On Ladyzhenskaya-Serrin condition sufficient for regular solutions to the Navier-Stokes equations. Periodic case
aa r X i v : . [ m a t h . G M ] S e p On Ladyzhenskaya-Serrin condition sufficientfor regular solutions to the Navier-Stokesequations. Periodic case
Wojciech M. Zaj¸aczkowski
Institute of Mathematics, Polish Academy of Sciences,´Sniadeckich 8, 00-656 Warsaw, Polande-mail:[email protected] of Mathematics and Cryptology, Cybernetics Faculty,Military University of Technology,S. Kaliskiego 2, 00-908 Warsaw, Poland
Abstract
We consider the Navier-Stokes equations in a bounded domainwith periodic boundary conditions. Let V = V ( x, t ) be the velocityof the fluid. The aim of this paper is to prove the bound k V ( t ) k H ≤ c for any t ∈ R + , where c depends on data. The proof is divided intotwo steps. In the first step the Lam´e system with a special versionof the convective term is considered. The system has two viscosities.Assuming that the second viscosity (the bulk one) is sufficiently largewe are able to prove the existence of global regular solutions to thissystem. The proof is divided into two steps. First the long timeexistence in interval (0 , T ) is proved, where T is proportional to thebulk viscosity. Having the bulk viscosity large we are able to showthat data at time T are sufficiently small. Then by the small data ar-guments a global existence follows. In this paper we are restricted toderive appropriate estimates only. To prove the existence we shoulduse the method of successive approximations and the continuationargument. Let v be a solution to it. In the second step we consider aproblem for u = v − V . Assuming that k u k H at t = 0 is sufficientlysmall we show that k u ( t ) k H is also sufficiently small for any t ∈ R + .Estimates for v and u in H imply estimate for k V ( t ) k H for any t ∈ R + . Key Words: Navier-Stokes equations, periodic boundary conditions, stabil-ity arguments, Ladyzhenskaya-Serrin condition1
Z132 — − − SC 2010: 35Q30, 35B35, 35D35, 35K40, 76D03, 76D05
We consider the Navier-Stokes equations with the periodic boundary con-ditions(1.1) V t + V · ∇ V − µ ∆ V + ∇ P = 0 , div V = 0 ,V | t =0 = V , where V = ( V ( x, t ) , V ( x, t ) , V ( x, t )) ∈ R is the velocity of the fluid, x =( x , x , x ) are the Cartesian coordinates, P = P ( x, t ) ∈ R is the pressure, µ > V ∈ H . Then we take a function v froma sufficiently small neighborhood of V in H . Next, we assume that v are initial data for solutions to the following problem with the periodicboundary conditions(1.2) v t + v H · ∇ v − µ ∆ v − ν ∇ div v = 0 ,v | t =0 = v , where v H is the divergence free part of v derived by the Helmholtz decom-position.Moreover, µ , ν are positive constant coefficients. We have to emphasizethat µ in (1.1) and in (1.2) are the same.However, v belongs to some neighborhood of V in H we shall needthat it is more regular. The regularity will be derived later.We have to emphasize that system (1.2) appeared from discussion be-tween the author and Piotr Mucha which we had during our stay in PragueNov. 19–23, 2019. We have to mention that system (1.2) replaces a systemof weakly compressible Navier-Stokes equations (density is close to a con-stant, the second viscosity is sufficiently large so the gradient part of velocityis sufficienty small) named by WCNSE and egzamined in [Z1, Z3, Z5]. TheWCNSE system was used in [Z2, Z4, Z6] to prove regularity of the weaksolutions to the Navier-Stokes equations. WCNSE is a physical system.2 Z132 — − − he proofs of existence of global regular solutions to WCNSE in [Z1, Z3,Z5] are very complicated. However, system (1.2) does not have a physicalmeaning it strongly simplifies considerations in [Z1]–[Z6].The crucial point of estimates in [Z1, Z3, Z5] is the estimate for velocity v in L ∞ (0 , T ; L (Ω)). Since we have three different proofs the three differentpapers are written.To simplify considerations we assume that(1.3) Z Ω v dx = 0 . Periodic boundary conditions in (1.2) and (1.3) imply that(1.4) Z Ω vdx = 0 . Moreover, (1.3) and definition of v give the restriction(1.5) Z Ω V dx = 0 . In view of (1.4) and the structure of system (1.2) we are looking for solutionsto (1.2) in the form(1.6) v = ∇ ϕ + rot ψ, where ϕ , ψ are periodic functions. In this case v H = rot ψ and problem(1.2) takes the form(1.7) v t + rot ψ · ∇ v − µ ∆ v − ν ∇ div v = 0 v | t =0 = v . Hence, problem (1.7) is treated as an auxiliary problem which helps us toprove regularity of solutions to the Navier-Stokes equations.To derive a global a priori estimate guaranteeing the existence of regularsolutions to problem (1.1) we have to linearize the Navier-Stokes equations.For this purpose we use solutions to problem (1.7) and derive a problem forthe difference(1.8) u = v − V which has the following form(1.9) u t + rot ψ · ∇ u + ( u − ∇ ϕ ) · ∇ ( v − u ) − µ ∆ u − ν ∇ div u − ∇ P = 0 ,u | t =0 = v − V ≡ u , where we used (1.6). 3 Z132 — − − owever, problem (1.9) is not linear it implies that sufficient smallness of k u k H (Ω) gives that k u ( t ) k H (Ω) is controlled for all time. This is possiblethanks to sufficient regularity of v and sufficiently large ν . Remark 1.1.
To justify (1.6) we have to find elliptic problems implyingexistence of potentials ϕ and ψ . Let v be given. Let ϕ , ψ satisfy the periodicboundary conditions. Then ϕ and ψ are solutions to the following ellipticproblems(1.10) ∆ ϕ = div v,ϕ satisfies the periodic boundary conditions , Z Ω ϕdx = 0and(1.11) rot ψ = rot v, div ψ = 0 ,ψ satisfies the periodic boundary conditions , Z Ω ψdx = 0 . Now we collect results of this paper. The used notation is described inNotations 1.6 and 1.7.From Theorem 3.6 and Lemma 4.1 we have
Theorem 1.2.
Consider problem (1.7). Assume that v is expressed in theform (1.6), where potentials ϕ , ψ are solutions to problems (1.10), (1.11),respectively. Assume that ∇ ϕ (0) ∈ Γ (Ω) , rot ψ (0) ∈ Γ (Ω) and the quan-tity √ ν k∇ ϕ (0) k Γ (Ω) + k rot ψ (0) k Γ (Ω) is bounded. Assume that there ex-ist constants c , c such that c /ν κ ≤ k ϕ (0) k L p (Ω) ≤ c /ν κ , p ∈ (3 , , κ = 3 / − /p ∈ (1 / , . Moreover, | ϕ (0) | ≤ c /ν κ .Then for ν sufficiently large there exists a constant A satisfying (3.54) suchthat solutions to problem (1.7) satisfy the bound (1.12) √ ν k∇ ϕ ( t ) k Γ (Ω) + k rot ψ ( t ) k Γ (Ω) + ν ( √ ν k∇ ϕ k L (0 ,t ;Γ (Ω)) + k rot ψ k L (0 ,t ;Γ (Ω)) ) + ν k∇ ϕ k L (0 ,t ;Γ (Ω)) ≤ A, where t ≤ T ≤ ν β , β < − κ ) . If cA ≤ µT , where c is the constant from(4.13), then (1.13) k∇ ϕ ( T ) k Γ (Ω) + k rot ψ ( T ) k Γ (Ω) ≤ k∇ ϕ (0) k Γ (Ω) + k rot ψ (0) k Γ (Ω) . Z132 — − − oreover, it is shown that (1.14) k ϕ ( T ) k + k v ( T ) k ≤ c ( e − µT | v (0) | + A/ √ ν ) ≡ cB. Then for T , ν sufficiently large the following inequality holds (1.15) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT | + k∇ ϕ k , , Ω tT ≤ cB, for any t ∈ ( T, ∞ ) . Theorem 1.3.
