Sobolev regular solutions for the incompressible Navier–Stokes equations in higher dimensions: asymptotics and representation formulae

In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain Ω ⊂ R with the dimension n ≥ 3 . We first establish asymptotic expan‐ sion formulae of Sobolev regular finite energy solutions in Ω . In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the reg‐ ular polyhedron Ω ∶= [0, T].


Introduction and main results
In this paper, we consider the steady incompressible Navier-Stokes equations: where x ∈ Ω , and Ω ⊂ ℝ n is a smooth bounded domain, U ∶ ℝ + × Ω → ℝ n is the fluid velocity, and it is of the form U(x) = (U 1 (x), U 2 (x), … , U n (x)) , P(x) ∶ ℝ + × Ω → ℝ stands for the pressure in the fluid, and the constant is the viscosity. We denote by f ∶= (f 1 , f 2 , … , f n ) an external force. The divergence free condition in second equations of (1.1) guarantees the incompressibility of the fluid.
We supplement the steady incompressible Navier-Stokes equations (1.1) with the Dirichlet boundary condition and the pressure takes the form In particular, when the external force f = 0 in equations (1.1), then our problem reduces to the steady incompressible Navier-Stokes equations The question of whether a solution of the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the Millennium Prize problems, see [8]. In 1934, Leray [22] showed that the 3D incompressible Navier-Stokes equations (1.1) admit global-forward-in-time weak solutions of the initial value problem. Caffarelli, Kohn and Nirenberg [6] established a -regularity criterion for equations (1.1). After that, Lin [23] gave a new and simpler proof for the result of Caffarelli, Kohn and Nirenberg. Koch and Tataru [21] proved the global well-posedness for the equations (1.1) in a space of arbitrary dimension with small initial data in BMO −1 space. Recently, Buckmaster and Vicol [5] proved that the Leray weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. We refer the readers to [2-4, 9, 19, 28, 34-34] for more related results on this equations.
Much attention attracted the existence and the regularity properties of stationary solutions for the incompressible Navier-Stokes equations. These solutions depend on the force f and the domain Ω . Gerhardt [15] proved that the steady four-dimensional problem admits a solution in W 2,p by assumption the force f ∈ L p . Frehse and Ruzicka [11] and Struwe [31] got the existence and regularity of the solutions in the fivedimensional case, respectively. Frehse and Ruzicka [10] also obtained the existence of regular solutions in a bounded domain of six-dimension. The fifteen dimensional torus case was given in [12]. Maz'ya and Rossmann [25] showed the existence of weak solutions in the three-dimensional case for a polyhedral domain. Kim [20] considered the existence of very weak solutions in a bounded domain of dimension d = 2, 3, 4 . Farwig and Sohr [7] proved the existence, uniqueness and very low regularity of solutions to the inhomogeneous Navier-Stokes equations with special external force in a bounded domain of d ≥ 3 dimension. Recently, Hou and Pei [18] got the existence of weak solutions in a bounded connected polygon or polyhedron of two or three dimension. Luo [24] obtained the non-uniqueness of weak solutions for this kind of problems in the case of the d-torus with d ≥ 4 . We cannot list all the contributions to this field, but we refer the readers to [12][13][14] and the references therein.
As pointed out in [24], the question of uniqueness of regular solutions to the steady incompressible Navier-Stokes equations (1.1) remains mostly an open problem. The nonuniqueness of the weak solution has been given in [24]. In this paper, under the assumption of the external force f being small and f ≢ 0 , we give asymptotic expansion formulae of Sobolev regular solution with finite energy for the steady incompressible Navier-Stokes equations (1.1) in a smooth bounded domain Ω of dimension n ≥ 3 . Next, we give explicit representation formulae of the Sobolev regular solution in a special domain, namely if Ω ∶= ([0, T]) n .
We now state the main result in this paper.
strongly on the initial approximation function U (0) (x) . Our proof is based on the Nash-Moser iteration scheme, by using some ideas developed in [37,38]. For the general Nash-Moser implicit function theorem, we refer to the seminal papers of Nash [27], Moser [26] and Hörmander [17], and to Rabinowitz [29] for a singular perturbation problem of elliptic equations by using the Nash-Moser implicit function theorem.
In particular, if we consider the domain Ω ∶= ([0, T]) n (a regular polyhedron) with the finite constant T > 0 , then we have the following explicit representation formulae. Moreover, the pressure is determined by Notations Throughout this paper, we assume that Ω ⊂ ℝ n with n ≥ 3 and we denote the usual norms of 2 (Ω) and ℍ s (Ω) by ‖ ⋅ ‖ 2 and ‖ ⋅ ‖ ℍ s , respectively. The norm of the Sobolev space H s (Ω) ∶= (ℍ s (Ω)) n is denoted by ‖ ⋅ ‖ H s . The symbol a ≲ b means that there exists a positive constant C such that a ≤ Cb . We denote by (x 1 , x 2 , x 3 , … , x n ) T the column vector in ℝ n . The letter C with subscripts to denote dependencies stands for a positive constant that might change its value at each occurrence.
The paper is organized as follows. In Sect. 2, we first give a class of initial approximation functions, then the Carleman-type estimate of solution for the linearized equations about the initial approximation functions is shown. Next, we prove the existence of the Sobolev regular solution for the linearized equations. In Sect. 3, we establish the general approximation step for the construction of the Nash-Moser iteration scheme. In the final section of this paper, we show how to construct a small Sobolev regular solution for the incompressible steady Navier-Stokes equations (1.1) by the proof of convergence for the Nash-Moser iteration scheme.

