Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 26, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO STEADY NAVIER-STOKES EQUATIONS VIA A MINIMAX APPROACH AMIN FEREIDOONI, ABBAS MOAMENI, ANANT GREWAL Communicated by Claudianor O. Alves Abstract. Our objective in this paper is to develop and utilize a minimax principle for proving the existence of symmetric solutions for the stationary Navier-Stokes equations. Notwithstanding its application to symmetric solu- tions in this paper, our minimax principle is broad enough to capture other types of solutions provided the equation and the external force are compati- ble under a family of operations including but not limited to being invariant by compact groups. The subset of functions compatible under this family of operations is not required to be a linear subspace, and being a closed convex set suffices for our purpose. 1. Introduction We are concerned with the following stationary Navier-Stokes equation with homogeneous boundary condition (u · ∇)u+ f(x) = ∆u−∇pu ∀x ∈ Ω, ∇ · u = 0 ∀x ∈ Ω, u = 0 ∀x ∈ ∂Ω, (1.1) where Ω is a bounded domain in Rm (m = 2, 3, 4), u is the vector-valued velocity function, pu is the scalar-valued pressure function associated with the velocity u, and f ∈ L2(Ω) is the external force function. We herein develop a minimax machin- ery to prove the existence of solutions to the above problem with specific properties based on the provided initial data Ω and the external force f . We then apply this machinery to several cases including the stationary Navier-Stokes equations under certain symmetric conditions. To be precise, for Ω ⊂ Rm (m = 2, 3, 4), set V = {u ∈ H1 0 (Ω) : ∇ · u = 0}, and define B : V × V → Rm as follows: B(u, v) = (u · ∇)v = m∑ j,k=1 uk ∂vj ∂xk ej , where ej is the unit vector along the jth axis. We set B(u, u) = Λu. The following theorem is the main abstract result of this paper. 2020 Mathematics Subject Classification. 35Q30, 37K58. Key words and phrases. Navier-Stokes equations; variational principles. ©2023. This work is licensed under a CC BY 4.0 license. Submitted August 14, 2022. Published March 7, 2023. 1 2 A. FEREIDOONI, A. MOAMENI, A. GREWAL EJDE-2023/26 Theorem 1.1. Let K be a closed convex subset of V , and assume one of the following two conditions hold: (i) For each u ∈ K, there exists v ∈ K such that Λu+ f(x) = ∆v −∇pv ∀x ∈ Ω, in a weak sense, that is∫ Ω Λu · η dx+ ∫ Ω f(x) · η dx = − ∫ Ω ∇v · ∇η dx ∀η ∈ V. (ii) For each u ∈ K, there exists v ∈ K such that B(u, v) + f(x) = ∆v −∇pv ∀x ∈ Ω, in a weak sense; then there exists ū ∈ K such that Λū+ f(x) = ∆ū−∇pū ∀x ∈ Ω, ∇ · ū = 0 ∀x ∈ Ω, ū = 0 ∀x ∈ ∂Ω. It is worthwhile emphasizing here that the primary consequence of this theorem centers on the choice of K, i.e., by choosing an appropriate K, one is able to establish the existence of a solution enjoying all the properties induced by the set K. For instance, in the case of the 3D stationary Navier-Stokes equations (1.1), choose K to be a subset of V containing all u = (u1, u2, u3) ∈ V with the following properties: u1(x1, x2, x3) = −u1(−x1, x2, x3), u2(x1, x2, x3) = u2(−x1, x2, x3), u3(x1, x2, x3) = u3(−x1, x2, x3). Correspondingly, let us define the maps π1, π2, π3 : Ω→ Ω as follows π1(x1, x2, x3) = (−x1, x2, x3), π2(x1, x2, x3) = (x1,−x2, x3), π3(x1, x2, x3) = (x1, x2,−x3). We shall show that if the domain Ω ⊂ R3 and the external force f are invariant under the maps π1, π2, π3 : Ω→ Ω, then the Navier-Stokes equations have a solution belonging to the set K. To illustrate our methodology, we have provided more examples throughout the paper. Historically, symmetry conditions of the form above have been imposed on the solution of the Navier-Stokes equations to address the existence problem of these equations in bounded domains, albeit with non-homogeneous boundary conditions, given by (u · ∇)u+ f(x) = ∆u−∇pu ∀x ∈ Ω, ∇ · u = 0 ∀x ∈ Ω, u = a(x) ∀x ∈ ∂Ω. (1.2) The bounded domain Ω is defined as Ω = Ω0 \ ( ∪Ni=1 Ωi ) , EJDE-2023/26 STEADY NAVIER-STOKES EQUATIONS 3 where Ωi ⊂ Ω0 for i = 1, . . . , N , and the C2 smooth boundary ∂Ω is composed of N + 1 disjoint components ∂Ωi, i.e., ∂Ω = ∪Ni=0∂Ωi. Note that the divergence free property of the flow (equation (1.1)) enforces the condition ∫ ∂Ω a(x) · n(x) ds = N∑ i=0 ∫ ∂Ωi a(x) · n(x) ds = 0, where n(x) is the unit outer normal to ∂Ω. Proving the existence of a solution for the above-mentioned stationary Navier-Stokes equations is commonly referred to as the Leray Problem. Although the 2D case is now solved [10], the general 3D case still remains an open problem. In the very first attempt to solve the problem, Leray in his seminal 1933 paper [13], proved the existence of a solution under the condition ∫ ∂Ωi a(x) · n(x) ds = 0. Solving the Leray Problem generally where the above condition is removed attracted lots of attention in the research community. For several decades, all the proposed solutions to the 2D case relied on some type of conditions; and this is still the case for the 3D problem [7]. In most attempts, this condition is imposed on a(x) at the boundary, i.e., N∑ i=0 ∣∣ ∫ ∂Ωi a(x) · n(x) ds ∣∣ < c. For some of the examples pertaining the major contributions to this line of research please refer to [11, 4, 6, 19, 12, 9, 8, 2, 17]. Some researchers, however, have tackled the problem where the required conditions are imposed on the entire domain Ω as symmetry conditions. Most notably, Amick [1] first studied the domain Ω ⊂ R2 invariant under the mapping π1, defined as: π1(x1, x2) = (−x1, x2). Using “reduction to absurdity”, Amick proved in 1984 that the steady Navier-Stokes equations (1.2) has a solution preserving the following symmetry condition, u1(−x1, x2) = −u1(x1, x2), u2(−x1, x2) = u2(x1, x2). In a similar effort, Sazonov [18] provided a proof of the existence problem in the presence of the aforementioned symmetry condition. By introducing the concept of ”Virtual drain”, Fujita [5] also proved the existence of a symmetric solution through constructing a symmetric solenoidal extension of the boundary