Decay of Strong Solutions for 4 D Navier-Stokes Equations Posed on Lipschitz Domains

Initial-boundary value problems for 4D Navier-Stokes equations posed on bounded and unbounded 4D parallelepipeds were considered. The existence and uniqueness of regular global solutions on bounded parallelepipeds and their exponential decay as well as the existence, uniqueness, and exponential decay of strong solutions on an unbounded parallelepiped have been established provided that initial data and domains satisfy some special conditions.


Introduction
This work concerns the existence and uniqueness of global strong solutions and sharp decay estimates of solutions to initial-boundary value problems for the 4D Navier-Stokes equations: ∇ ⋅  = 0 in Ω, (, 0) =  0 () , where Ω is either a bounded or an unbounded parallelepiped in R 4 with the homogeneous Dirichlet condition on the boundary of Ω.
The question of decay of the energy for weak solutions had been stated by J. Leray in [1] and attracts till now attention of many pure and applied mathematicians [2][3][4][5][6][7][8][9].In all of these papers, the decay rate of ‖‖()  2 (Ω) was controlled by the first eigenvalue of the operator  = −Δ, where  is the projection operator on the solenoidal subspace of  2 (Ω).Obviously, this approach does not work in unbounded domains; see [6,7,9].
It is well known that solutions of the 2D Navier-Stokes equations posed on smooth bounded domains with the Dirichlet boundary conditions are globally regular [4,[6][7][8][9].On the other hand, the question of regularity for 3D and 4D NSE with arbitrary initial data is till now an open problem even for smooth domains; see [6,7,9].Small initial data help to solve this problem [6,7,9] as well as the so-called "thin" domains when some size of a domain is small [10,11].The question of regularity becomes more difficult while a domain is Lipschitzian [10,12,13].
Our goal here is making some geometrical restrictions, to prove the existence and uniqueness of strong global solutions in 4D Lipschitz domains for arbitrary regular initial data as well as exponential decay of solutions.
In this work, making use of ideas of [15], we have established that  ∈  2,4/3 (Ω) for a 4D bounded parallelepiped.The following inequality holds: where  > 0. Introduction.Section 2 contains notations and auxiliary facts.In Section 3, existence, uniqueness, and decay of global strong solutions on a bounded 4D parallelepiped have been established.In Section 4, the existence, uniqueness, and decay of regular solutions on bounded 4D parallelepipeds and strong solutions on 4D unbounded parallelepipeds have been demonstrated.

Notations and Auxiliary Facts
Let  = ( 1 ,  2 ,  3 ,  4 ) and Ω be a domain in R 4 .Define as in [9], p.2-4 We denote for scalar functions () the Banach space   (Ω), 1 <  < +∞ with the norm For  = 2,  2 (Ω) is a Hilbert space with the scalar product The Sobolev space  , (Ω) is a Banach space with the norm When  = 2,  ,2 (Ω) =   (Ω) is a Hilbert space with the following scalar product and the norm: Let D(Ω) or D(Ω) be the space of  ∞ functions with compact support in Ω or Ω.The closure of  ∞ functions in  , (Ω) is denoted by  , 0 (Ω) and (  0 (Ω) when  = 2).Define the auxiliary spaces which are projections for the solenoidal vector functions, The space  is equipped with the natural  2 inner product.
The space  will be equipped with the scalar product when Ω is bounded.If Ω is unbounded, we define the inner product as the sum of the inner products as follows: We use the usual notations of Sobolev spaces  , ,   , and   for vector functions and the following notations for the norms: (i) For vector functions () = ( 1 (),  2 (),  3 (),  4 ()), The closures of V in  2 (Ω) and in  1 0 (Ω) are the basic spaces in our study.We denote them by  and , respectively.Remark 1.By definition,  is a proper subspace of  1 0 (Ω).
The next lemmas will be used in estimates.

Uniqueness of the Strong
that can be rewritten as Acting in the same manner as by the proof of Estimate II, we come to the inequality By conditions of Theorem 5, for  = 0. Taking into account Estimates (28) and using standard arguments, we get for all  > 0 Hence, (53) becomes This implies  ≡ 0 that proves uniqueness of the strong solution and completes the proof of Theorem 5.

More Regularity
Consider the Poisson problem in a bounded domain Ω ∈ R  : In [15] Theorem 11, p. 120-123, the following has been proved.

Theorem 7. The problem (57) posed in a parallelepiped
Returning to the original problem for the Navier-Stokes equations, where () is a vector function from R 4 into R 4 and  is a real function from R 4 into R, and making use of Galerkin approximations, we establish the following result.Theorem 8. Given  0 ∈  2 (Ω)∩ and a domain Ω satisfying (25), then problem (61) has a unique regular solution (, ) such that which for all Φ() ∈  satisfies the following integral identity: Moreover, where Proof (decay of  2,4/3 (Ω)-norm).Taking into account that conditions of Theorem 8 and of Theorem 5 are the same, by Theorem 5, we have a unique strong solution of (61).Hence, to prove Theorem 8, it is sufficient to establish that ‖‖  2,4 /3 (Ω) ().

Advances in Mathematical Physics
Returning to (65) and having  ∈  4/3 (Ω), we obtain, due to Theorem 7, By the Sobolev theorems, The proof of Theorem 8 is complete.
In some sense, this is the superior regularity for the problem (61).It looks like  = 4 is the critical case of the Navier-Stokes system.
Conclusions.In our work, we tried to respond to some questions posed by J. Leray [1], namely, regularity of global solutions of the Navier-Stokes equations and their decay.Therefore, our results can be divided into two parts: the first one concerns decay of global regular solutions of the 4D Navier-Stokes equations posed on bounded 4D parallelepipeds.It is known that there exist global regular solutions for the 2D Navier-Stokes equations posed on smooth bounded domains [4,6,8,9], but regularity in nonsmooth (Lipschitz) domains is not obvious.For bounded 4D parallelepipeds, we have established the existence of a unique global regular solution which decays exponentially as  → +∞ provided that initial data satisfies (25).We demonstrated that the decay rate is different for different norms; see (77), where  is defined by the geometrical characteristics of a domain Ω.
The second part of our work concerns decay of solutions for the 4D Navier-Stokes equations posed on an unbounded parallelepiped.In existing publications [3,4,6,9], the decay rate of ‖‖  2 (Ω) () is controlled by the first eigenvalue of the operator  = −Δ, where  is the projection operator on a solenoidal subspace of  2 (Ω).It is clear that this approach does not work in unbounded domains.
On the other hand, our approach based on the Steklov inequalities allowed us to estimate the decay rate of a strong solution for the 4D Navier-Stokes equations posed on an unbounded 4D parallelepiped.
We must emphasize that this estimate is the first one which gives an explicit value of the decay rate for unbounded 4D domains.Results established in our work can be used in constructing of numerical schemes for solving initialboundary value problems for the Navier-Stokes equations appearing in Mechanics of viscous liquid.From the physical point of view, decay estimates show that the decay rate of perturbations of solutions caused by the initial data is bigger for bigger values of viscosity ] and smaller sizes of 4D parallelepipeds.
My interest for the 4D Navier-Stokes equations is purely mathematical and, on my opinion, can not be extended to higher dimensions beyond 4. I must also note that there are publications on the existence of weak solutions for 4D Navier-Stokes equations [7], [9] p. 189-197.