The first approximation step
We introduce a family of smooth operators possessing the following properties. Lemma 2.1 [1,17]

The initial approximation function
Let s ≥ 1 be a fixed finite constant and 0 < 0 < 2 ≪ 1 . For any x ∈ Ω , we choose the initial approximation functions Meanwhile, we require Moreover, for any fixed constant s ≥ 1 and x ∈ Ω and i, j = 1, 2, … , n , we also need the condition and the initial error term By direct computations, it follows that hence and Moreover, we observe that these functions decay to 0 as |x| → +∞ , and for every fixed integer p, the assumptions (2.4)-(2.6) are fulfilled.
Here, the initial approximation pressure satisfies

The Carleman estimate of linearized equations
We now construct the first approximation solution denoted by U (1) (x) of (2.3). The first approximation step between the initial approximation function and first approximation solution is denoted by Then we linearize the nonlinear system (2.3) around U (0) to get the linearized operators as follows where D U (0) denotes the Fréchet derivatives on U (0) . By (1.3), we obtain We now consider the linear system and the boundary condition The solution of this problem gives the first approximation step of the steady incompressible Navier-Stokes equations (1.1). Before we carry out some a priori estimates, for j = 1, 2, 3, … , n , we rewrite equations of (2.9) into a coupled system as follows with the Dirichlet boundary condition We now derive the Carleman-type estimate of the solution to the linear system (2.11). Proof Let (x 1 ) be a function with x 1 ∈ Ω such that and e − (x 1 ) is bounded for x 1 ∈ Ω . Here the constant belongs to (0, 1 4 ) . The condition (2.14) implies �� (x 1 ) > 1 4 > 0 . In fact, there are many functions satisfying these conditions. We give in what follows a simple example. Since x 1 ∈ Ω , there exists a suitable positive constant b > 1 such that the function is well defined. Here, we take the constant a ∶= ( 1 4 − ) − 1 2 . Direct computations give that thus the ODE inequality (2.14) holds for a suitable b > 1 . Meanwhile, there exists a positive constant C 0 such that

Lemma 2.2 Let
Multiplying both sides of equations in (2.11) by e − (x 1 ) h (1) j , respectively, then integrating over Ω , by noticing the Dirichlet boundary condition (2.10), for j = 1, 2, 3, … , n , it follows that We sum up (2.15) from j = 1 to j = n , then it follows that On the one hand, note that we have chosen the initial approximation function U (0) satisfying (2.4)-(2.6). We integrate by parts to get since the initial approximation function Combining this estimate with (2.14) and the fact that ≥ 1 , we deduce that there exists a positive constant C , ,n depending on , , n such that where ∈ (0, 1 4  (∀i = 1, 2, 3, … , n) to both sides of (2.11), it follows that with the boundary condition where 1 ≤ l ≤ s and Next, we derive higher derivative estimates of solutions to the coupled system (2.11).