value. Fur- thermore, Morimoto [14] presented a different proof by invoking the concept of stream functions. In extending the previous works to R3, Punhnachev [16, 15] and subsequently Korobkov et al. [10] proved an existence theorem for the axially sym- metric problem in a domain with a multiply connected boundary. Note that the function h = (hr, hθ, hz) in the cylindrical coordinate is called axially symmetric if hθ = 0, and hr and hz are not dependent on θ. 4 A. FEREIDOONI, A. MOAMENI, A. GREWAL EJDE-2023/26 2. Proof of Theorem 1.1 We shall need some preliminary results before proving our abstract Theorem 1.1. We define V = {u ∈ H1 0 (Ω) : ∇ ·u = 0}, and assume K is a closed convex subset of V . Furthermore, define B : V × V → Rm (m = 2, 3, 4) as follows: B(u, v) = (u · ∇)v = m∑ j,k=1 uk ∂vj ∂xk ej , (2.1) where ej is the unit vector along the jth axis. Note that in particular B(u, u) = Λu. Lemma 2.1. The function M1 : K × K → R defined by M1(u, v) = 〈Λu, v〉 is weakly lower semi-continuous on K ×K for each v ∈ K. Proof. Let v ∈ C1(Ω) ∩ K and un ⇀ u weakly in V . Using Rellich-Konrachov Compactness Theorem, one can prove that un → u strongly in Lp(Ω) for 1 ≤ p < 2m/(m− 2). Applying Lemma 4.3 in the Appendix results in M1(un, v) = ∫ Ω (un · ∇)un · v dx = − ∫ Ω (un · ∇)v · undx. (2.2) Therefore, |M1(un, v)−M1(u, v)| = ∣∣∣ m∑ j,k=1 ∫ Ω ( unk ∂vj ∂xk unj − uk ∂vj ∂xk uj ) dx ∣∣∣ ≤ ‖v‖C1(Ω) m∑ j,k=1 ∫ Ω ∣∣unkunj − ukuj∣∣dx ≤ ‖v‖C1(Ω) m∑ j,k=1 (∫ Ω |unkunj − ukunj |+ ∫ Ω |ukunj − ukuj | ) dx ≤ ‖v‖C1(Ω) m∑ j,k=1 ( ‖unj ‖L2(Ω)‖unk − uk‖L2(Ω) + ‖uk‖L2(Ω)‖unj − uj‖L2(Ω) ) . Therefore, M1(un, v) converges strongly to M1(u, v) on K for every v ∈ C1(Ω)∩K. Using Lemma 4.2 in the Appendix, we know that M1(u, v) is strongly continuous on H1(Ω); hence, by using the density argument we can conclude that M1(u, v) is weakly lower semi-continuous on K ×K for each v ∈ K. � Next, we define M : V × V → R as M(u, v) = 1 2 ∫ Ω |∇u|2dx− 1 2 ∫ Ω |∇v|2dx+ ∫ Ω Λu · (u− v)dx+ ∫ Ω f(x) · (u− v)dx, where f ∈ L2(Ω). Lemma 2.2. The function M(u, v) is lower semi-continuous on K ×K, where K is a convex and closed subset of V . Proof. Assume that un ⇀ u weakly in V , • It follows from the lower semi continuity of the norm that∫ Ω |∇u|2dx ≤ lim inf n→∞ ∫ Ω |∇un|2dx; EJDE-2023/26 STEADY NAVIER-STOKES EQUATIONS 5 • we have Λu · (u) = 0 resulting from Lemma 4.3 in the Appendix, and M1(u, v) = 〈Λu, v〉 is weakly lower semi-continuous as proven in Lemma 2.1; • since f ∈ L2(Ω), applying the strong convergence of un → u in L2(Ω) leads to ∫ Ω f(x)u dx = lim n→∞ ∫ Ω f(x)undx. This proves that M(u, v) is lower semi-continuous on K ×K. � Proof of Theorem 1.1. Part 1: Assume condition (i) holds. Set M : V × V → R as follows: M(u, v) = 1 2 ∫ Ω |∇u|2dx− 1 2 ∫ Ω |∇v|2dx+ ∫ Ω Λu · (u− v)dx+ ∫ Ω f(x) · (u− v)dx, where f ∈ L2(Ω). Note that M : K ×K → R satisfies all the conditions of the Ky Fan’s Min-Max Principle presented in Theorem 4.1 in the Appendix: (1) For each v ∈ K, the map u 7→M(u, v) is weakly lower semi-continuous on