Decay of weak solutions for the navier stokes equations in general domains

Let u be a weak solution of the in-stationary Navier-Stokes equations in a completely general domain in R3. Firstly, we prove that the time decay rates of the weak solution u in the L2-norm like ones of the solutions for the homogeneous Stokes system taking the same initial value in which the decay exponent is less than 34 . Secondly, we show that under some additive conditions on the initial value, then u coincides with the solution of the homogeneous Stokes system when time tends to infinity.

ISSN: 1859-2171 e-ISSN: 2615-9562 TNU Journal of Science and Technology 225(02): 45 - 51 L2 DECAY OF WEAK SOLUTIONS FOR THE NAVIER-STOKES EQUATIONS IN GENERAL DOMAINS Vu Thi Thuy Duong1*, Dao Quang Khai2 Quang Ninh University of Industry - Quang Ninh - Viet Nam Institute of Mathematics - Ha Noi - Viet Nam ABSTRACT Let u be a weak solution of the in-stationary Navier-Stokes equations in a completely general domain in R3 Firstly, we prove that the time decay rates of the weak solution u in the L2-norm like ones of the solutions for the homogeneous Stokes system taking the same initial value in which the decay exponent is less than 34 Secondly, we show that under some additive conditions on the initial value, then u coincides with the solution of the homogeneous Stokes system when time tends to infinity Our proofs use the theory about the uniqueness arguments and time decay rates of strong solutions for the Navier-Stokes equations in the general domain when the initial value is small enough Keywords: Navier-Stokes equations, Decay , Weak solutions, Stokes equations, Uniqueness of solution Received: 13/02/2020; Revised: 21/02/2020; Published: 26/02/2020 DÁNG ĐIỆU TIỆM CẬN CỦA NGHIỆM YẾU CHO HỆ PHƯƠNG TRÌNH NAVIER-STOKES TRONG MIỀN TỔNG QUÁT VỚI CHUẨN L2 Vũ Thị Thùy Dương1*, Đào Quang Khải2 Trường Đại học Công nghiệp Quảng Ninh - Việt Nam Viện Tốn học Việt Nam TĨM TẮT Giả sử u nghiệm yếu hệ phương trình Navier-Stokes khơng dừng miền tổng quát R3 Trước hết, chứng minh tốc độ hội tụ theo thời gian nghiệm yếu u với chuẩn L2 giống tốc độ hội tụ theo thời gian nghiệm hệ Stokes với giá trị ban đầu số mũ hội tụ nhỏ 34 Thứ hai, với số điều kiện giá trị ban đầu u trùng với nghiệm hệ Stokes thời gian dần tới vô Phần chứng minh kết báo dựa lý thuyết tính tốc độ hội tụ theo thời gian nghiệm mạnh cho hệ phương trình Navier-Stokes miền tổng quát giá trị ban đầu đủ nhỏ Từ khóa: Hệ phương trình Navier-Stokes, Dáng điệu tiệm cận, Nghiệm yếu, Hệ phương trình Stokes, Tính nghiệm Ngày nhận bài: 13/02/2020; Ngày hoàn thiện: 21/02/2020; Ngày đăng: 26/02/2020 * Corresponding author Email: vuthuyduong309@gmail.com https://doi.org/10.34238/tnu-jst.2020.02.2617 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 45 Introduction and main result and Sobolev Lq (Ω), W k,p (Ω), with norms · = · q and · W k,p (Ω) = · k,p FurLq (Ω) ther, we use the Bochner spaces Ls 0, T ; Lp (Ω) , ≤ s, p ≤ ∞ with the norm We consider the in-stationary problem of the 1/s Navier-Stokes system T s  · s := · dτ = · p,s,T p L 0,T ;Lp (Ω)   ut − ∆u + u · ∇u + ∇p = 0,  div u = 0, (1) To deal with solenoidal vector fields we introu| = 0,    ∂Ω duce the spaces of divergence - free smooth