Lemma 2.3 Let ≥ 1. Assume the initial approximation function U (0) satisfying (2.4)-(2.6). Then the solution (1) (x) of the linear system (2.11) satisfies
Proof This proof is based on the induction. Let s = 1 , by (2.27), it follows that with the boundary condition Let us choose the weighted function satisfies (2.14). We multiply both sides of (2.30) by , respectively, then integrating over Ω by noticing (2.31), and summing up those equalities from j = 1 to j = n , it follows that where (2.29) We now estimate each term in (2.32). On the one hand, since we have chosen the initial approximation function U (0) satisfying (2.4)-(2.6), using the similar method of getting (2.17)-(2.20), we obtain By the incompressible condition ∇ ⋅ (1) = 0 and integrating by parts, we find from which, by the standard Calderon-Zygmund theory and Young's inequality we find and (2.38) We notice that the main term of A i (x) (i = 1, 2, 3, … , n) is given by Combining this fact with the assumption (2.5), we deduce that Thus, by (2.14), there exists a positive constant C , ,n depending on , , n such that where ∈ (0, 1 4 ) . Similarly, it follows that and Therefore Furthermore, one can see that the last two terms in the right-hand side of (2.42) can be controlled by using (2.13). Thus, it follows that Assume that the 2 ≤ l ≤ s − 1 derivative case holds, that is, We now prove that the sth derivative case holds. Multiplying both sides of equations (2.27) by D s i h (1) j e − (x 1 ) , then integrating over Ω by using the boundary condition (2.28), and summing up those equalities from j = 1 to j = n , it follows that We notice that where By assumptions of (2.4)-(2.5), one can see that the coefficients C i (x) (i = 1, 2, 3, … , n) have the same main terms with A i (x) . It follows that By this estimate and (2.14), one can deduce that there exists a positive constant C , ,n depending on , , n such that where ∈ (0, 1 4

The existence of first approximation step
Based on the previous a priori estimates, we are ready to prove the existence of the first approximation step, by using the classical theory of elliptic equations; see [16,30]. Then the linearized elliptic system admits a Sobolev regular solution (1) Moreover, it follows that where This is also the nonlinear term in approximation problem (2.3) at U (m−1) (x).
The following result establishes how to construct the m-th approximation solution.
Then we will find the m-th (m ≥ 2) approximation solution U (m) (x) , which is equivalent to find (m) (x) such that Substituting (3.10) into (2.3), it follows that Setting we supplement it with the boundary conditions (3.3).
Since we assume U (m−1) (x) ∈ B , there is the same structure between the linear system (2.9) and the linear system of mth approximation solutions. Thus, by means of the same proof process in Proposition 2.1, we can show that the above problem admits a solution (m) (x) ∈ H s (Ω) . For this purpose, we should use (2.2). Furthermore, similar to (2.50), we can use (3.8)-(3.9) to derive where one can see the (m − 1)-th error term E (m−1) such that Moreover, by (3.5) and the standard Calderon-Zygmund theory, it follows that The proof is now complete. ◻

Convergence of the approximation scheme
Our target is to prove that U (∞) (x) is a global solution of nonlinear equations (1.1). This is equivalent to show that the series For a fixed constant s ≥ 1 , let 1 ≤ s =k < k 0 ≤ k and which gives that Assume that the case of (m − 1) holds, that is, We now prove that the case of m holds. Using (2.2), (3.6) and the second inequality of (4.3), we obtain Combining this estimate with (2.2), (3.7) and (4.1), we obtain (4.1)  This means that U (m) ∈ B . Hence we conclude that (4.2) holds. Therefore, the steady incompressible Navier-Stokes equations (1.1) with the Dirichlet boundary condition admit global solutions from which, one can see the solution depends on the initial approximation function U (0) (x) strongly.
Finally, by (1.3) and the standard Calderon-Zygmund theory (that is, for the Riesz operator R ), we have ‖RU‖ s 0 ≤ ‖U‖ s 0 with 1 < s 0 < ∞ . We conclude that and this completes the proof.