K as proved in Lemma 2.2. (2) For each u ∈ V , the map v 7→M(u, v) is concave on K: note that M(u, v) is a linear functional with respect to v except for 1 2 ∫ Ω |∇v|2dx, which is in fact convex. (3) Note that M(u, u) = 0 = γ for every u ∈ K. (4) As required in Theorem 4.1, we should show that there exists v0 ∈ K such that the set {u ∈ K : M(u, v0) ≤ γ} is bounded. Set v0 = 0, we show that such that K0 = {u ∈ K : M(u, v0) ≤ γ} is bounded. Take u ∈ K0, using Hölder’s inequality, we have 1 2 ‖∇u‖2L2(Ω) = 1 2 ∫ Ω |∇u|2dx ≤ − ∫ Ω f(x) · (u)dx ≤ ‖f‖L2(Ω)‖u‖L2(Ω). Using Sobolev embedding results ‖u‖L2(Ω) ≤ c‖∇u‖L2(Ω) on the right hand side, we obtain ‖∇u‖L2(Ω) ≤ C‖f‖L2(Ω). Therefore, the set K0 is bounded under the ‖ · ‖H1(Ω). We now apply the Ky Fan’s Min-Max Principle to conclude that there exists ū ∈ K such that M(ū, v) ≤ 0 ∀v ∈ K; that is 1 2 ∫ Ω |∇ū|2dx− 1 2 ∫ Ω |∇v|2dx+ ∫ Ω Λū · (ū− v)dx+ ∫ Ω f(x) · (ū− v)dx ≤ 0 (2.3) for all v ∈ K. By assumption (i), there exits v̄ ∈ K such that∫ Ω Λū · η dx+ ∫ Ω f(x) · η dx = − ∫ Ω ∇v̄ · ∇η dx ∀η ∈ V. (2.4) Now, choose η = ū− v̄, we have∫ Ω Λū · (ū− v̄) dx+ ∫ Ω f(x) · (ū− v̄) dx = − ∫ Ω ∇v̄ · ∇ (ū− v̄) dx. (2.5) On the other hand, equation (2.3) holds for v̄ ∈ K, i.e., 1 2 ∫ Ω |∇ū|2dx− 1 2 ∫ Ω |∇v̄|2dx+ ∫ Ω Λū · (ū− v̄)dx+ ∫ Ω f(x) · (ū− v̄)dx ≤ 0. 6 A. FEREIDOONI, A. MOAMENI, A. GREWAL EJDE-2023/26 Replacing the last two terms of the above inequality with the right-hand side of equation (2.5) results in the inequality 1 2 ∫ Ω |∇ū|2dx− 1 2 ∫ Ω |∇v̄|2dx− ∫ Ω ∇v̄ · ∇ (ū− v̄) dx ≤ 0. Therefore, 1 2 ∫ Ω |∇ū−∇v̄|2dx ≤ 0. Hence, we have ∇ū = ∇v̄, and since ū = v̄ = 0 on ∂Ω, we conclude that ū = v̄ on Ω. Substituting ū = v̄ in equation (2.4) results in∫ Ω Λū · η dx+ ∫ Ω f(x) · η dx = − ∫ Ω ∇ū · ∇η dx ∀η ∈ V. or equivalently Λū+ f(x) = ∆ū−∇pū ∀x ∈ Ω, ∇ · ū = 0 ∀x ∈ Ω, ū = 0 ∀x ∈ ∂Ω. Part 2: Assume condition (ii) holds. Using Lemma 4.3 in the Appendix, we have Λu · (u− v) = (u · ∇)u · (u− v) = (u · ∇)v · (u− v) = B(u, v) · (u− v). (2.6) The rest of the proof is identical to Part 1. � 3. Applications In this section, we demonstrate how Theorem 1.1 can be used for proving the existence of symmetric solutions to the Navier-Stokes equations in dimension three. The less involved two dimensional cases can be addressed using a similar approach; thus, they are not repeated here. In light of this objective, let us define the maps π1, π2, π3 : Ω→ Ω as follows π1(x1, x2, x3) = (−x1, x2, x3), π2(x1, x2, x3) = (x1,−x2, x3), π3(x1, x2, x3) = (x1, x2,−x3). Theorem 3.1. Consider the 3D stationary Navier-Stokes equations presented in equation (1.1). Assume that Ω is invariant under the map π1 : Ω→ Ω. Moreover, assume that K is a subset of V containing all u ∈ V with the following properties: u1(x1, x2, x3) = −u1(−x1, x2, x3), u2(x1, x2, x3) = u2(−x1, x2, x3), u3(x1, x2, x3) = u3(−x1, x2, x3). (3.1) Furthermore, assume that f(x) ∈ L2(Ω) also holds the same properties; i.e., f1(x1, x2, x3) = −f1(−x1, x2, x3), f2(x1, x2, x3) = f2(−x1, x2, x3), f3(x1, x2, x3) = f3(−x1, x2, x3). Then, the