Consider problem (1.9) with coefficients dependent on so-lutions to problem (1.7). Let the assumptions of Theorem 1.2 hold. Let u (0) ∈ H (Ω) and k u (0) k H (Ω) ≤ γ , where γ is sufficiently small. Then, for ν sufficiently large, we have (1.16) k u ( t ) k H (Ω) ≤ γ for any t ∈ R + . Theorem 1.4.
Let the assumptions of Theorems 1.2 and 1.3 hold. Then,by (1.12), (1.15) and (1.16), we have (1.17) k V k L ∞ ( R + ; H (Ω)) ≤ c ( A + γ ) . Remark 1.5.
The aim of this paper is to prove estimate (1.17) for solutionsto the Navier-Stokes equations (1.1). We do not know how (1.17) can beproved directly for solutions to (1.1). Therefore, we consider first problem(1.7), where v is used in form (1.6). Since (1.7) is considered with largeparameter ν decomposition (1.6) implies smallness of ϕ in Γ (Ω) if ϕ (0)in Γ (Ω) is small too. Next, for ν sufficiently large, we are able to provethe global existence of solutions to (1.7) without any restrictions on themagnitude of ψ in suitable norms. It is important for comparison V with v because the magnitude of divergence free V can be restricted by dataonly. This implies that function u defined by (1.8) and satifying (1.9) canbe show small in H (Ω) if the difference rot ψ | t =0 − V | t =0 in H (Ω) and k∇ ϕ (0) k ≤ c/ν are small too. We have to mention that ν is large butalways finite. Any passage with ν to infinity makes the considerations inthis paper wrong.The first step in the proof of global existence of solutions to problem(1.7) is derivation of estimate (1.12). The proof of (1.12) is divided intothree main steps. First we derive inequality (2.27) of the form(1.18) | v | , ∞ , Ω t ≤ D , where D is an increasing positive function of Ψ /ν α , α > Z132 — −−
The aim of this paper is to prove estimate (1.17) for solutionsto the Navier-Stokes equations (1.1). We do not know how (1.17) can beproved directly for solutions to (1.1). Therefore, we consider first problem(1.7), where v is used in form (1.6). Since (1.7) is considered with largeparameter ν decomposition (1.6) implies smallness of ϕ in Γ (Ω) if ϕ (0)in Γ (Ω) is small too. Next, for ν sufficiently large, we are able to provethe global existence of solutions to (1.7) without any restrictions on themagnitude of ψ in suitable norms. It is important for comparison V with v because the magnitude of divergence free V can be restricted by dataonly. This implies that function u defined by (1.8) and satifying (1.9) canbe show small in H (Ω) if the difference rot ψ | t =0 − V | t =0 in H (Ω) and k∇ ϕ (0) k ≤ c/ν are small too. We have to mention that ν is large butalways finite. Any passage with ν to infinity makes the considerations inthis paper wrong.The first step in the proof of global existence of solutions to problem(1.7) is derivation of estimate (1.12). The proof of (1.12) is divided intothree main steps. First we derive inequality (2.27) of the form(1.18) | v | , ∞ , Ω t ≤ D , where D is an increasing positive function of Ψ /ν α , α > Z132 — −− −−
The aim of this paper is to prove estimate (1.17) for solutionsto the Navier-Stokes equations (1.1). We do not know how (1.17) can beproved directly for solutions to (1.1). Therefore, we consider first problem(1.7), where v is used in form (1.6). Since (1.7) is considered with largeparameter ν decomposition (1.6) implies smallness of ϕ in Γ (Ω) if ϕ (0)in Γ (Ω) is small too. Next, for ν sufficiently large, we are able to provethe global existence of solutions to (1.7) without any restrictions on themagnitude of ψ in suitable norms. It is important for comparison V with v because the magnitude of divergence free V can be restricted by dataonly. This implies that function u defined by (1.8) and satifying (1.9) canbe show small in H (Ω) if the difference rot ψ | t =0 − V | t =0 in H (Ω) and k∇ ϕ (0) k ≤ c/ν are small too. We have to mention that ν is large butalways finite. Any passage with ν to infinity makes the considerations inthis paper wrong.The first step in the proof of global existence of solutions to problem(1.7) is derivation of estimate (1.12). The proof of (1.12) is divided intothree main steps. First we derive inequality (2.27) of the form(1.18) | v | , ∞ , Ω t ≤ D , where D is an increasing positive function of Ψ /ν α , α > Z132 — −− −− nequality (1.18) follows from (2.2) and (1.19), where we needed that c /ν κ ≤ | ϕ (0) | p ≤ c /ν κ , c < c , κ ∈ (1 / , p ∈ (3 , D depends on somenorms of v multiplied by Ψ /ν α . Moreover, t ≤ ν β , β < − κ ) yields thattime t does not appear in D .In the next step we derive inequality (2.28) of the form(1.19) | v t ( t ) | + k v t k , , Ω t ≤ D , where D is an increasing function of D .Denote the l.h.s. of (1.12) by X ( t ).In the third step by applying the energy method and (1.18), (1.19) we deriveinequality (3.51) of the form(1.20) X ≤ φ ( D ( X ) , D ( X ) , X/ν, X/ √ ν, X (0)) , where φ is an increasing positive function of polynomial type. In (1.20) itis assumed that the quantity(1.21) X (0) = √ ν k∇ ϕ (0) k Γ (Ω) + k rot ψ (0) k Γ (Ω) is finite. This implies the foolowing smallness condition(1.22) k∇ ϕ (0) k Γ (Ω) ≤ c/ √ ν. However, to show (1.18) we needed that(1.23) c /ν κ ≤ | ϕ (0) | p ≤ c /ν κ , κ ∈ (1 / , , p ∈ (3 , . We have to emphasize that the energy method does not work without (1.18),(1.19), because estimates can not be closed.Since ν is finite estimate (1.12) derived from (1.20) implies long timeexistence of solutions to problem (1.7). To prove global existence we derive(1.14), where B is small for large ν .This implies that initial data for problem (1.7) at time t = T are small.Hence the small data technique implies global existence of regular solutionsto (1.7) in ( T, ∞ ).We have to emphasize that we are restricted to derive appropriate es-timates only. Then the local solution can be proved by the method ofsuccessive approximations and the continuation argument implies globalexistence.Theorem 1.3 follows from Lemmas 5.1–5.4 by the stability argument whichworks for ν sufficiently large.Theorem 1.4 follows directly from Theorems 1.2 and 1.3.We use the simplified notation 6 Z132 — − − otation 1.6. k u k L p (Ω) = | u | p , k u k H s (Ω) = k u k s , k u k W sp (Ω) = k u k s,p ,H s (Ω) = W s (Ω) , k u k W sp (Ω) = (cid:18) X | α |≤ s Z Ω | D αx u ( x ) | p dx (cid:19) /p ,D αx = ∂ α x ∂ α x ∂ α x , | α | = α + α + α , s ∈ N , p ∈ [1 , ∞ ] , | u | k,l = l X i =0 k ∂ it u k k − i , | u | k,l,r, Ω t = (cid:18) t Z | u ( t ′ ) | rk,l dt ′ (cid:19) /r , | u | r,q, Ω t = (cid:18) t Z | u ( t ′ ) | qr dt ′ (cid:19) /q , r, q ∈ [1 , ∞ ] , k u k L r (0 ,t ; H s (Ω)) = k u k s,r, Ω t , k u k L r (0 ,t ; W sp (Ω)) = k u k s,p,r, Ω t , s ∈ N , r, p ∈ [1 , ∞ ] . Introduce the spaceΓ kl (Ω) = { u : | u | k,l < ∞} , l ≤ k, l, k ∈ N = N ∪ { } . For t fixed we have k u ( t ) k Γ kl (Ω) = k u ( t ) k H k (Ω) + l X i =1 k ∂ it u ( t ) | t = t k H k − i (Ω) . By φ , φ σ , σ ∈ N , we always denote increasing positive continuous functionsof their argumens.Finally, we introduce some notation