comu(0, x) = u0 , ∞ pactly supported functions C0,σ (Ω) = {u ∈ ∞ in a general domain Ω ⊆ R , i.e a non-empty C (Ω), div(u) = 0}, and the spaces L2 (Ω) = σ · · connected open subset of R3 , not necessarily 1,2 ∞ (Ω) C0,σ , W01,2 (Ω) = C0∞ (Ω) W , and bounded, with boundary ∂Ω and a time interval · W 1,2 (Ω) 1,2 ∞ (Ω) [0, T ), < T ≤ ∞ and with the initial value u0 , W0,σ (Ω) = C0,σ where u = (u1 , u2 , u3 ); u · ∇u = div(uu), uu = Let P : L2 (Ω) −→ L2σ (Ω) be the Helmholtz pro(ui uj )i,j=1 , if div u = In this paper we discuss the behavior as t → ∞ jection Let the Stokes operator of weak solutions of the Navier-Stokes equations in space L2 (Ω), which goes to zero with explicit rates The L2 -decay problem for Navier-Stokes system was first posed by Leray [1] in R3 The first (affirmative) answer was given by Kato [2] in case D = Rn , n = 3, 4, through his study of strong solutions in general spaces Lp , see also [3, 4, 5] The idea of Schonbek was then applied by [6, 7] to the case where D is a halfspace of Rn , n ≥ or an exterior domain of Rn , n ≥ W Borchers and T Miyakawa [8] developed the method in [3, 6, 7] for the case of an arbitrary unbounded domain They showed that if e−tA u0 = O(t−α ) for some α ∈ (0, 12 ), then u(t) = O(t−α ) Our purpose in this paper is to improve and generalize the result of [8] Firstly, we obtain the same result as that of them but under more general condition on α, in which the condition α ∈ (0, 21 ) is replaced by α ∈ (0, 34 ) Secondly, we obtain the stronger result than theirs by assuming some additive conditions on the initial value We recall some well-known function spaces, the definitions of weak and strong solutions to (1) and introduce some notations before describing the main results Throughout the paper, we sometimes use the notation A B as an equivalent to A ≤ CB with a uniform constant C The notation A B means that A B and B A The expression ·, · Ω denotes the pairing of functions, vector fields, etc on Ω and ·, · Ω,T means the corresponding pairing on [0, T ) × Ω For ≤ q ≤ ∞ we use the well-known Lebesgue A = −P∆ : D(A) −→ L2σ (Ω) with the domain of definition 1,2 D(A) = {u ∈ W0,σ (Ω), ∃f ∈ L2σ (Ω) : ∇u, ∇ϕ Ω = f, ϕ Ω, 1,2 ∀ ϕ ∈ W0,σ (Ω)} be defined as Au = −P∆u = f, u ∈ D(A) As in [9], we define the fractional powers Aα : D(Aα ) −→ L2σ (Ω), −1 ≤ α ≤ We have D(A) ⊂ D(Aα ) ⊂ L2σ (Ω) for α ∈ (0, 1] It is known that for any domain Ω ⊆ R3 the operator A is self-adjoint and generates a bounded analytic semigroup e−tA , t ≥ on L2σ (Ω) The following embedding properties play a basic role in the theory of the Navier-Stokes system β A− Pu ≤ C u q , u ∈ Lqσ (Ω) (2) where 21 ≤ β < 32 , 1q = 12 + β Furthermore, we mention the Stokes semigroup estimates Aα e−tA u ≤ t−α u , (3) with u ∈ L2σ (Ω), ≤ α ≤ Now we recall the definitions of weak and strong solutions to (1) Definition 1.1 (See [9].) Let u0 ∈ L2σ (Ω) A vector field 1,2 u ∈ L∞ 0, T ; L2σ (Ω))∩L2loc ([0, T ); W0,σ (Ω) (4) is called a weak solution in the sense of Leray- Theorem 1.3 Let Ω ⊆ R3 be a genHopf of the Navier-Stokes system (1) with the eral domain, u0 ∈ L2σ (Ω) and u is a weak solution of the Navier-Stokes system (1) satisinitial value u(0, x) = u0 if the relation fying strong energy inequality (8) If there exist − u, wt Ω,T + ∇u, ∇w Ω,T − uu, ∇w Ω,T positive constants t0 , C1 , and C2 such that = u0 , w Ω (5) C1 t−α1 ≤ e−tA u0 ≤ C2 t−α2 f or t ≥ t0 , is satisfied for all test functions ∞ w ∈ C0∞ [0, T ); C0,σ (Ω) , and additionally the where α , and α are constants satisfying energy inequality 1 t 1 ≤ α2 < and α2 ≤ α1 < α2 + , 2 u(t) + ∇u(τ ) 22 dτ ≤ u0 (6) 2 then u coincides with the solution of the homois satisfied for all t ∈ [0, T ) A weak solution u is called a strong solution of geneous Stokes system with the initial value u0 the Navier-Stokes equation (1) if additionally lo- when time tends to infinity in the sense that cal Serrin’s condition u∈ Lsloc q [0, T ); L (Ω) (7) t→∞ is satisfied with < s < ∞, < q < ∞ where + ≤ s q As is well known, in the case the domain Ω is bounded, it is not difficult to prove the existence of a weak solution u as in Definition 1.1 which additionally satisfies the strong energy inequality u(t) t 2 ∇u(τ ) 22 dτ ≤ + t u(t ) 2 u(t) − e−tA u0 u(t) lim = (9) Proof of main theorems Let us construct a weak solution of the following integral equation t 1 A e−(t−τ )A A− P(u · ∇u)dτ u(t) = e−tA u0 − (8) (10) We know that for almost all t ∈ [0, T ) and all t ∈ [t , T ), see 1,2 [9], p 340 For further results in this context for u ∈ L∞ 0, T ; L2σ (Ω)) ∩ L2loc ([0, T ); W0,σ (Ω) unbounded domains we refer to [10] is a weak solution of the Navier-Stokes system Now we can state our main results (1) iff u satisfies the integral equation (10), see Theorem 1.1 Let Ω ⊆ R3 be a gen[9] In order to prove the main theorems, we need eral domain, u0 ∈ L2σ (Ω) and u is a weak the following lemmas solution of the Navier-Stokes system (1) satisfying strong energy inequality (8) Then Lemma 2.1 Let γ, θ ∈ R and t > 0, then −tA −α (a) If e u0 = O(t ) for some ≤ α < , (a) If θ < 1, then then u(t) = O(t−α ) as t → ∞ t −tA −α (t − τ )−γ τ −θ dτ = K1 t1−γ−θ (b) If e u0 = o(t ) for some ≤ α < , then u(t) = o(t−α ) as t → ∞ −γ −θ Theorem 1.2 Let Ω ⊆ R3 be a gen- where K1 = (1 − τ ) τ dτ < ∞ eral domain, u0 ∈ L2 (Ω) and u is a weak (b) If γ < 1, then σ solution of the Navier-Stokes system (1) satisfying strong energy inequality (8) If u0 ∈ Lq (Ω) ∩ L2σ (Ω), < q ≤ 2, then u(t) = o t− 1 q−2 t (t − τ )−γ τ −θ dτ = K2 t1−γ−θ t as t → ∞ where K2 = 1 (1 − τ )−γ τ −θ dτ < ∞ The proof of this lemma is elementary and may Consider the weak solution of the Navier-Stokes be omitted system (1) satisfying the energy inequality Lemma 2.2 Let u ∈ L2 (Ω) and ∇u ∈ L2 (Ω) Then where β is ≤β< 2 β− 12 β e−tA P(u · ∇u) t− u positive ∇u constant u(t) −β that β β β u ∇u β 2 (12) u ∇u t− u β− 21 ∇u β t− for all t ≥ t0 2 The proof of Lemma 2.4 is complete β t− u · ∇u Proof of Theorem 1.1 q β− 21 t ∇u(t) β = A e−tA A− P(u · ∇u) ≤ t− A− P(u · ∇u) −β t0 u(t0 ) D(A ) e−tA P(u · ∇u) t ∇u(τ ) 22 dτ ≤ Combining Lemma 2.3, inequality (12), and Serrin’s uniqueness criterion [9, 12], we obtain Proof Applying inequalities (6), (3), Holder inequality, interpolation inequality, and Lemma 2.1, we obtain −β t + for all t ∈ [0, ∞) \ N with N is a null set Let δ be a positive constant in Lemma 2.3 Since (11) and (12), it follows that there exists the large enough t0 ∈ [0, ∞) \ N such that ≤ δ u(t0 ) such 2 −β −β ∇u (a) Consider the weak solution of