_{1}

^{*}

Navier-Stokes equation has for a long time been considered as one of the greatest unsolved problems in three and more dimensions. This paper proposes a solution to the aforementioned equation on R
^{3}. It introduces results from the previous literature and it proves the existence and uniqueness of smooth solution. Firstly, the concept of turbulent solution is defined. It is proved that turbulent solutions become strong solutions after some time in Navier-Stokes set of equations. However, in order to define the turbulent solution, the decay or blow-up time of solution must be examined. Differential inequality is defined and it is proved that solution of Navier-Stokes equation exists in a finite time although it exhibits blow-up solutions. The equation is introduced that establishes the distance between the strong solutions of Navier-Stokes equation and heat equation. As it is demonstrated, as the time goes to infinity, the distance decreases to zero and the solution of heat equation is identical to the solution of N-S equation. As the solution of heat equation is defined in the heat-sphere, after its analysis, it is proved that as the time goes to infinity, solution converges to the stationary state. The solution has a finite τ time and it exists when τ → ∞ that implies that it exists and it is periodic. The aforementioned statement proves the existence and smoothness of solution of Navier-Stokes equation on R
^{3} and represents a major breakthrough in fluid dynamics and turbulence analysis.

In this paper, the following form of Navier-Stokes equations in R^{3} is studied:

With initial conditions

Here

applied force, v is a positive coefficient (the viscosity) and

Euler equations are considered, then the same set of equation must be applied with the condition that viscosity is equal to zero.

The following conditions must be satisfied as it is wanted to make sure that

And

The accepted solution of N-S is physically reasonable if it only satisfies:

And

At the same time, it is possible to look at spatially periodic solutions. We can assume the following conditions:

Under the condition that

The solution is then accepted if it satisfies:

And

The problem is to find and analyze whether a strong, physically reasonable solution exists for the Navier- Stokes equation.

The statement that will be proved is existence and smoothness of Navier-Stokes solutions on R^{3}. Take v > 0 and n = 3. Let ^{3} x[0, ∞] and the above conditions and equations are satisfied.

Firstly, the definition of turbulent solutions (Oliver and Titti) [^{r} norm^{2}. L^{r} stands for the usual L^{r}-space over R^{n},

When X is a Banach space,

Def 1. (Oliver and Titti) [

1)

The relation

2)

Holds for almost all T and all

Strong energy inequality

3)

Holds for almost all

It is necessary to introduce the Stokes operator

where

The existence of turbulent solutions for n = 3 and n = 4 is given by Leray and Kato. In order to derive the next results, theorem from Takahiro Okabe will be introduced.

Theorem 1. Let

For

For

Suppose that

holds for all

Def 2. Let

where

Takahiro Okabe [

For

From (16) to (18), the following is obtained:

Afted dividing the both sides of (19) by

By (17), the following is obtained

By introducing the new theorem that is proved in Takahiro Okabe’s paper [

Theorem 2. Let

1)

2)

If

As

The following theorem can be proved by using well-known Leray’s structure theorem, every turbulent solution of N-S becomes the strong solution after some time. Although Kato proves that the strong solution decays in the same way as the Stokes flow

By introducing the above mentioned theorem, the following result is obtained and it proves Theorem 1.

This result proves that energy of the molecules of fluid moving is smaller than some value determined by C, n, r, m and it proves asymptotic energy concentration. In order to prove that turbulent solutions are at the same time strong solutions, blow-up time of solutions must be analyzed.

It is demonstrated that Navier-Stokes equation enter some class as it was already proved

Theorem 3. Let

The theorem is proved by using Plancherel theorem, the triangle inequality, the inequality

Firstly, we assume the existence of solutions

where

Then the Gevrey norm is used to find the following result:

The contribution of pressure term is zero because A commutes with the Leray projection onto divergence free vector fields. Note that:

By using Theorem 3 and Cauchy-Schwarz inequality, the following result is obtained.

In order to proceed, we introduce the Theorem 4.

Theorem 4. For all nonnegative p, q and τ we have the following:

The proof is similar to that in Theorem 3, just it should be noted that for every

After introducing the theorem and interpolating

where

This proves that there exists a

Now the result of differential inequality for longer time will be derived. The radius of uniform analyticity

If first two terms of Equation (26) are considered and it is assumed that only contribution from linear terms is included, interpolation can be used as well as Young inequality while breaking the second term in several fractions. Theorem 3 provides the following:

we all together obtain:

New theorem is introduced, it is already proved by using Plancherel theorem:

Theorem 5. Provided that

Combining Theorem 5 with

If we set

So that the first two terms on the right of equation (33) are nonpositive and can be neglected. The main task is now to analyze the nonlinear terms and if possible prove that these nonlinear solutions do not affect the decay properties of the solution to infinite order. Applying the estimate on nonlinear term and by interpolating

The following differential inequality is obtained.

As we are considering global asymptotics and blow-up profiles, they are only possible in the presence of a critical controlled quantity or the combination of a subcritical and a supercritical controlled quantity. It turns out that the Navier-Stokes equation according to differential inequality tends to contract these quantities, in that way leading to a useful way to force finite time blow-up. The idea of using minimal surface area as controlled quantities originates from Hamilton. In order to discuss the blow-up time, we introduce the following well known proposition:

Assume that

Each such sphere has an energy

the infimum of

where

If the above mentioned state holds, then the differential inequality, in order to make nonlinear terms of lower order, has to satisfy the following form:

where

where

pend on

tion

The integrating factor for linear differential inequality is:

So the following is obtained.

If we fix

If the condition (39) is satisfied for all t, estimate (44) will be global in time. It is sufficient to show the following:

for some non-increasing function

which satisfies the above mentioned conditions and it proves the existence of a solution. As

It is obtained that:

The upper bound of decay is calculated and given below:

where

This proves that solution is existent even when

Now in order to proceed and analyze the blow-up time, v as the solution of the heat equation will be introduced. It should be proved that the solution

First an estimate on the difference w in

As the heat equation preserves the divergence condition, the following equation is obtained

And repeating the steps, the following result is obtained:

The second of nonlinear terms arises from (47) by using and choosing the smallest possible

The following differential inequality is obtained:

And the following is obtained:

After having proved that solution for

Now the heat solution equation Cannon [

Satisfies a mean-value property

Precisely if u solves

And

Then

where

Notice that

So that

The previous assumptions and results prove the existence of smooth and strong Navier-Stokes solution of equation in R^{3} and represent the solution of millennium problem in R^{3}.

It is proved that the strong solution of Navier-Stokes equation is smooth, existent and unique. Firstly, turbulent solutions are defined and it is proved that they are strong solution, but as the turbulent solutions are only possible for small time intervals, it is tried to extend the time interval by using the Equation (39) and it is proved that the differential inequality (40) holds at the same time for some ^{3} and represents a breakthrough in fluid dynamics analysis.

I would like to thank my family, my favourite aunt, my grandparents who had a tremendous influence on my love towards mathematics. I want to thank my aunt Cica, my aunt Sonja and her husband Voja, her daughters, my uncle Nemanja and his family and all other relatives who provided me immense support. Love you all.