Notation 1.7. Ψ( t ) = ν |∇ ϕ | , , , Ω t χ = √ ν |∇ ϕ | , , ∞ , Ω t ,X ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , + µ ( ν |∇ ϕ | , , , Ω t + | rot ψ | , , , Ω t ) + ν |∇ ϕ | , , , Ω t ,X (0) = ν |∇ ϕ (0) | , + | rot ψ (0) | , ,Y ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , . Let Ω tT = Ω × ( T, t ), t > T . Let Ψ T , χ T , X T contain Ω tT instead of Ω t .Then X T ( T ) = ν |∇ ϕ ( T ) | , + | rot ψ ( T ) | , . Z132 — − − = | v (0) | , D = c Ψ / φ (Ψ / /ν / + 1 /ν ( κ − / / ) + c ( | v (0) | + A ) ,φ = exp (cid:20) c | v | / (1 − κ )2 p/ ( p − , / (1 − κ ) , Ω t [( µ + ν ) κ min t ( | ϕ (0) | p − √ tν Ψ)] / (1 − κ ) (cid:21) , p ∈ (3 , , κ = 3 / − /p ∈ (1 / , , D = | v t (0) | exp( D A ) . There is a huge literature concerning the regularity problem of weak solu-tions to the Navier-Stokes equations. Therefore we are not able to presentall papers devoted to this problem. Moreover, we are not able to close thelist of mathematicians trying to solve it. Hence, we concentrate the pre-sentation on some directions and recall mathematicians working in theseareas.1. Formulation of sufficient conditions guaranteeing regularity of weaksolutionsThe first who formulated such conditions was J. Serrin [S]. This ap-proach was continued by D. Chae, H.J. Choe, H. Kozono, H. Sohr, J.Neustupa, M. Pokorny, P. Penel and the references of their papers arecited in [Z8, Z9]. We have to recall results of G. Seregin, V. ˇSver´ak,T. Shilkin, A. Mikhaylov (see [S1, S2, S3, S4, SSS, SS1, ESS, MS]).2. Local regularity theory.The direction of examining regularity of weak solutions of the Navier-Stokes equations was initiated in the celebrated paper of L. Caffarelli,R. Kohn, L. Nirenberg (see [CKN]). The famous mathematiciansworking in this directions are G. Seregin [S1, S2, S3, S4], V. ˇSver´ak[SS1].3. Rotating Navier-Stokes equations.The existence of global regular solutions to the rotating Navier-Stokesequations was strongly examined by A. Babin, A. Mahalov, B. Nico-laenko (see [BMN1, BMN2, BMN3, MN]).4. Global regular solutions to the Navier-Stokes equations with somespecial properties. We have to distinguish the following directionsa. Thin domains (see [RS1, RS2, RS3]).b. Small variations of vorticity (see [CF]).c. Motions in cylindrical domains with data close to data of 2dsolutions (see [Z10, Z11, Z12, Z13, NZ]).d. Motions in axisymmetric domains with data close to data of ax-isymmetric solutions (see [Z3, Z4]).8
Z132 — − − Notation and auxiliary results
First we obtain the energy type estimate for solutions to (1.2).
Lemma 2.1.
Let v be a solution to (1.7). Assume that v (0) ∈ L (Ω) .Then the following estimate holds (2.1) | v ( t ) | + µ k v k , , Ω t + ν | ∆ ϕ | , Ω t ≤ c | v (0) | ≡ A . Proof.
Multiply (1.7) by v , integrate over Ω, integrate by parts, exploit(1.4), (1.6) and integrate with respect to time. Then we obtain (2.1) andconclude the proof. Lemma 2.2.
Let v be a solution to (1.7). Assume that v (0) ∈ L r (Ω) , r > .Let ν | div v | r/ ( r +1) ,r, Ω t be finite.Then there exists a constant c = c ( r, µ, c ) , where c is the constant fromimbedding (2.7), such that (2.2) | v ( t ) | r + | v | r,r, Ω t + |∇| v | r/ | /r , Ω t ≤ c ( r, µ, c )[ ν | div v | r/ ( r +1) ,r, Ω t + | v (0) | r + A ] where c = max { ¯ c, } and ¯ c is introduced in (2.9).Proof. Multiplying (1.7) by v | v | r − and integrating over Ω yield(2.3) 1 r ddt | v | rr + µ Z Ω ∇ v · ∇ ( v | v | r − ) dx + ν Z Ω div v div ( v | v | r − ) dx = 0 . First we consider J = µ Z Ω ∇ v · ∇ ( v | v | r − ) dx = µ Z Ω |∇ v | | v | r − dx + µ Z Ω v k ∇ v k ( r − | v | r − ∇| v | dx ≡ I + I . Using that v k ∇ v k = ∇| v | = | v |∇| v | we have I = ( r − µ Z Ω | v | r − |∇| v | | dx = ( r − µ Z Ω | | v | r − ∇| v | | dx = 4( r − µr Z Ω |∇| v | r/ | dx. Z132 — − − o examine I we use the formula(2.4) |∇ u | = | u | (cid:12)(cid:12)(cid:12)(cid:12) ∇ u | u | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| u | | . Then I takes the form I = µ Z Ω (cid:20) | v | (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| v | | (cid:21) | v | r − dx = µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + 4 µr Z Ω |∇| v | r/ | dx. Hence, J = 4( r − r µ Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx. Next, we consider J = ν Z Ω div v div ( v | v | r − ) dx = ν Z Ω | div v | | v | r − dx + ν Z Ω div vv · ∇| v | r − dx ≡ I + I , where | I | ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx. Employing the above expressions in (2.3) one gets(2.5) 1 r ddt | v | rr + 4( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx = ν ( r − Z Ω | div v | | v | r − | v | r − |∇| v | | dx ≤ ε Z Ω | | v | r − ∇| v | | dx + ν ( r − ε Z Ω | div v | | v | r − dx = 2 εr Z Ω |∇| v | r/ | dx + ν ( r − ε Z Ω | div v | | v | r − dx. Z132 — − − etting ε = ( r − µ we obtain from (2.5) the inequality(2.6) 1 r ddt | v | rr + 2( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − r − µ Z Ω | div v | | v | r − dx. Consider the second integral on the l.h.s. of (2.6). Omitting the coefficientwe write it in the form I ≡ Z Ω |∇| v | r/ | dx = 13 |∇| v | r/ | + 13 |∇| v | r/ | + 13 |∇| v | r/ | . Let u = | v | r/ . Then we use the Poincar´e inequality Z Ω (cid:12)(cid:12)(cid:12)(cid:12) u − — Z u (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ c p |∇ u | . Hence | u | ≤ c p |∇ u | + (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω udx (cid:12)(cid:12)(cid:12)(cid:12) . Recalling that u = | v | r/ we have13 |∇| v | r/ | ≥ c p (cid:18) | v | rr − (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . Therefore I ≥ c p | v | rr + 16 c p | v | rr + 13 |∇| v | r/ | + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Let ¯ c = min (cid:0) c P , (cid:1) . Next we use the inequality(2.7) ¯ c | v | r r ≤ |∇| v | r/ | + | v | rr . Then we have I ≥ c p | v | rr + ¯ c ¯ c | v | r + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Z132 — −−
Let v be a solution to (1.7). Assume that v (0) ∈ L r (Ω) , r > .Let ν | div v | r/ ( r +1) ,r, Ω t be finite.Then there exists a constant c = c ( r, µ, c ) , where c is the constant fromimbedding (2.7), such that (2.2) | v ( t ) | r + | v | r,r, Ω t + |∇| v | r/ | /r , Ω t ≤ c ( r, µ, c )[ ν | div v | r/ ( r +1) ,r, Ω t + | v (0) | r + A ] where c = max { ¯ c, } and ¯ c is introduced in (2.9).Proof. Multiplying (1.7) by v | v | r − and integrating over Ω yield(2.3) 1 r ddt | v | rr + µ Z Ω ∇ v · ∇ ( v | v | r − ) dx + ν Z Ω div v div ( v | v | r − ) dx = 0 . First we consider J = µ Z Ω ∇ v · ∇ ( v | v | r − ) dx = µ Z Ω |∇ v | | v | r − dx + µ Z Ω v k ∇ v k ( r − | v | r − ∇| v | dx ≡ I + I . Using that v k ∇ v k = ∇| v | = | v |∇| v | we have I = ( r − µ Z Ω | v | r − |∇| v | | dx = ( r − µ Z Ω | | v | r − ∇| v | | dx = 4( r − µr Z Ω |∇| v | r/ | dx. Z132 — − − o examine I we use the formula(2.4) |∇ u | = | u | (cid:12)(cid:12)(cid:12)(cid:12) ∇ u | u | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| u | | . Then I takes the form I = µ Z Ω (cid:20) | v | (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| v | | (cid:21) | v | r − dx = µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + 4 µr Z Ω |∇| v | r/ | dx. Hence, J = 4( r − r µ Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx. Next, we consider J = ν Z Ω div v div ( v | v | r − ) dx = ν Z Ω | div v | | v | r − dx + ν Z Ω div vv · ∇| v | r − dx ≡ I + I , where | I | ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx. Employing the above expressions in (2.3) one gets(2.5) 1 r ddt | v | rr + 4( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx = ν ( r − Z Ω | div v | | v | r − | v | r − |∇| v | | dx ≤ ε Z Ω | | v | r − ∇| v | | dx + ν ( r − ε Z Ω | div v | | v | r − dx = 2 εr Z Ω |∇| v | r/ | dx + ν ( r − ε Z Ω | div v | | v | r − dx. Z132 — − − etting ε = ( r − µ we obtain from (2.5) the inequality(2.6) 1 r ddt | v | rr + 2( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − r − µ Z Ω | div v | | v | r − dx. Consider the second integral on the l.h.s. of (2.6). Omitting the coefficientwe write it in the form I ≡ Z Ω |∇| v | r/ | dx = 13 |∇| v | r/ | + 13 |∇| v | r/ | + 13 |∇| v | r/ | . Let u = | v | r/ . Then we use the Poincar´e inequality Z Ω (cid:12)(cid:12)(cid:12)(cid:12) u − — Z u (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ c p |∇ u | . Hence | u | ≤ c p |∇ u | + (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω udx (cid:12)(cid:12)(cid:12)(cid:12) . Recalling that u = | v | r/ we have13 |∇| v | r/ | ≥ c p (cid:18) | v | rr − (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . Therefore I ≥ c p | v | rr + 16 c p | v | rr + 13 |∇| v | r/ | + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Let ¯ c = min (cid:0) c P , (cid:1) . Next we use the inequality(2.7) ¯ c | v | r r ≤ |∇| v | r/ | + | v | rr . Then we have I ≥ c p | v | rr + ¯ c ¯ c | v | r + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Z132 — −− −−
Let v be a solution to (1.7). Assume that v (0) ∈ L r (Ω) , r > .Let ν | div v | r/ ( r +1) ,r, Ω t be finite.Then there exists a constant c = c ( r, µ, c ) , where c is the constant fromimbedding (2.7), such that (2.2) | v ( t ) | r + | v | r,r, Ω t + |∇| v | r/ | /r , Ω t ≤ c ( r, µ, c )[ ν | div v | r/ ( r +1) ,r, Ω t + | v (0) | r + A ] where c = max { ¯ c, } and ¯ c is introduced in (2.9).Proof. Multiplying (1.7) by v | v | r − and integrating over Ω yield(2.3) 1 r ddt | v | rr + µ Z Ω ∇ v · ∇ ( v | v | r − ) dx + ν Z Ω div v div ( v | v | r − ) dx = 0 . First we consider J = µ Z Ω ∇ v · ∇ ( v | v | r − ) dx = µ Z Ω |∇ v | | v | r − dx + µ Z Ω v k ∇ v k ( r − | v | r − ∇| v | dx ≡ I + I . Using that v k ∇ v k = ∇| v | = | v |∇| v | we have I = ( r − µ Z Ω | v | r − |∇| v | | dx = ( r − µ Z Ω | | v | r − ∇| v | | dx = 4( r − µr Z Ω |∇| v | r/ | dx. Z132 — − − o examine I we use the formula(2.4) |∇ u | = | u | (cid:12)(cid:12)(cid:12)(cid:12) ∇ u | u | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| u | | . Then I takes the form I = µ Z Ω (cid:20) | v | (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) + |∇| v | | (cid:21) | v | r − dx = µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + 4 µr Z Ω |∇| v | r/ | dx. Hence, J = 4( r − r µ Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx. Next, we consider J = ν Z Ω div v div ( v | v | r − ) dx = ν Z Ω | div v | | v | r − dx + ν Z Ω div vv · ∇| v | r − dx ≡ I + I , where | I | ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx. Employing the above expressions in (2.3) one gets(2.5) 1 r ddt | v | rr + 4( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − Z Ω | div v | | v | r − |∇| v | | dx = ν ( r − Z Ω | div v | | v | r − | v | r − |∇| v | | dx ≤ ε Z Ω | | v | r − ∇| v | | dx + ν ( r − ε Z Ω | div v | | v | r − dx = 2 εr Z Ω |∇| v | r/ | dx + ν ( r − ε Z Ω | div v | | v | r − dx. Z132 — − − etting ε = ( r − µ we obtain from (2.5) the inequality(2.6) 1 r ddt | v | rr + 2( r − µr Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − r − µ Z Ω | div v | | v | r − dx. Consider the second integral on the l.h.s. of (2.6). Omitting the coefficientwe write it in the form I ≡ Z Ω |∇| v | r/ | dx = 13 |∇| v | r/ | + 13 |∇| v | r/ | + 13 |∇| v | r/ | . Let u = | v | r/ . Then we use the Poincar´e inequality Z Ω (cid:12)(cid:12)(cid:12)(cid:12) u − — Z u (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ c p |∇ u | . Hence | u | ≤ c p |∇ u | + (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω udx (cid:12)(cid:12)(cid:12)(cid:12) . Recalling that u = | v | r/ we have13 |∇| v | r/ | ≥ c p (cid:18) | v | rr − (cid:12)(cid:12)(cid:12)(cid:12) — Z Ω | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . Therefore I ≥ c p | v | rr + 16 c p | v | rr + 13 |∇| v | r/ | + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Let ¯ c = min (cid:0) c P , (cid:1) . Next we use the inequality(2.7) ¯ c | v | r r ≤ |∇| v | r/ | + | v | rr . Then we have I ≥ c p | v | rr + ¯ c ¯ c | v | r + 13 |∇| v | r/ | − c p (cid:12)(cid:12)(cid:12)(cid:12) — Z | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) . Z132 — −− −− inally, we estimate the expression (cid:12)(cid:12)(cid:12)(cid:12) Z Ω | v | r/ dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω | v | r/ − µ | v | µ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω | v | r − µ dx Z Ω | v | µ dx ≤ | Ω | µr (cid:18) Z Ω | v | r dx (cid:19) ( r − µ ) /r Z Ω | v | µ dx ≡ J, where | Ω | = volume of Ω. Setting µ = 1 we have J = | Ω | r | v | r − r | v | ≤ ε | v | rr + c/ε | Ω | | v | r . Using the estimates in the lower bound for I we obtain for sufficiently small ε the inequality I ≥ c p | v | rr + ¯ c ¯ c | v | r r + 13 |∇| v | r/ | − c | v | r ≡ c | v | rr + c | v | r r + 13 |∇| v | r/ | − c | v | r . Then (2.6) takes the form(2.8) 1 r ddt | v | rr + ( r − µr c | v | rr + ( r − µc r | v | r r + ( r − r Z Ω |∇| v | r/ | dx + µ Z Ω | v | r (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) dx + ν Z Ω | div v | | v | r − dx ≤ ν ( r − r − µ | div v | r/ ( r +1) | v | r − r + c | v | r ≤ ε r/ ( r − r/ ( r − | v | r r + 1 ε r/ r/ (cid:18) ν ( r − r − µ (cid:19) r/ | div v | r r/ ( r +1) + c | v | r . Setting ε r/ ( r − r/ ( r −
2) = ( r − µc r we have that ε = (cid:18) ( r − µc r − r (cid:19) ( r − /r . Then the coefficient near | div v | r r/ ( r +1) from the r.h.s. of (2.8) equals(2.9) ( r − (cid:20) r − r − µ (cid:21) r − · r r/ − c r/ − ν r ≡ ¯ c ( r, µ, c ) ν r . Z132 — − − sing the above expressions in the r.h.s. of (2.8) and integrating the resultwith respect to time we obtain(2.10) 1 r | v ( t ) | rr + ( r − µc r | v | r r,r, Ω t + ( r − µ r |∇| v | r/ | , Ω t + µ (cid:12)(cid:12)(cid:12)(cid:12) | v | r/ (cid:12)(cid:12)(cid:12)(cid:12) ∇ v | v | (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) , Ω t + ν (cid:12)(cid:12)(cid:12)(cid:12) | div v | | v | r − (cid:12)(cid:12)(cid:12)(cid:12) , Ω t ≤ ¯ cν r | div v | r r/ ( r +1) ,r, Ω t + 1 r | v (0) | rr + A r , where we used that t Z | v | r dt ′ ≤ | v | r − , ∞ , Ω t | v | , Ω t ≤ A r . Inequality (2.10) implies (2.2). This concludes the proof.
