubes each containing an r-ball. The union of cubes is the 3-Torus since it is compact. The dimension of the cubes is such that by periodicity the forcing is singular at the centres of the cubes.

The following system of PNS initial value problems is considered with singular forcing as given above,

$\begin{array}{l}{L}_F{v}_{3_i}=F_i,i=1,\cdots ,n\\ \nabla \cdot v=0\\ \Psi \circ \Psi \underset{n-1}{\underset{︸}{\circ \Psi \circ \cdots \circ }}\Psi {v_{3_i}|}_{s=0}=0\end{array}$(14)

where $L_F$ is the nonlinear partial differential operator defined by Equations (11)-(12) and Equation (13). The different uncoupled IVPs starting with $n=1$ and indexed by n are solved. Each IVP will produce solutions such that the initial conditions are not smooth due to non-smooth forcing. The idea is to solve these IVPs for each n and see if Hölder continuous functions exist for each n. The Ψ are related to the form of the solution for $v_{3_i}$ and will be shown to exist and one can see how the n compositions of Ψ affect the solution once the $n=1$ PDE is solved. The question that arises is, what happens when n approaches infinity? Can we use a problem with singularities to obtain in the limit a globally smooth problem? It has been proven that for initial conditions and singular forcing existing in a unique space of tempered distributions that a finite energy solution exists in $L^2{\left({ℝ}^3\right)}^3$ for all $t\ge 0$ [2] . So then in a top down approach one can take a sequence in $L^2$ converging to a $C^{\infty }$ function of the above initial value problems. However one does not know what the form of the functions are here. So a bottom up approach must be adhered to solve the PNS system. Next continuing with the solution procedure consider the $\left(a_i\mathrm{,}{b}_i\mathrm{,}{c}_i\right)$ as the center of each cell of the lattice in ${\mathbb{T}}^3$, upon substituting the WeierstrassP functions and their reciprocals into Equations. (11)-(12) together with the forcing terms given by Λ (in Equation (13)), it has been determined that a major simplification of Equations. (11)-(12) occurs. The torus ${\mathbb{T}}^3={\left[-\alpha \mathrm{,}\alpha \right]}^3$ is considered which contains a union of the $ϵ$ balls. Here ${\left[-\alpha \mathrm{,}\alpha \right]}^3=\cup C_i$ throughout the lattice for all cells or boxes $C_i$ in it with side lengths of $2ϵ$. In [17] solutions of the Navier Stokes equations are posed on large periodic domains $Q_{\alpha }\mathrm{<mo>\; :\; </mo>}={\left(-\alpha \mathrm{,}\alpha \right)}^3$ and on the whole space ${ℝ}^3$. One would expect, when the initial velocity is sufficiently localized that the solutions on a large enough domain should mimic those in ${ℝ}^3$. In this paper, we primarily focus on each cell $C_i$ of the Torus. For functions $P_i$ there exists a compact set about zero, i.e. $Q_1\mathrm{<mo>\; :\; </mo>}={\left[-L\mathrm{,}L\right]}^3\subset {\mathbb{T}}^3$ where the forcing functions have mean zero and on a suitable subspace the solutions u of the PNS equations also are mean zero functions. (at least in one direction $y_i$) The purpose of the m parameter in the P function is to vary it according to decreasing arbitrarily small amounts so that $Q_1={\mathbb{T}}^3$ for arbitrarily large $\alpha$. The form of the full Navier Stokes equations is expressed as a perturbation equation valid in an arbitrary $ϵ$ ball of ${\mathbb{T}}^3$. This is precisely $U+ϵV$, where U and V are differential operators making up the full 3D Navier Stokes equations. Two PDEs are being considered here. The first one is listed as in Equations. (11)-(13). For this one the solution in [7] is proven to be

locally Hölder continuous with $\alpha =\frac{1}{3}$ and the second one here is V,

$V=\frac{1}{2P^5\left({\left(W\left(F\left(y_1,y_2,y_3,s\right)\right)+1\right)}^{-1},3m^2,m^3\right){\left(F\left(y_1,y_2,y_3,s\right)\right)}^2{\left(W\left(F\right)+1\right)}^6}\left(V_1+V_2\right)=0$

