# Bi-Theory Method & Navier Stokes Problem

farzad didehvar

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farzad didehvar. Bi-Theory Method & Navier Stokes Problem. 2024. hal-04652398

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# Bi-Theory Method & Navier Stokes Problem

Farzad Didehvar

didehvar@aut.ac.ir

Amir Kabir University of Technology (Tehran Polytechnic)

23June 2024

**Abstract.** Here, we try to show that the method used in [2] is a general method and sheds light on different types of problems. The author chooses the Navier Stokes problem, to check the possibility of solving this problem by this method.

This text is a guideline of possible proof of this problem.

**Keywords.** Bi-theory method, Navier Stokes problem, Fuzzy time, P vs NP problem, Tao bilinear operator

In [2] we use fuzzifying time as a method to conclude
$TC + CON(TC^*) \notin NP$. Here, firstly we introduce
the generalization of this method, we call it bi-theory
method. The special case of this method is applying fuzzy
time to have the second type of models as we see in [2].
We show the possibility of solving Navier Stokes problem
by this method. We wish to present bi-theory method as an approach to different types of problems.

We are able to demonstrate the method which is applied in [2] by following simple lemma.

**LEMMA.** For any two theories $T, T^*$ and $P \in T$, suppose that there exists $\varphi, P^*$,

$$ \varphi: T \to T^* ; \varphi(P) = P^*; $$

$$ T^* \vdash P^* \quad \rightarrow \quad T + CON(T^*) \vdash P . $$

Now if $T^* \vdash P^*$ we have $T + CON(T^*) \vdash P$.

The above simple lemma shows how the method works. (Bi theory method).

In our examples, the theories more specifically are $T = TH(M), T^* = TH(M^*)$. In this case, the bi-theory method is called here bi-model method.

In [2], we considered $TC, TC^*$ as our two chosen theories, we associate to any statement and formula $\varphi$, its fuzzy form. In brief, in any term $t$ we have in $TC$, we replaced it by $t^*$.

In special case, if $T = TC, T = TC^*$ we have fuzzifying time method.

**Navier Stokes problem**
We studied the impact of fuzzy time on P vs NP problem [2]. Seemingly, Bi theory method is more convenient for Navier Stokes rather P vs NP problem. Here, we present the reasons of this claim. To do this, we study Terrance Tao paper [1] and the Major results of it. Before this we quote a version of Navier Stokes from [1]. page 3

Conjecture 1.2 (Navier Stokes global regularity, again, quoted in [1]). Let

$u_0: R^3 \to R^3$ be a divergence-free vector field in the Schwartz class. Then there exists a

mild solution $u: [0, \infty) \to H_{df}^{10}(R^3)$ to the Navier-Stokes equations with initial data $u_0$.

Here, we call the above Navier Stokes conjecture, as NS.

The numbers used in the above lemma and following theorem are the numbers in the article [1].

Theorem 1. (Finite time blowup for an averaged Navier Stokes equation, quoted in [1]). There exists a symmetric averaged Euler bilinear operator $\tilde{B}: H_{df}^{10}(R^3) \times H_{df}^{10}(R^3) \to H_{df}^{10}(R^3)^*$
Obeying the cancellation property (1.16) for all $u \in H_{df}^{10}(R^3)$, and a Schwartz divergence free vector field $u_0$, such that here is no global-in-time mild solution

$u: [0, +\infty) \to H_{df}^{10}(R^3)$ to the averaged Navier Stokes equation (1.9) with initial data $u_0$.

**Definition.** Tao bilinear operator is a bilinear operator which satisfies the conditions in above theorem.

We need a lemma to show the above assumption is equivalent to the normal form of NS when, we have a specific fuzzy time. We call it $NS^{*1}$, in contrast to $NS^*$ for fuzzy time in general.

**Conjecture\*.** There is a Tao bilinear operator $u$, respect to it, there is a fuzzy time Function $v_u$, such that solving Navier Stokes equation by it, is equivalent to have a solution for the associated averaged Navier Stokes equation as we see in theorem 1 in above, for $u$.