Remark 2.3.
The first term on the r.h.s. of (2.2) equals | ∆ ϕ | r/ ( r +1) ,r, Ω t .To examine it we first consider the interpolation(2.11) | ∆ ϕ | r/ ( r +1) ≤ c |∇ ∆ ϕ | θ |∇ ϕ | − θ , where θ = 3 / − / r , 1 − θ = 1 / / r . Therefore, we have(2.12) | ∆ ϕ | r/ ( r +1) ,r, Ω t ≤ c |∇ ∆ ϕ | / − / r , ∞ , Ω t (cid:18) t Z |∇ ϕ ( t ′ ) | (1 / / r ) r dt ′ (cid:19) /r , where (1 / / r ) r ≤ r ≤
6. For r = 6 (2.12) takes the form(2.13) | ∆ ϕ | / , , Ω t ≤ c |∇ ∆ ϕ | / , ∞ , Ω t |∇ ϕ | / , Ω t . Introduce the quantity(2.14) Ψ = ν |∇ ϕ | , , , Ω t . Using the imbeddings k ∆ ϕ k , , Ω t ≤ k∇ ϕ k , , Ω t ≤ |∇ ϕ | , , , Ω t , | ∆ ϕ t | , Ω t ≤ k ∆ ϕ t k , , Ω t ≤ |∇ ϕ | , , , Ω t , |∇ ∆ ϕ | , ∞ , Ω t ≤ c k ∆ ϕ k W , (Ω t ) ≤ c |∇ ϕ | , , , Ω t we obtain from (2.13) the inequality(2.15) ν | ∆ ϕ | / , , Ω t ≤ c Ψ / ν / |∇ ϕ | / , Ω t ≡ I. Z132 — − − ur aim is to find the estimate(2.16) I ≤ c X α,β Ψ α ν β , where α , β are positive constants. This ends the Remark.To show (2.16) we derive from (1.2) the equation(2.17) ∆ ϕ t − ( µ + ν )∆ ϕ = − div (rot ψ · ∇ v ) . Applying the operator ∆ − we obtain(2.18) ϕ t − ( µ + ν )∆ ϕ = − ∆ − ∂ x i ∂ x j ( v i v j ) + ∆ − ∂ x i ∂ x j ( ϕ x i v j ) ,ϕ | t =0 = ϕ (0) . Lemma 2.4.
Assume that v ∈ L p/ ( p − , / (1 − κ ) (Ω t ) , κ = 3 / − /p ∈ (1 / , , p ∈ (3 , , v (0) ∈ L (Ω) , ϕ (0) ∈ L p (Ω) . Assume that there ex-ist positive constants c , c , c such that ( µ + ν ) κ (cid:18) | ϕ (0) | p − √ tν Ψ (cid:19) ≥ c > , c ν κ ≤ | ϕ (0) | p ≤ c ν κ . Then (2.19) | ϕ ( t ) | + ( µ + ν ) |∇ ϕ | , Ω t ≤ exp (cid:20) ( c | v | / (1 − κ )2 p/ ( p − , / (1 − κ ) , Ω t ) / (cid:20) ( µ + ν ) κ min t (cid:18) | ϕ (0) | p − √ tν Ψ (cid:19)(cid:21) / (1 − κ ) (cid:21) ·· (cid:20) c Ψ ν + c ν κ (cid:21) ≡ φ · (cid:18) c Ψ ν + c ν κ (cid:19) and |∇ ϕ | , Ω t ≤ φ · (cid:18) c Ψ ν + c ν κ +1 (cid:19) where κ + 1 > because κ > / and φ = exp (cid:20) c | v | / (1 − κ )2 p/ ( p − , / (1 − κ ) , Ω t [( µ + ν ) κ min t ( | ϕ (0) | p − √ tν Ψ)] / (1 − κ ) (cid:21) . Proof.
Multiplying (2.18) by ϕ and integrating over Ω yield(2.20) 12 ddt | ϕ | + ( µ + ν ) |∇ ϕ | = Z Ω ( − ∆ − ∂ x i ∂ x j ( v i v j ) ϕ ) dx + Z Ω ∆ − ∂ x i ∂ x j ( ϕ x i v j ) ϕdx ≡ Z Ω ¯ D ϕdx + Z Ω ¯ D ϕdx ≡ I + I . Z132 — −−
Multiplying (2.18) by ϕ and integrating over Ω yield(2.20) 12 ddt | ϕ | + ( µ + ν ) |∇ ϕ | = Z Ω ( − ∆ − ∂ x i ∂ x j ( v i v j ) ϕ ) dx + Z Ω ∆ − ∂ x i ∂ x j ( ϕ x i v j ) ϕdx ≡ Z Ω ¯ D ϕdx + Z Ω ¯ D ϕdx ≡ I + I . Z132 — −− −−
Multiplying (2.18) by ϕ and integrating over Ω yield(2.20) 12 ddt | ϕ | + ( µ + ν ) |∇ ϕ | = Z Ω ( − ∆ − ∂ x i ∂ x j ( v i v j ) ϕ ) dx + Z Ω ∆ − ∂ x i ∂ x j ( ϕ x i v j ) ϕdx ≡ Z Ω ¯ D ϕdx + Z Ω ¯ D ϕdx ≡ I + I . Z132 — −− −− irst we examine I . By the H¨older inequality we have | I | ≤ | ¯ D | p/ ( p − | ϕ | p = | ¯ D | p/ ( p − | ϕ | p | ϕ | p ≡ I , where 1 < p <
6. Let α = | ¯ D | p/ ( p − | ϕ | p . Then we use the interpolation I / = α / | ϕ | p ≤ α / ( ε / κ |∇ ϕ | + cε − / (1 − κ ) | ϕ | ) , where κ = 3 / − /p , p ∈ (3 , ε / κ α / = (cid:0) µ + νk (cid:1) / , k ∈ N , wehave ε = (cid:0) µ + ναk (cid:1) κ / . Then α / ε − / (1 − κ ) = ( αk κ ) / − κ ) ( µ + ν ) κ / − κ ) . Therefore,(2.21) I ≤ k ( µ + ν ) |∇ ϕ | + c ( k κ α ) / (1 − κ ) ( µ + ν ) κ / (1 − κ ) | ϕ | = 1 k ( µ + ν ) |∇ ϕ | + ck κ / (1 − κ ) | ¯ D | / (1 − κ ) p/ ( p − [( µ + ν ) κ | ϕ | p ] / (1 − κ ) | ϕ | , where | ¯ D | q ≤ c P i,j =1 | v i v j | q for any q ∈ (1 , ∞ ).By the H¨older and Young inequalities we get(2.22) | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω ∆ − ∂ x j ( ϕ x i v j ) ϕ x i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ + νk |∇ ϕ | + ckµ + ν X i,j =1 | ∆ − ∂ x j ( ϕ x i v j ) | ≤ µ + νk |∇ ϕ | + ckµ + ν X i,j =1 k ∆ − ∂ x j ( ϕ x i v j ) k , / ≤ µ + νk |∇ ϕ | + ckµ + ν X i,j =1 ( |∇ ∆ − ∂ x j ( ϕ x i v j ) | / + | ∆ − ∂ x j ( ϕ x i v j ) | / ) ≤ µ + νk |∇ ϕ | + ckµ + ν X i,j =1 |∇ ∆ − ∂ x j ( ϕ x i v j ) | / ≤ µ + νk |∇ ϕ | + ckµ + ν X i,j =1 | ϕ x i v j | / ≤ µ + νk |∇ ϕ | + ckµ + ν | ϕ x | | v | , Z132 — − − here in the fourth inequality the Poincar´e inequality is used.Using (2.21) and (2.22) in (2.20), assuming that k = 4 and integrating theresult with respect to time yield(2.23) | ϕ ( t ) | + ( µ + ν ) |∇ ϕ | , Ω t ≤ exp (cid:18) | v | / (1 − κ )2 p/ ( p − , / (1 − κ ) , Ω t [( µ + ν ) κ min t | ϕ ( t ) | p ] / (1 − κ ) (cid:19) ·· (cid:20) cµ + ν | ϕ x | , , Ω t | v | , ∞ , Ω t + | ϕ (0) | (cid:21) . From Lemma 2.1 we have that | v | , ∞ , Ω t ≤ A . Moreover | ϕ x | , , Ω t ≤ Ψ ν . Toguarantee that the argument of exp is finite we consider ϕ ( t ) = ϕ (0) + t Z ϕ t ′ dt ′ so | ϕ ( t ) | p ≥ | ϕ (0) | p − (cid:12)(cid:12)(cid:12)(cid:12) t Z ϕ t ′ dt ′ (cid:12)(cid:12)(cid:12)(cid:12) p ≥ | ϕ (0) | p − t Z | ϕ t ′ | p dt ′ ≥ | ϕ (0) | p − √ tν Ψ . To have the argument of exp finite we assume existence of a positive constant c such that(2.24) ( µ + ν ) κ (cid:18) | ϕ (0) | p − √ tν Ψ (cid:19) ≥ c . Such constant exists at least for the local solution. Moreover, we need(2.25) c ν κ ≤ | ϕ (0) | p ≤ c ν κ , where c , c are positive constants. The above considerations imply (2.19)and conclude the proof. Remark 2.5.
Using (2.19) in (2.15) yields(2.26) ν | ∆ ϕ | / , , Ω t ≤ c Ψ / φ · (cid:18) Ψ / ν / + 1 ν ( κ − / / (cid:19) . Using (2.26) in (2.2) gives(2.27) | v ( t ) | + | v | , , Ω t + |∇| v | | / , Ω t ≤ c Ψ / φ · (Ψ / /ν / + 1 /ν ( κ − / / ) + c ( | v (0) | + A ) ≡ D , Z132 — − − here φ = exp (cid:20) c | v | / (1 − κ )2 p/ ( p − , / (1 − κ ) , Ω t [( µ + ν ) κ min t ( | ϕ (0) | p − √ tν Ψ)] / (1 − κ ) (cid:21) and c is defined in (2.2). Lemma 2.6.
Assume that v t (0) ∈ L (Ω) and the assumptions of Lemmas2.1 and 2.4 hold. Let D , defined by (2.27), be finite.Then (2.28) | v t ( t ) | + µ k v t k , , Ω t + ν | ∆ ϕ t | , Ω t ≤ | v t (0) | exp( D A ) ≡ D . Proof.
Differentiate (1.7) with respect to t , multiply by v t and integrateover Ω. Then we have(2.29) 12 ddt | v t | + µ |∇ v t | + ν | div v t | = − Z Ω rot ψ · ∇ v t · v t dx − Z Ω rot ψ t · ∇ v · v t dx ≡ J + J . Integration by parts implies that J = 0. Next | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇ v t · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ v t | + c/ε | rot ψ t | | v | . Using the estimate in (2.29) yields(2.30) ddt | v t | + µ k v t k + ν | ∆ ϕ t | ≤ c | rot ψ t | | v | ≤ c |∇ rot ψ t | | rot ψ t | | v | ≤ ε |∇ v t | + c/ε | rot ψ t | | v | Continuing, we have(2.31) ddt | v t | + µ k v t k + ν | ∆ ϕ t | ≤ c | v t | | v | . Integration with respect to time yields | v t ( t ) | + exp (cid:18) c t Z | v ( t ′ ) | dt ′ (cid:19) t Z ( µ k v t ′ ( t ′ ) k + ν | ∆ ϕ t ′ ( t ′ ) | ) ·· exp (cid:18) − c t ′ Z | v ( t ′′ ) | dt ′′ (cid:19) dt ′ ≤ | v t (0) | exp (cid:18) c t Z | v ( t ′ ) | dt ′ (cid:19) . Simplifying we get (2.28). This concludes the proof.17
Z132 — − − Estimates and existence
First we derive estimates for solutions to (1.7) by applying the energymethod.
Lemma 3.1.
Assume that A , D are finite, ∇ ϕ (0) , rot ψ (0) ∈ L (Ω) .Then (3.1) |∇ ϕ ( t ) | + 1 ν | rot ψ ( t ) | + µ (cid:18) |∇ ϕ | , Ω t + 1 ν |∇ rot ψ | , Ω t (cid:19) + ν | ∆ ϕ | , Ω t ≤ cν A D + |∇ ϕ (0) | + 1 ν | rot ψ (0) | . Proof.
Multiply (1.7) by ∇ ϕ and integrate over Ω. Then we have(3.2) 12 ddt |∇ ϕ | + Z Ω rot ψ · ∇ v · ∇ ϕdx + µ |∇ ϕ | + ν | ∆ ϕ | = 0 . Integration by parts in the second term yields (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · v · ∇ ϕdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ | + c/ε | v | | rot ψ | . Using this in (3.2) implies(3.3) ddt |∇ ϕ | + µ |∇ ϕ | + ν | ∆ ϕ | ≤ cν | rot ψ | | v | ≤ cν | rot ψ | D . Inegrating with respect to time gives(3.4) |∇ ϕ | + µ |∇ ϕ | , Ω t + ν | ∆ ϕ | , Ω t ≤ cν | rot ψ | , , Ω t D + |∇ ϕ (0) | . Multiply (1.7) by rot ψ and integrate over Ω. Then we obtain(3.5) 12 ddt | rot ψ | + µ |∇ rot ψ | = − Z Ω rot ψ · ∇ v · rot ψdx = Z Ω rot ψ · ∇ rot ψ · vdx and the r.h.s. is bounded by ε |∇ rot ψ | + c/ε | rot ψ | | v | . Z132 — − − sing this in (3.5) and integrating the result with respect to time we obtain(3.6) | rot ψ ( t ) | + µ |∇ rot ψ | , Ω t ≤ c | rot ψ | , , Ω t D + | rot ψ (0) | . Multiplying (3.6) by 1 /ν and adding to (3.4) give(3.7) |∇ ϕ ( t ) | + 1 ν | rot ψ ( t ) | + µ (cid:18) |∇ ϕ | , Ω t + 1 ν |∇ rot ψ | , Ω t (cid:19) + ν | ∆ ϕ | , Ω t ≤ cν | rot ψ | , , Ω t D + |∇ ϕ (0) | + 1 ν | rot ψ (0) | . In view of interpolation the first term on the r.h.s. of (3.7) is bounded by cν |∇ rot ψ | , Ω t | rot ψ | , Ω t D ≤ εν |∇ rot ψ | , Ω t + cνε | rot ψ | , Ω t D where the last expression is bounded by cνε | v | , Ω t D ≤ cνε A D , where Lemma 2.1 is exploited.Using the estimates in (3.7) we derive (3.1). This concludes the proof. Lemma 3.2.