$\begin{array}{c}{V}_1=4\rho [\left(\left(\frac{1}{2}{v}_3{\delta }^3-\frac{1}{2}{\delta }^2\right)\frac{\partial v_3}{\partial y_3}-v_3\left(\frac{\partial v_3}{\partial y_1}+\frac{\partial v_3}{\partial y_2}\right)\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\\ +\left(\frac{\partial v_3}{\partial s}\right)(v_3\frac{{\partial }^2{v}_3}{\partial y_3\partial y_1}+v_3\frac{{\partial }^2{v}_3}{\partial y_3\partial y_2}+\left(-\frac{1}{2}{v}_3{\delta }^3+\frac{1}{2}{\delta }^2\right)\frac{{\partial }^2{v}_3}{\partial y_3^2}\\ +\left(\frac{\partial v_3}{\partial y_3}\right)\left(-\frac{1}{2}\left(\frac{\partial v_3}{\partial y_3}\right){\delta }^3+\frac{\partial v_3}{\partial y_1}+\frac{\partial v_3}{\partial y_2}\right))]\\ \times \delta \rho F^2{\left(W\left(F\right)+1\right)}^6{P}^2\left({\left(W\left(F\right)+1\right)}^{-1},3m^2,m^3\right)\end{array}$

$\begin{array}{c}{V}_2=-6W\left(F\right)P^{\prime }\left({\left(W\left(F\right)+1\right)}^{-1},3m^2,m^3\right)F{v}_3\left(\frac{\partial F}{\partial y_3}\right){\delta }^2{\left(W\left(F\right)+1\right)}^2\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\\ +[-F{v}_3{\delta }^2{\left(W\left(F\right)+1\right)}^2\frac{{\partial }^{2}F}{\partial y_3^2}+(\frac{\partial F}{\partial y_3}(W\left(F\right){\delta }^2{v}_3\left(W\left(F\right)+4\right)\frac{\partial F}{\partial y_3}\\ -2F{\left(W\left(F\right)+1\right)}^2(\left(-1/2{\delta }^3{v}_3+{\delta }^2\right)\frac{\partial v_3}{\partial y_3}+v_3\left(\frac{\partial v_3}{\partial y_1}+\frac{\partial v_3}{\partial y_2}\right))))\frac{\partial v_3}{\partial s}]\\ \times P\left({\left(W\left(F\right)+1\right)}^{-1},3m^2,m^3\right)\rho \left(W\left(F\right)+1\right)\delta \end{array}$

For this second part $V\ne 0$. There are two situations that may occur here. First the asymptotic limit as $\delta \to 0$ gives $V=0$. Secondly, in Section 9 due to the Laplacian, an expression named $M=\frac{10}{{ϵ}^2}-1$ which is infinite at $ϵ=0$ allows us to divide by M so that it can be shown that V is not infinite. The solutions of $V=0$ as the first order perturbation, $v_i,i=1,2,3$ due to isotropic blowup [18] are in the form of a Hölder continuous function with parameter $\alpha =1/3$. Equations (11)-(13) are solved first and then $v_3$ is substituted into the later equation to obtain the condition of arbitrarily large data on a suitable subspace to be described. The idea here is to use the LambertW (or Ψ function) dependent solution obtained in the first step and substitute it into the second step equation. The arbitrary large data is in the form $\zeta \left(s\right)$ where $s\in R^{+}$. The $y_3$ component in the PNS system is set as $y_3=\frac{1}{\zeta \left(s\right)}+C$ where C is a constant. When this solution is substituted in, algebraically we can solve for $\frac{\text{d}\zeta \left(s\right)}{\text{d}s}=S\left(\zeta \left(s\right)\right)$ where S is a function associated with the $V=0$ equation. If $S\left(\zeta \left(s\right)\right)>0$ then the function

$\zeta \left(s\right)$ is increasing in s as s gets arbitrary large. It turns out that $\zeta$ is in general linear in s. Here $\zeta$ progressively approaches $\pm \infty$.