Navier-Stokes equation has a solution in K. EJDE-2023/26 STEADY NAVIER-STOKES EQUATIONS 7 Proof. Step 1: It can be shown that the set K is convex and closed in V . To be precise, since K ⊂ V , the identities in (3.1) are to be understood almost every where in Ω. If {un} is a sequence in K such that un converges weakly in V to a function u ∈ V, then un converges strongly in L2(Ω). Therefore, up to a sub- sequence, un(x) → u(x) for a.e. x ∈ Ω. This implies that u satisfies the identities in (3.1) almost every where in Ω. On the other hand since K is a linear subset of V it is clearly convex. Step 2: Fix u ∈ K. We now show that there exits v ∈ V such that Λu− f(x) = ∆v −∇pv ∀x ∈ Ω, (3.2) in a weak sense. To this end, define the the functional I : V → R as follows: I(w) = 1 2 ∫ Ω |∇w|2dx+ ∫ Ω Λu · w dx+ ∫ Ω f(x) · w dx. The functional I is coercive, lower semi-continuous and strictly convex; thus, there exist a unique v ∈ V such that I(v) = inf w∈V I(w), and satisfies equation (3.2). Step 3: We then need to show that v ∈ K. Define v̄(x) as follows: v̄1(x1, x2, x3) = −v1(−x1, x2, x3), v̄2(x1, x2, x3) = v2(−x1, x2, x3), v̄3(x1, x2, x3) = v3(−x1, x2, x3). (3.3) Now by calculations, we have I(v̄) = 1 2 ∫ Ω |∇v̄(x)|2dx+ ∫ Ω Λu(x) · v̄(x) dx+ ∫ Ω f(x) · v̄(x) dx. To rewrite I(v̄) in terms of v, we set x̄ = (−x1, x2, x3). We first show that Λu(x) · v̄(x) = Λu(x̄) · v(x̄). (3.4) For simplicity of notation, Di denotes derivative with respect to the ith variable of a given function u(x). Therefore, Du(x) = D1u1(x) D2u1(x) D3u1(x) D1u2(x) D2u2(x) D3u2(x) D1u3(x) D2u3(x) D2u3(x)  = +D1u1(x̄) −D2u1(x̄) −D3u1(x̄) −D1u2(x̄) +D2u2(x̄) +D3u2(x̄) −D1u3(x̄) +D2u3(x̄) +D2u3(x̄)  . (3.5) Now we expand the left-hand side of (3.4) as follows: Λu(x) · v̄(x) = [u1(x)D1u1(x) + u2(x)D2u1(x) + u3D3u1(x)] v̄1(x) + [u1(x)D1u2(x) + u2(x)D2u2(x) + u3D3u2(x)] v̄2(x) + [u1(x)D1u3(x) + u2(x)D2u3(x) + u3D3u3(x)] v̄3(x). Using the relationships in (3.1), (3.3) and (3.5), we have Λu(x) · v̄(x) = [ (−u1(x̄))(+D1u1(x̄)) + (+u2(x̄))(−D2u1(x̄)) + (+u3(x̄))(−D3u1(x̄)) ] (−v1(x̄)) + [ (−u1(x̄))(−D1u2(x̄)) 8 A. FEREIDOONI, A. MOAMENI, A. GREWAL EJDE-2023/26 + (+u2(x̄))(+D2u2(x̄)) + (+u3(x̄))(+D3u2(x̄)) ] (+v2(x̄)) + [ (−u1(x̄))(−D1u3(x̄)) + (+u2(x̄))(+D2u3(x̄)) + (+u3(x̄))(+D3u3(x̄)) ] (+v3(x̄)) = Λu(x̄) · v(x̄). Moreover, one can similarly prove that f(x) · v̄(x) = f(x̄) · v(x̄) |∇v̄(x)|2 = |∇v(x̄)|2. Since |J | = |∂x/∂x̄| = 1, we can equivalently write I(v̄) = 1 2 ∫ Ω |∇v(x̄)|2dx̄+ ∫ Ω Λu(x̄) · v(x̄) dx̄+ ∫ Ω f(x̄) · v(x̄) dx̄. Finally, we conclude that I(v̄) = I(v). Step 4: Note that ∇ · v̄(x) = ∇ · v(x) = 0. Therefore, v̄(x) ∈ V . Since v is the unique minimizer of I, we can conclude that v̄(x) = v(x); therefore, there exits v ∈ K such that equation (3.2) is satisfied for a fixed u ∈ K. Step 5: Note that the existence of v ∈ K (as proved above) satisfies condition (i) of Theorem 1.1; therefore, a solution of the Navier-Stokes equations exist in the set K; i.e., there exists ū ∈ K that satisfies the following