the NavierStokes system (1), then u holds the integral equation t The proof of Lemma 2.2 is complete u(t) = e−tA u0 − e−(t−s)A P u · ∇u ds (13) Lemma 2.3 There exists a positive constant 1 From Lemma 2.2, we have δ such that if u0 ∈ D(A ) and A u0 ≤ δ, then the Navier-Stokes system (1) has a e−tA u0 u(t) strong solution with the initial value u0 satisfying t β ∇u(t) t− for all t ≥ + (t − s)− u(s) β− 21 ∇u(s) −β ds for all ≤ β < We divide the above integral 2 Lemma 2.4 Let u be a weak solution of the into two different parts as follow Navier-Stokes system (1) with the initial value t β β− −β u0 ∈ L2σ (Ω) Then there exists the positive value I= (t − s)− u(s) 2 ∇u(s) 22 ds t0 large enough such that ∇u(t) t− for all t β β− −β t ≥ t0 = (t − s)− u(s) 2 ∇u(s) 22 ds Proof See [11] t Proof Applying Holder inequality, we have t ∞ A4 u ∞ λ d Eλ u = ∇u(s) −β ds = I1 + I2 2 ∞ 1 d Eλ u 22 ) (11) We consider the following three cases: β− 21 λ d Eλ u 22 ) ( ≤( β (t − s)− u(s) + = A u 0≤α≤ u 1 1 , ≤ α < , and ≤ α < 4 2 Case 1: ≤ α ≤ Applying the energy inequality and Holder inequality, we obtain I1 t β− 12 − β t 2 u0 −β ∇u(s) It is not difficult to show that there exists a number β such that β 1 + ≥ α and ≤β< 2 Therefore, choose one of such β, it follows that ds t β β− u0 2 t− ( 2β−1 ds) ∇u(s) ( β− 21 − β t 2 u0 u(t) t 2 ds) 5−2β ≤α< Applying Case of part (a), we have Case 3: 2β−1 t 5−2β u0 = O(t− ) u(t) From Lemma 2.4 and Lemma 2.1(b), we have I2 u0 t β− 21 β (t − s)− s− −β ds t −β (s−γ )β− ∇u(s) ds where t0 is the constant in Lemma 2.4 It follows that t −β t −β t s −2γ ds t 2β−1 ∇u(s) + I ≤ O(t−α ) + O(t− ) t β t− I1 e−tA u0 (14) Ap2 plying inequality (14) and Holder inequality, we obtain t−γ for t ≥ 0, where γ is a constant such that ≤ γ < = O(t− ) for t ≥ 2t0 u(t) = O(t−α ) as t → ∞ = O(t−α ) as t → ∞ t 2 ds 5−2β −2γ+1 2β−1 = O(t γ −γβ− 14 ) Moreover, from Lemma 2.4 and Lemma 2.1(b), 1 we have Case 2: ≤ α < t β −β β− 12 (t − s)− s−γ s− 2 ds Applying the above inequality for α = − and I2 t Holder inequality, we obtain t β I1 t (s − 14 β− 12 ) ∇u(s) −β β t− ( s− ds) ds t t t 2β−1 −β β ∇u(s) 2β−1 −β −8 = O(t (1 − s)− s−γ(β− ) ds γ 2 ds = O(t −γβ− ) for t ≥ 2t0 5−2β It follows that t t t− −γ(β− )− ( −β)+1 t −β u(t) ) e−tA u0 γ +I ≤ O(t−α )+O(t −γβ− ) for t ≥ 2t0 Similar to the above case, it is not On the other hand, from Lemma 2.4 and Lemma difficult to show that there exist γ and β such 2.1(b), we have that 1 γ t − γβ − ≤ −α, ≤ β < , and ≤ γ < −β − 14 β− − 12 −β s ds I2 (t − s) s 2 t Choose ones of such γ and β, we conclude that t β β β u(t) = O(t−α ) as t → ∞ (t − s)− s − s − ds t (b) This is deduced from the proof of part (a) β The proof of Theorem is complete = O(t− − ) for t ≥ 2t0 So, we have u(t) e−tA u0 Corollary 2.1 Let Ω ⊆ R3 be a general domain Given u0 and u as in Theorem 1.1 If u(t) = o(t−γ ) for some γ ∈ [0, 21 ), then u(t) − e−tA u0 = o(t−(γ+θ) ) for all θ ∈ [0, 41 ) +I β ≤ O(t−α ) + O(t− − ) for t ≥ 2t0 Proof The proof is derived directly from the proof of Case of Theorem 1.1 where t1 = max t0 , Proof of Theorem 1.2 2M1 C1 4(α2 −α1 )+1 From the above two estimates, we obtain that u(t) − e−tA u0 u(t) Theorem 1.2 is an immediate consequence of Theorem 1.1(b) and the following lemma ≤ M1 t−( α1 +α2 + 18 ) C1 −α1 t Lemma 2.5 Let u0 ∈ L2σ (Ω) Then +1 2M1 − α2 −α = t → as t → ∞ (a) e−tA u0 → as t → ∞ C1 q (b) If u0 ∈ Lσ (Ω) ∩ L (Ω) for some < q ≤ 2, The proof of Theorem is complete then e−tA u0 1 q−2 = o t− as t → ∞ (15) References Proof (a) See Lemma 1.5.1 in [9], p 204 (b) Applying inequality (3), we obtain e−tA u0 = e −tA = A t e −tA 1 q−2 − 12 1 q−2 u0 e −tA e −tA −tA 1 q−2 A 1 q−2 A− − 12 1 q−2 u0 u0 (16) On the other hand, using inequality (2), we get A− e [1] J Leray, "Sur le mouvement d’un liquide visqueux emplissant l’espace", Acta Math, vol 63, pp 193-248, 1934 u0 ∈ L2σ (Ω) [3] R Kajikiya, T Miyakawa, "On L2 decay of weak solutions of the Navier-Stokes equations in Rn ", Math Z., vol 192, pp 135148, 1986 (17) Property 15 is deduced from Lemma 2.5(a), (16), and (17) [4] M E Schonbek, "Large time behaviour of solutions to the Navier-Stokes equations", Commun In Partial Diff Eq., vol 11, pp 733-763, 1986 Proof of Theorem 1.3 [5] M Wiegner, "Decay results for weak solutions of the Navier-Stokes equations in Rn ", J London Math Soc., vol 35, pp 303-313, 1987 Proof Applying Corollary 2.1 for γ = α2 , θ = α1 − α2 + , there exists a positive constant M1 such that u(t) − e−tA u0 = M1 t−( ≤ M1 t α1 +α2 + 18 ) α −α −(α2 + 2 [6] W Borchers, T Miyakawa, " L2 -Decay for the Navier-Stokes flows in halfspaces", Math Ann., vol 282, pp 139-155, 1988 + 18 ) for t ≥ t0 [7] W Borchers, T Miyakawa, "Algebraic L2 - decay for Navier-Stokes flows in exterior domains", Acta Math, vol 165, pp 189-227, 1990 It follows from the above inequality that u(t) ≥ u(t) ≥ C1 t −α1 − u(t) − e−tA u0 − M1 t ≥ C1 − M1 t−( ≥ −( α1 +α2 α2 −α1 2 [8] W Borchers, T Miyakawa, " L2 -Decay for Navier-Stokes flows in unbounded domains with application to Exterior Stationary Flows", Arch Rational Mech Anal., vol 118, pp 273-295, 1992 + 18 ) + 18 ) [2] T Kato, "Strong Lp solutions of the Navier-Stokes equation in Rm , with applications to weak solutions", Math Z., vol 187, pp 471-480, 1984 t−α1 C1 −α1 t for t ≥ t1 , [9] H Sohr, The Navier-Stokes Equations An Elementary Functional Analytic Approach, Birkhă auser Advanced Texts, Birkhăauser Verlag, Basel, 2001 solution and its decay properties for the Navier-Stokes equations in three dimensional domains with non-compact boundaries", Mathematische Zeitschrift, vol 216, pp 1-30, 1994 [10] R Farwig, H Kozono, and H Sohr, "An Lq -approach to Stokes and Navier–Stokes equations in general domains", Acta Math, [12] J Serrin, The initial value problem for the Navier-Stokes equations, Univ Wisconvol 195, pp 21-53, 2005 sin Press, Nonlinear problems, Ed R E [11] H Kozono, T Ogawa, "Global strong Langer, 1963 ... of solutions to the Navier- Stokes equations" , Commun In Partial Diff Eq., vol 11, pp 733-763, 1986 Proof of Theorem 1.3 [5] M Wiegner, "Decay results for weak solutions of the Navier- Stokes equations. .. -approach to Stokes and Navier Stokes equations in general domains" , Acta Math, [12] J Serrin, The initial value problem for the Navier- Stokes equations, Univ Wisconvol 195, pp 21-53, 2005 sin Press,... → ∞ jection Let the Stokes operator of weak solutions of the Navier- Stokes equations in space L2 (Ω), which goes to zero with explicit rates The L2 -decay problem for Navier- Stokes system was