Calculating the tensor product term in Equation (9) see [14] and using Equation (21) in [14] shows its volume integral to approach zero due to ${\Vert \nabla u_z\Vert }_2^2$ approaching zero where the Hölder, Preköpa-Leindler and Gagliardo-Nirenberg Inequalities have been used. Also $\nabla \cdot \left(\frac{1}{\delta \rho }{u}_z^2{\nabla }_{xy}P+\stackrel{\to }{b}\frac{1}{\rho }{u}_z\frac{\partial P}{\partial z}\right)={\left|\stackrel{\to }{r}\right|}^4\Phi \left(t\right)$ where $\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}+z\stackrel{\to }{k}$. In Equation (6) of [14] , for the vector $\stackrel{\to }{B}=u_z\nabla \cdot \left(u_z\stackrel{\to }{b}\right)\stackrel{\to }{b}$, $L\stackrel{\to }{u}=\stackrel{\to }{B}$. Furthermore in Equation (10) [14] there the term ${\Omega }_4$ is the divergence of the vector $\stackrel{\to }{B}$. Using Ostrogradsky’s formula in terms of the vorticity $\stackrel{\to }{\omega }$ and velocity $\stackrel{\to }{b}$, ${\displaystyle {\int }_{\Omega }{u}_z\stackrel{\to }{b}\nabla \cdot \left({\stackrel{\to }{u}}_z\stackrel{\to }{\omega }\left|\stackrel{\to }{r}\right|\right)\text{d}\stackrel{\to }{x}}=-{\displaystyle {\int }_{\Omega }\left|\stackrel{\to }{r}\right|u_z\stackrel{\to }{\omega }\cdot \nabla \left(\stackrel{\to }{b}{u}_z\right)\text{d}\stackrel{\to }{x}}$. Now for a specific pressure P on an R-sphere, ${\displaystyle {\int }_{\Omega }{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}}\text{d}\stackrel{\to }{x}={\left|\stackrel{\to }{r}\right|}^2\Phi \left(t\right)$, where $\Phi \left(t\right)$ assumed to

be bounded and contain the pressure terms in Equation (9). The sphere is $\left|\stackrel{\to }{r}\right|=R$. Since the 3-Torus is compact there are m closed sets covering it. The outer measure is used where the infimum is taken over all finite subcollections $ℳ$ of closed balls ${\left\{E_j\right\}}_{j=1}^n$ covering a specific subspace of ${\mathbb{T}}^3$. The specific subspace $S_{{\mathbb{T}}^3}$ is within $ϵ$ measure of the 3-Torus and is obtained by minimally smoothening the vertices of ${\left[-L\mathrm{,}L\right]}^3$ and slightly puffing out its facets. Also inner measure is used where the supremum is taken over all finite subcollections $\mathcal{N}$ of closed balls ${\left\{F_j\right\}}_{j=1}^p$ inside $S_{{\mathbb{T}}^3}$. Generally by Hölder’s inequality, ${\mathcal{G}}_{\delta 3}$ is shown to be positive for pressure defined to be in $L^2$. (The pressure will turn out to be unbounded from below as t approaches infinity). Furthermore, u will be shown to have derivatives that blow up in finite time on a specific subspace of the solution space. The following string of inequalities occurs bounding the PNS operator equation on the interval $\left[t_0\mathrm{,}\infty \right]$ where $t_0$ is proposed to be the first blowup of PNS equations:

$\underset{\mathcal{N}}{\sup }\left|{\displaystyle {\int }_{t_0}^t}\left[{\displaystyle {\int }_{F={\displaystyle \underset{j=1}{\overset{p}{\cup }}}{F}_j}}\stackrel{\to }{r}\times {\nabla }_D^{-1}\left[\mathcal{G}-{\mathcal{G}}_3\right]\text{d}\stackrel{\to }{x}\right]\text{d}t\right|\le \underset{ℳ}{\inf }\left|{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{\Omega }_R={\displaystyle \underset{j=1}{\overset{m}{\cup }}}{E}_j}}\stackrel{\to }{r}\times {\nabla }_D^{-1}\left[\mathcal{G}-{\mathcal{G}}_3\right]\text{d}\stackrel{\to }{x}\text{d}t\right|$