equations: Λū+ f(x) = ∆ū−∇pū ∀x ∈ Ω, ∇ · ū = 0 ∀x ∈ Ω, ū = 0 ∀x ∈ ∂Ω. � One can generalize the aforementioned theorem to encompass a variety of prob- lems that follow the same structure. In order to achieve this, let us define the maps γ1, γ2, γ3 : L2(Ω)→ L2(Ω) as follows: γ1(u1(x), u2(x), u3(x)) = (−u1(x), u2(x), u3(x)), γ2(u1(x), u2(x), u3(x)) = (u1(x),−u2(x), u3(x)), γ3(u1(x), u2(x), u3(x)) = (u1(x), u2(x),−u3(x)). We denote the the group generated by γ1, γ2 and γ3 as Gγ and it isomorphic counterpart by Gπ which is generated by the elements π1, π2 and π3. The two groups correspond to each other by the isomorphism g : Gπ → Gγ , as follows: g(πi) = γi, g(πi ◦ πj) = γi ◦ γj , g(πi ◦ πj ◦ πk) = γi ◦ γj ◦ γk, where i, j, k = 1, 2, 3. Theorem 3.2. Consider the 3D stationary Navier-Stokes equations presented in equation (1.1). Define the groups Gπ and Gγ and their isomorphism g as above. Assume that Ω is invariant under the map π̄1, . . . , π̄m ∈ Gπ, and K is a subset of V containing all u ∈ V with the property that when g(π̄1) = γ̄1, . . . , g(π̄m) = γ̄m we have u(x) = γ̄1(u(π̄1(x))), . . . , u(x) = γ̄m(u(π̄m(x))). Furthermore, assume that EJDE-2023/26 STEADY NAVIER-STOKES EQUATIONS 9 f(x) ∈ H1 0 (Ω) also holds the same property; i.e., f(x) = γ̄1(f(π̄1(x))), . . . , f(x) = γ̄m(f(π̄m(x))). Then, the Navier-Stokes equation has a solution in K. Proof. The proof of this theorem follows the steps presented in the previous example except that Step 3 needs to be verified for the pair of functions π̄1, γ̄1 to π̄m, γ̄m instead of π1, γ1. � The following two corollaries, whose 2D versions have been solved in the litera- ture using other techniques, are also worthwhile pointing out herein. Corollary 3.3. Consider the 3D stationary Navier-Stokes equations presented in equation (1.1). Assume that Ω is invariant under the map π : Ω → Ω, which is defined as follows: π(x) = π(x1, x2, x3) = (−x1,−x2,−x3) = −x. (3.6) Moreover, assume that K is a subset of V containing all u ∈ V with the property u(x1, x2, x3) = −u(−x1,−x2,−x3). Furthermore, assume that f(x) ∈ H1 0 (Ω) also holds the same property; i.e., f1(x1, x2, x3) = −f(−x1,−x2,−x3). (3.7) Then, the Navier-Stokes equation has a solution in K. Proof. Applying Theorem 3.2 for the case m = 1, we set π̄1 = π3 ◦ π2 ◦ π1 and γ̄1 = γ1 ◦ γ2 ◦ γ3. � Corollary 3.4. Consider the 3D stationary Navier-Stokes equations presented in equation (1.1). Assume that Ω is invariant under the maps π1, π2, π3 : Ω → Ω. Moreover, assume that K is a subset of V containing all u ∈ V with the property ui(x) = { −ui(πj(x)) i = j, ui(πj(x)) otherwise, where u(x) = (u1(x), u2(x), u3(x)). Furthermore, assume that f(x) ∈ H1 0 (Ω) also holds the same property; i.e., fi(x) = { −fi(πj(x)) i = j, fi(πj(x)) otherwise, where f(x) = (f1(x), f2(x), f3(x)). Then, the Navier-Stokes equation has a solution in K. Proof. Applying Theorem 3.2 for the case m = 3, we set π̄i = πi and γ̄i = γi for i = 1, 2, 3. � 4. Appendix The following is the well-known Ky Fan’s Min-Max Principle by