Assume that ν ∈ (0 , ∞ ) , D , D , χ , Ψ are finite, ∇ ϕ t (0) , rot ψ t (0) ∈ L (Ω) . Then (3.8) |∇ ϕ t ( t ) | + 1 ν | rot ψ t ( t ) | + µ (cid:18) |∇ ϕ t | , Ω t + 1 ν |∇ rot ψ t | , Ω t (cid:19) + ν | ∆ ϕ t | , Ω t ≤ cν (cid:20)(cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) + D (cid:18) D + χ ν (cid:19)(cid:21) + |∇ ϕ t (0) | + 1 ν | rot ψ t (0) | ≡ cν φ (cid:18) D , D , χ √ ν , Ψ ν (cid:19) + |∇ ϕ t (0) | + 1 ν | rot ψ t (0) | . Proof.
Differentiate (1.7) with respect to time, multiply by ∇ ϕ t and inte-grate over Ω. Then we have(3.9) 12 ddt |∇ ϕ t | + µ |∇ ϕ t | + ν | ∆ ϕ t | = − Z Ω rot ψ · ∇ v t · ∇ ϕ t dx − Z Ω rot ψ t · ∇ v · ∇ ϕ t dx ≡ I + I . Z132 — −−
Differentiate (1.7) with respect to time, multiply by ∇ ϕ t and inte-grate over Ω. Then we have(3.9) 12 ddt |∇ ϕ t | + µ |∇ ϕ t | + ν | ∆ ϕ t | = − Z Ω rot ψ · ∇ v t · ∇ ϕ t dx − Z Ω rot ψ t · ∇ v · ∇ ϕ t dx ≡ I + I . Z132 — −− −−
Differentiate (1.7) with respect to time, multiply by ∇ ϕ t and inte-grate over Ω. Then we have(3.9) 12 ddt |∇ ϕ t | + µ |∇ ϕ t | + ν | ∆ ϕ t | = − Z Ω rot ψ · ∇ v t · ∇ ϕ t dx − Z Ω rot ψ t · ∇ v · ∇ ϕ t dx ≡ I + I . Z132 — −− −− ow we estimate the particular terms from the r.h.s. of (3.9). | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇ ( ∇ ϕ t + rot ψ t ) · ∇ ϕ t dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇ rot ψ t · ∇ ϕ t dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇∇ ϕ t · rot ψ t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ t | + c/ε | rot ψ t | | rot ψ | ≤ ε |∇ ϕ t | + c/ε | rot ψ t | (cid:18) D + χ ν (cid:19) , where we used that | rot ψ | = | rot ψ + ∇ ϕ − ∇ ϕ | ≤ | v | + |∇ ϕ | ≤ D + χ √ ν ,and | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ t · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ t | + c/ε | rot ψ t | D . Using the estimates in (3.9) and integrating the result with respect to timeyield(3.10) |∇ ϕ t ( t ) | + µ |∇ ϕ t | , Ω t + ν | ∆ ϕ t | , Ω t ≤ cν (cid:18) D + χ ν (cid:19) | rot ψ t | , , Ω t + |∇ ϕ t (0) | . Differentiate (1.7) with respect to time, multiply by rot ψ t and integrateover Ω. Then we obtain(3.11) 12 ddt | rot ψ t | + µ |∇ rot ψ t | = − Z Ω rot ψ · ∇ v t · rot ψ t dx − Z Ω rot ψ t · ∇ v · rot ψ t dx ≡ J + J , where | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇ rot ψ t · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ t | + c/ε | rot ψ | | v t | and | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇ rot ψ t · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ t | + c/ε | rot ψ t | | v | . Z132 — − − sing the estimates in (3.11), integrating the result with respect to timeand employing (2.27), Lemma 2.6 we obtain(3.12) | rot ψ t | + µ |∇ rot ψ t | , Ω t ≤ c | rot ψ | , ∞ , Ω t D + c | rot ψ t | , , Ω t D + | rot ψ t (0) | . Multiplying (3.12) by 1 /ν and adding to (3.10) yield(3.13) |∇ ϕ t ( t ) | + 1 ν | rot ψ t ( t ) | + µ (cid:18) |∇ ϕ t | , Ω t + 1 ν |∇ rot ψ t | , Ω t (cid:19) + ν | ∆ ϕ t | , Ω t ≤ cν (cid:18) D + χ ν (cid:19) | rot ψ t | , , Ω t + cν | rot ψ | , ∞ , Ω t D + |∇ ϕ t (0) | + 1 ν | rot ψ t (0) | . Employing the inequalities | rot ψ t | , , Ω t = | rot ψ t + ∇ ϕ t − ∇ ϕ t | , , Ω t ≤ c ( | v t | , , Ω t + |∇ ϕ t | , , Ω t ) ≤ c (cid:18) D + Ψ ν (cid:19) , | rot ψ | , ∞ , Ω t = | rot ψ + ∇ ϕ − ∇ ϕ | , ∞ , Ω t ≤ c ( | v | , ∞ , Ω t + |∇ ϕ | , ∞ , Ω t ) ≤ c (cid:18) D + χ ν (cid:19) in (3.13) implies (3.8) and concludes the proof. Lemma 3.3.
Assume that ν ∈ (0 , ∞ ) , A , D are finite, ∇ ϕ (0) , rot ψ (0) ∈ H (Ω) . Then (3.14) |∇ ϕ x ( t ) | + 1 ν | rot ψ x ( t ) | + µ (cid:18) |∇ ϕ x | , Ω t + 1 ν |∇ rot ψ x | , Ω t (cid:19) + ν | ∆ ϕ x | , Ω t ≤ cν A (cid:18) D + χ ν (cid:19) + cν D (cid:20) A + 1 ν A D + |∇ ϕ (0) | + 1 ν | rot ψ (0) | (cid:21) + |∇ ϕ x (0) | + 1 ν | rot ψ x (0) | ≡ cν φ (cid:18) A , D , |∇ ϕ (0) | + 1 ν | rot ψ (0) | (cid:19) + |∇ ϕ x (0) | + 1 ν | rot ψ x (0) | . Proof.
