Classical Fundamental Unique Solution for the Incompressible Navier-Stokes Equation in RN

We present a class of non-convective classical solutions for the multidimensional incompressible Navier-Stokes equation. We validate such class as a representative for solutions of the equation in bounded and unbounded domains by investigating the compatibility condition on the boundary, the smoothness of the solution inside the domain and the boundedness of the energy. Eventually, we show that this solution is indeed the unique classical solution for the problem given some appropriate and convenient assumptions on the data.


Introduction
In this article, a well known model is to be investigated that represents the flow of an incompressible fluid in both bounded and unbounded domains of Newton's second law of motion, the balance of momentum and the mass conservation, which eventually leads to the definition of the Cauchy stress tensor. In Newtonian fluids, this stress tensor is a function in the pressure, the viscosity and the gradient of the velocity. For a convenient physical background about the basics of continuum mechanics and how we derive the Navier-Stokes equation we propose [1] [2]. Also a very interesting work from a physical point of view can be found in [3] [4]. In particular, the work of Kambe in [4] was the source of inspiration for the ideas in this article.
This model poses a serious challenge when it comes to proving the existence and the smoothness of its solution. This problem was perfectly addressed by Ladyzhenskaya in two dimensional spaces among many other issues in higher dimensional spaces [5]. However, a decisive answer in the three-dimensional space or higher remains unavailable. It is almost impossible to enlist all the results obtained for this equation. Therefore, we suggest for the interested reader to review the monographs [5] [6] [7] and the references within for much more details.
Recently, the interest in this equation is not fading at all. There are persistent efforts to clarify the properties of the solution, especially its smoothness. Among many respectful results, we mention the outstanding analysis by Tao in [8], the work of Constantin in [9] [10] [11]. A very interesting result for partial regularity of suitable weak solutions was obtained by Caffarelli in [12].
In this article the idea is simple. A class of possible solutions is proposed and then it is proved that it indeed represents the unique classical solution of the problem. Most of the results are obtained by considering standard theories of partial differential equations. Some of the results in the monograph [7] are also used repeatedly. In the next section a statement of the problem is introduced along with some definitions, notations and employed functional spaces. Afterwards, the proofs of the main results are established.

Statement of the Problem
The spatial domain is Ω which is either a bounded region in N  or the whole of N  and this point shall be specified explicitly. For the sake of con- x .
Clearly, such notation should not be taken to imply a moving boundary. The where the last equation in the above model is what many authors commonly refer to as the incompressibility condition or the solenoidal condition. The first term in the first equation is the acceleration of the fluid's flow in time, the second is the convective term that represents the acceleration of the flow in space, the third represents the diffusion scaled by the kinematic viscosity constant µ , the fourth is the pressure, and the last one represents the total of the external body forces. The initial profile is denoted by 0 v and the boundary datum is denoted by * v . The solution v is the vector field representing the velocity of the flow in each direction, and its rotation ω is the vorticity. Note that in t Ω by compatibility. The well known Lebesgue spaces ( ) q L Ω will be used repeatedly to represent the functions with bounded mean of order q. The Sobolev space H Ω is used to represent functions with bounded derivatives such that for a vector field ∈ Ω for every 1, , m α =  and 1, , This motivates the usage of the space ( ) m V Ω , which is a well known space of functions in the theory of incompressible fluids as a representative for divergence free (solenoidal) bounded vector fields such that By laws of classical mechanics, the energy generated by a moving object is proportional to the square of its velocity. Hence, the energy ( ) E t generated by the flow v is defined as follows (2) Recall that the above integral represents the norm of v in the Lebesgue space The smoothness of the boundary datum ( ) ) and ,~for any 1.
Finally, the forcing term f is smooth in space and time such that , 0, ; and ,~for any 0.
Note that the intersections in the above conditions are not really required in the case of bounded domain since boundedness of the domain and continuity of the functions are enough to imply boundedness in the sense of the mean. However, these requirements are of significant importance in the case of unbounded domain as will be shown later.
The target is to define a class of possible solutions to Model Equation (1) from which v , ω and p can be concluded. Once v is obtained, then p can easily be recovered from the main model. The validity of this solution as a meaningful physical solution will be investigated when inserted in the main model.
A meaningful solution is a unique and smooth solution that vanishes as t → ∞ , and in the case of unbounded domain it vanishes as → ∞ x as well. Remark. The curl operator or the rotation of a vector field has a physical meaning only in three dimensional space. However, it will be used in N  for the sake of generality. Most of the results depend on the curl operator in the sense of a differential operator without direct exposure to its definition. Note that the main interest is to find the velocity, which means that any use of the rotation, in spite of its significance in this work, is nothing more than a transient step. It can always be assumed that the space is three dimensional when needed, and a generalization becomes possible by reverting back to the results of the velocity. In particular, some of the vorticity ideas introduced in [7] are adopted in this study.

Main Results
In this section a class of possible solutions is proposed and the insertion of these solutions in the main model is investigated to check where they will lead to. This is initiated by the statement of the following claim.
. An important question in the theory of Navier-Stokes equation is the ability to verify the compatibility condition on the boundary with minimum restrictions on the flux passing through the boundary especially if ∂Ω is divided into several parts ( [6], p. [4][5][6][7][8]. This condition is a natural consequence of the incompressibility of the flow. Hence, it takes the form where n  is the outward unit vector normal to ∂Ω . This motivates the introduction of the following lemma. Lemma 1 (Tangential flow). Let Ω be an arbitrary domain,  be any vector field independent of x . The Compatibility Condition (6) is satisfied for every divergence free vector field ( ) x u . In particular, on every part of ∂Ω , v and its rotation ω are tangents to ∂Ω such that 0 Identity (7) can be used to deduce that ( ) ( ) and when integrated over Ω for any which is a true identity for every arbitrary Ω , ψ and u and for every On the other hand, given that ( ) 0 which is true for arbitrary Ω , ψ and u . Now, since Identity (10) implies that 2 ∇ v is orthogonal to the space spanned by v and ω . Combine Identities (9), (11) and (12), and exclude the cases by the arbitrariness of the choice to deduce that it is necessary that ; that is to say v is tangential to every part of ∂Ω . It can also be deduced that either = v ω on every part of ∂Ω (this actually means  (3), (4) and (5) respectively. If

( )
,t v x is in the form proposed in Claim 1, then Model Equation (1) has a clas- In particular, the exact solution is given by solving the following system where p ∇ can be defined uniquely in terms of the values of v and f on the boundary. Moreover, if Proof. The proof is quite simple and it depends mostly on classical results and the standard theory of linear parabolic and elliptic equations of second order. If  (14) Apply the divergence operator to get where the incompressibility condition which is the first equation in System (13). Finally, given the incompressibility of v and applying a simple vector identity lead to the third equation in System (13) that is The fundamental solution ω to Equation (16) with initial profile 0 ω and force ∇ × f is given formally by that v is unique, which is our main concern, to conclude the uniqueness of ω. Now, go back to Equation (17). By virtue of the results obtained above for ω and the standard theory of elliptic equations one directly concludes that v ∆v and f on the boundary. This provides a form of boundary conditions for p which consequently guarantees its uniqueness up to a constant, for details see [14]. The infinite differentiability of p also follows from the main model and the fact that x for any 0 δ > . One can argue that some of the results in the literature require a rate of decay higher than that for the surface integrals to vanish; these restrictions can be dropped because these integrals already vanish by virtue of Lemma 1, (see [7], Lemma 1.5]). However, this does not mean that rapid rates of decay are not achievable, they are achievable as demonstrated next.
In order to derive such an estimate one goes back to Formula (19) that represents the fundamental solution for v . If C ≤ ω then it is expected that the outcome of this integral will provide nothing less than a linear rate of growth for v , which is a bad answer to the problem in hand. Therefore, Formula (18) shall be used to help us estimate some rates of decay for ω and consequently for v so that boundedness in 2 ( ) L Ω can be proved for every 0. for Model Equation (1) represented by the System (13) and defined as It follows that all the conditions of ( [7], Theorem 3.4, Theorem 3.6) are satisfied such that v exists globally in time and such that ( ) which implies the boundedness of the energy ( ) Take the absolute value of both sides, perform some manipulation to the integrands, use ⋅ ≤ x y x y and maximize the time integral (the integrand is a decreasing function in time) so that one finally gets the term with the highest power for x as follows where we used the Divergence theorem in the right hand side, the facts that 0