$=\left|{\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\displaystyle {\int }_{\Omega }}\left|\stackrel{\to }{r}\right|\left|{\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}\right]\right|\sin \left(\theta \right)\stackrel{\to }{n}\text{d}\stackrel{\to }{x}\text{d}t\right|\le \left|{\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\displaystyle {\int }_{\Omega }}\left|\stackrel{\to }{r}\right|\left|{\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 3}+{\mathcal{G}}_{\delta 4}\right]\right|\text{d}\stackrel{\to }{x}\text{d}t\right|$

$=\left|{\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}-{\displaystyle {\int }_{\Omega }}{\left|\stackrel{\to }{r}\right|}^2{\stackrel{\to }{\omega }}_1\cdot \nabla \left(u_z\stackrel{\to }{b}\right)\text{d}\stackrel{\to }{x}\text{d}t\right|$

$=\left|{\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\displaystyle {\int }_{\Omega }}{\left|\stackrel{\to }{r}\right|}^2{\stackrel{\to }{\omega }}_1\cdot \nabla \left(u_z\stackrel{\to }{b}\right)\text{d}\stackrel{\to }{x}\text{d}t\right|\le {\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\displaystyle {\int }_{\Omega }}\left|{\left|\stackrel{\to }{r}\right|}^2{\stackrel{\to }{\omega }}_1\cdot \nabla \left(u_z\stackrel{\to }{b}\right)\right|\text{d}\stackrel{\to }{x}\text{d}t$

$\le {\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\left({\displaystyle {\int }_{\Omega }}{\left|\stackrel{\to }{r}\right|}^4\text{d}\stackrel{\to }{x}\right)}^{\frac{1}{4}}\left[{\left[{\left({\displaystyle {\int }_{\Omega }}{\left|{\stackrel{\to }{\omega }}_1\cdot \left(\nabla u_z\otimes \stackrel{\to }{b}\right)\right|}^2\text{d}\stackrel{\to }{x}\right)}^{1/2}\right]}^{\frac{3}{2}}+{\left[{\left({\displaystyle {\int }_{\Omega }}{\left|u_z\nabla \left(\stackrel{\to }{b}\right)\cdot {\stackrel{\to }{\omega }}_1\right|}^2\text{d}\stackrel{\to }{x}\right)}^{1/2}\right]}^{\frac{3}{2}}\right]\text{d}t$

$\le {\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\left({\displaystyle {\int }_{\Omega }}{\left|\stackrel{\to }{r}\right|}^4\text{d}\stackrel{\to }{x}\right)}^{\frac{1}{4}}\left[{\left[{\displaystyle {\int }_{\Omega }}{\left|{\stackrel{\to }{\omega }}_1\right|}^2{\left|\stackrel{\to }{u}\right|}^2{\left|\nabla u_z\right|}^2\text{d}\stackrel{\to }{x}\right]}^{\frac{3}{4}}+{\left[{\displaystyle {\int }_{\Omega }}{\left|u_z\right|}^2{\left|\nabla \stackrel{\to }{b}\right|}^2{\left|{\stackrel{\to }{\omega }}_1\right|}^2\text{d}\stackrel{\to }{x}\right]}^{\frac{3}{4}}\right]\text{d}t$

$\le {\displaystyle {\int }_{\left[t_0\mathrm{,}t\right]}}{\left({\displaystyle {\int }_{\Omega }}{\left|\stackrel{\to }{r}\right|}^4\text{d}\stackrel{\to }{x}\right)}^{\frac{1}{4}}\left[{\left[{\left({\displaystyle {\int }_{\Omega }}{\left|{\stackrel{\to }{\omega }}_1\cdot \left(\nabla u_z\otimes \stackrel{\to }{b}\right)\right|}^2\text{d}\stackrel{\to }{x}\right)}^{1/2}\right]}^{\frac{3}{2}}+{\left[{\left({\displaystyle {\int }_{\Omega }}{\left|u_z\nabla \left(\stackrel{\to }{b}\right)\cdot {\stackrel{\to }{\omega }}_1\right|}^2\text{d}\stackrel{\to }{x}\right)}^{1/2}\right]}^{\frac{3}{2}}\right]\text{d}t$