Brezis-Nirenberg- Stampacchia [3]. Theorem 4.1. Let E be a closed convex subset of a reflexive Banach space Z, and consider M : E × E → R̄ to be a function such that (1) For each y ∈ E, the map x→M(x, y) is weakly lower semi-continuous on E; (2) For each x ∈ E, the map y →M(x, y) is concave on E; 10 A. FEREIDOONI, A. MOAMENI, A. GREWAL EJDE-2023/26 (3) There exists γ ∈ R such that M(x, x) ≤ γ for every x ∈ E; (4) There exists a y0 ∈ E such that E0 = {x ∈ E : M(x, y0) ≤ γ} is bounded. Then, there exits x̄ ∈ E such that M(x̄, y) ≤ γ for all y ∈ E. We have made frequent use of the following standard result. Now we provide a short proof, for the convenience of the reader. Lemma 4.2. Let f(u, v, w) in Rm (m = 2, 3, 4) be defined as f(u, v, w) = 〈(u · ∇) · v, w〉 = m∑ j,k=1 uk ∂vj ∂xk wj . (4.1) Then, f(u, v, w) is continuous on H1 ×H1 ×H1. Proof. Using Hölder’s inequality, we have |f(u, v, w)| ≤ ‖u‖L4‖∇v‖L2‖w‖L4 . (4.2) Using the Sobolev embedding H1(Ω) ⊂ L 2m m−2 (Ω), we have that |f(u, v, w)| ≤ C‖u‖H1‖v‖H1‖w‖H1 , (4.3) for an appropriate constant C. This proves that that f(u, v, w) is strongly contin- uous. � Lemma 4.3. Let u ∈ V and v, w ∈ H1. Then f(u, v, w) = 〈(u · ∇) · v, w〉 = −〈(u · ∇) · w, v〉 = −f(u,w, v), (4.4) and in particular, f(u, v, v) = 〈(u · ∇) · v, v〉 = 0. (4.5) Proof. Assume u ∈ C∞ c (Ω) ∩ V and v, w ∈ C1(Ω). Using integration by parts, we have 〈(u · ∇)v, w〉 = ∫ Ω m∑ j,k=1 uk ∂vj ∂xk wjdx = − ∫ Ω m∑ j,k=1 ∂uk ∂xk vjwjdx− ∫ Ω m∑ j,k=1 ukvj ∂wj ∂xk dx = −〈(u · ∇) · w, v〉. Since f(u, v, w) is continuous on H1 × H1 × H1 (proven in Lemma 4.2), we use the density argument to extend the above conclusion to u ∈ V and v, w ∈ H1. Furthermore, note that 〈(u · ∇)v, v〉 = −〈(u · ∇)v, v〉, therefore, 〈(u · ∇)v, v〉 = 0. (4.6) � Acknowledgements. The authors acknowledge the support from the National Research Council of Canada. We thank the anonymous reviewer for the careful reading of our manuscript and the insightful comments and suggestions. EJDE-2023/26 STEADY NAVIER-STOKES EQUATIONS 11 References [1] Charles J. Amick; Existence of solutions to the nonhomogeneous steady Navier-Stokes equa- tions, Indiana University mathematics journal, 33 (1984), no. 6, 817–830. [2] W. Borchers, K. 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[18] Leonid Ivanovich Sazonov; On the existence of a stationary symmetric solution of the two- dimensional fluid flow problem, Mathematical Notes 54 (1993), no. 6, 1280–1283. [19] Iosif Izrailevich Vorovich, Victor Iosifovich Yudovich; Steady flow of a viscous incompressible fluid, Matematicheskii Sbornik 95 (1961), no. 4, 393–428. Amin Fereidooni National Research Council Canada, Ottawa, ON, Canada Email address: Amin.Fereidooni@nrc-cnrc.gc.ca Abbas Moameni School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada Email address: momeni@math.carleton.ca Anant Grewal National Research Council Canada, Ottawa, ON, Canada Email address: Anant.Grewal@nrc-cnrc.gc.ca 1. Introduction 2. Proof of Theorem 1.1 3. Applications 4. Appendix Acknowledgements References