The existence of $v_u$ in above is called $NS^*$ . By Conjecture\* and the above theorem fails to be true.

**Theorem2.** By Conjecture\* & Theorem 1, NS fails to be true.
## Discussions around and about the Proof of the above theorem.

To apply a similar strategy which we used in P vs NP problem we should remind that, one of the key points in that proof is $P = NP$ is equivalent to $P^* = NP^*$.

The above theorem besides the existence of random generator in fuzzy time provides a situation allows us to apply the method. To apply the method for Navier Stokes theorem we have similar ideas. In the sequel, we give the hint of proof

1. The first is about Navier Stokes theorem in real time and fuzzy time.

Here, we have two versions of it.

In real time, we represent the version of this theorem by $NS$ and in Fuzzy time $NS^*$ (and $NS^{*1}$).

We need to prove

$$ (NS \to NS^{*1}) $$

Similar to $P \neq NP \to P^* \neq NP^*$.
(Or for the specific fuzzy time related to the
Conjecture* $NS \to NS^*$).

Actually, we prove $\sim NS^{*1} \to \sim NS$.

Similar to the proof of $TC + CON(TC^*) \not\vdash NP \neq NP$ [2] we apply fuzzy time, in the new proof, from $\sim NS^{*1}$ we have different possible worlds. One of them is Classical world with classical time. So $\sim NS$ comes true by similar technic in proof of $TC + CON(TC^*) \not\vdash NP \neq NP$ [2]. If the above claims hold, the bi-theory method shows the NS fails to be true.

Now, by the Conjecture*, and major theorem (theorem 1) we have $\sim NS^{*1}$. This and the above result implies $\sim NS$. □

As the major result of this article, by proving the
Conjecture*, we prove Navier Stokes conjecture.

**Comment.** The usage of Fuzzy time in solving some paradoxes first was introduced in 2018 in Shiraz Conference of Mathematics [5] and in a series of papers the author tries to show the impact of these hypothesis
on Theory of computation and Physics. In this topic, we
have two aspects. First, fuzzy time as a physical reality
and the second as a mathematical method. Here, we
consider the second approach. We have results about
some problems in Complexity Theory like P vs NP. It is
shown by Shoenfield, problems like P vs NP is not
solvable by Forcing method (Shonfield absoluteness,
thank the individual who reminds me that.)

. Also, it is well-known that the methods like
Parallelization, Algebrization and natural proofs are not
able to solve this problem. This makes the methods used
in [2] more interesting.

In the first chapter, we try to generalize the method; in
the second chapter we use the fuzzy time method as an
approach to attack the other type of problem, the Navier
Stokes conjecture. It is a part of an attempt, a brief of
story is written in the last comment of [2]. In [2], also in
[3],[4],[5] is shown how the author reaches the idea of
fuzzy time from 2018-2019 or as some friends proposed
recently "Temporal Continuum" []!

By the way, the author faces a morality problem:
if in some texts they don't reference you in a subject,
must you reference them in the same subject? Does the
answer depend on the case?

Reference

1. Terence Tao, FINITE TIME BLOWUP FOR AN AVERAGED THREE-DIMENSIONAL NAVIER-STOKES EQUATION, arXiv:1402.0290v3 [math.AP] 1 Apr 2015

2. Farzad Didehvar, Theory of Fuzzy Time Computation (TC* vs TC, TQC), hal-0433081

3. Farzad Didehvar, Is classical Mathematics appropriate for Theory of Computation?, 2017, Vixra, Philpapers 2019

4. Farzad Didehvar, Zeno Paradox, Unexpected Hanging Paradox (Modeling of Reality & Physical Reality, a Historical-Philosophical View), 2022, SSRN

5. Farzad Didehvar, Fuzzy Time, from Paradox to Paradox, December 2019, Philpapers, The article is presented in 50th Annual Iranian Conference of Mathematics.
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