Differentiate (1.7) with respect to x , multiply by ∇ ϕ x and integrateover Ω. Then we obtain(3.15) 12 ddt |∇ ϕ x | + µ |∇ ϕ x | + ν | ∆ ϕ x | = − Z Ω rot ψ · ∇ v x · ∇ ϕ x dx − Z Ω rot ψ x · ∇ v · ∇ ϕ x dx ≡ I + I . Z132 — − − ntegration by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇∇ ϕ x · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ x | + c/ε | rot ψ | | v x | . Similarly, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ x · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ x | + c/ε | rot ψ x | | v | . Using the estimates in (3.15) and integrating the result with respect to timeimply(3.16) |∇ ϕ x ( t ) | + µ |∇ ϕ x | , Ω t + ν | ∆ ϕ x | , Ω t ≤ cν | rot ψ | , ∞ , Ω t | v x | , , Ω t + cν | rot ψ x | , , Ω t D + |∇ ϕ x (0) | . To estimate the first term on the r.h.s. of (3.16) we use the estimate(3.17) | rot ψ | , ∞ , Ω t = | rot ψ + ∇ ϕ − ∇ ϕ | , ∞ , Ω t ≤ c ( | v | , ∞ , Ω t + |∇ ϕ | , ∞ , Ω t ) ≤ c (cid:18) D + χ √ ν (cid:19) . Then (3.16) takes the form(3.18) |∇ ϕ x ( t ) | + µ |∇ ϕ x | , Ω t + ν | ∆ ϕ x | , Ω t ≤ cν | v x | , , Ω t (cid:18) D + χ ν (cid:19) + cν | rot ψ x | , , Ω t D + |∇ ϕ x (0) | . Differentiate (1.7) with respect to x , multiply by rot ψ x and integrate overΩ. Then we obtain(3.19) 12 ddt | rot ψ x | + µ |∇ rot ψ x | = − Z Ω rot ψ · ∇ v x · rot ψ x dx − Z Ω rot ψ x · ∇ v · rot ψ x dx ≡ J + J . Integrating by parts in J yields | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇ rot ψ x · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ x | + cε | rot ψ | | v x | . Z132 — − − imilarly, we have | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇ rot ψ x · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ x | + cε | rot ψ x | | v | . Using the estimates and (3.17) in (3.19), taking into account Lemma 2.4and integrating the result with respect to time we obtain the inequality(3.20) | rot ψ x ( t ) | + µ |∇ rot ψ x | , Ω t ≤ c | v x | , , Ω t (cid:18) D + χ ν (cid:19) + c | rot ψ x | , , Ω t D + | rot ψ x (0) | . Multiplying (3.20) by 1 /ν and summing up with (3.18) we have(3.21) |∇ ϕ x ( t ) | + 1 ν | rot ψ x ( t ) | + µ (cid:18) |∇ ϕ x | , Ω t + 1 ν |∇ rot ψ x | , Ω t (cid:19) + ν | ∆ ϕ x | , Ω t ≤ cν | v x | , , Ω t (cid:18) D + χ ν (cid:19) + cν | rot ψ x | , , Ω t D + |∇ ϕ x (0) | + 1 ν | rot ψ x (0) | . By interpolation and Lemma 2.1 we have(3.22) | v x | , , Ω t ≤ c |∇ v x | , Ω t | v x | , Ω t ≤ ε |∇ v x | , Ω t + c/εA and(3.23) | rot ψ x | , , Ω t ≤ c |∇ rot ψ x | , Ω t | rot ψ x | , Ω t ≤ ε |∇ rot ψ x | , Ω t + c/ε | rot ψ x | , Ω t , where | rot ψ x | , Ω t = | rot ψ x + ∇ ϕ x − ∇ ϕ x | , Ω t ≤ c ( | v x | , Ω t + |∇ ϕ x | , Ω t ) ≤ cA + c |∇ ϕ x | , Ω t and |∇ ϕ x | , Ω t is estimated by (3.1). Hence, we have |∇ ϕ x | , Ω t ≤ cν A D + |∇ ϕ (0) | + 1 ν | rot ψ (0) | . Using the above estimates in (3.21) implies (3.14). This concludes the proof.23
Z132 — − − emma 3.4. Assume that ν ∈ (0 , ∞ ) , D , D , χ are finite, ∇ ϕ (0) , rot ψ (0) ∈ H (Ω) , ∇ ϕ t (0) , rot ψ t (0) ∈ H (Ω) , rot ψ x ∈ L ∞ (0 , t : L (Ω)) .Then (3.24) |∇ ϕ xt ( t ) | + 1 ν | rot ψ xt ( t ) | + µ (cid:18) |∇ ϕ xt | , Ω t + 1 ν |∇ rot ψ xt | , Ω t (cid:19) + ν | ∆ ϕ xt | , Ω t ≤ cν D | rot ψ x | , ∞ , Ω t + cν (cid:18) D + χ ν (cid:19) ·· ( φ + ν |∇ ϕ t (0) | + | rot ψ t (0) | ) + cν (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + cν (cid:18) D + χ ν (cid:19) Ψ ν + |∇ ϕ xt (0) | + 1 ν | rot ψ xt (0) | . Proof.
Differentiate (1.7) with respect to x and t , multiply the result by ∇ ϕ xt and integrate over Ω. Then we have(3.25) 12 ddt |∇ ϕ xt | + µ |∇ ϕ xt | + ν | ∆ ϕ xt | = − Z Ω (rot ψ · ∇ v ) xt · ∇ ϕ xt dx ≡ − I. Performing differentiations with respect to x and t in I implies I = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) ·∇ ϕ xt dx ≡ X i =1 I i . Integrating by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇∇ ϕ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ x | | v t | . Next, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ t | | v x | . Z132 — −−
Differentiate (1.7) with respect to x and t , multiply the result by ∇ ϕ xt and integrate over Ω. Then we have(3.25) 12 ddt |∇ ϕ xt | + µ |∇ ϕ xt | + ν | ∆ ϕ xt | = − Z Ω (rot ψ · ∇ v ) xt · ∇ ϕ xt dx ≡ − I. Performing differentiations with respect to x and t in I implies I = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) ·∇ ϕ xt dx ≡ X i =1 I i . Integrating by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇∇ ϕ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ x | | v t | . Next, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ t | | v x | . Z132 — −− −−
Differentiate (1.7) with respect to x and t , multiply the result by ∇ ϕ xt and integrate over Ω. Then we have(3.25) 12 ddt |∇ ϕ xt | + µ |∇ ϕ xt | + ν | ∆ ϕ xt | = − Z Ω (rot ψ · ∇ v ) xt · ∇ ϕ xt dx ≡ − I. Performing differentiations with respect to x and t in I implies I = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) ·∇ ϕ xt dx ≡ X i =1 I i . Integrating by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇∇ ϕ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ x | | v t | . Next, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ t | | v x | . Z132 — −− −− inally, we consider I = Z Ω rot ψ · ∇ ( ∇ ϕ xt + rot ψ xt ) · ∇ ϕ xt dx = Z Ω rot ψ · ∇ rot ψ xt · ∇ ϕ xt dx = − Z Ω (rot ψ ) i (rot ψ xt ) j ∂ x i ∂ x j ϕ xt dx. Then, we have | I | ≤ ε |∇ ϕ xt | + c/ε | rot ψ | | rot ψ xt | . Using the above estimates in (3.25) we obtain after integration with respectto time and with the help of (2.27), (2.28) and (3.17) the inequality(3.26) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν D | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t | v x | , , Ω t + cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + |∇ ϕ xt (0) | . Using (3.14) we have(3.27) | v x | , , Ω t ≤ c k v x k , , Ω t ≤ c ( ν k∇ ϕ x k , , Ω t + k rot ψ x k , , Ω t ) ≤ cφ + c ( ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Therefore, (3.26) takes the form(3.28) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + |∇ ϕ xt (0) | . Differentiate (1.7) with respect to x and t , multiply the result by rot ψ xt and integrate over Ω. Then we have(3.29) 12 ddt | rot ψ xt ( t ) | + µ |∇ rot ψ xt | = − Z Ω (rot ψ · ∇ v ) xt · rot ψ xt dx ≡ − J. Performing differentiations with respect to x and t in J yields J = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) · rot ψ xt dx ≡ X i =1 J i . Z132 — −−
Differentiate (1.7) with respect to x and t , multiply the result by ∇ ϕ xt and integrate over Ω. Then we have(3.25) 12 ddt |∇ ϕ xt | + µ |∇ ϕ xt | + ν | ∆ ϕ xt | = − Z Ω (rot ψ · ∇ v ) xt · ∇ ϕ xt dx ≡ − I. Performing differentiations with respect to x and t in I implies I = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) ·∇ ϕ xt dx ≡ X i =1 I i . Integrating by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇∇ ϕ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ x | | v t | . Next, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ t | | v x | . Z132 — −− −− inally, we consider I = Z Ω rot ψ · ∇ ( ∇ ϕ xt + rot ψ xt ) · ∇ ϕ xt dx = Z Ω rot ψ · ∇ rot ψ xt · ∇ ϕ xt dx = − Z Ω (rot ψ ) i (rot ψ xt ) j ∂ x i ∂ x j ϕ xt dx. Then, we have | I | ≤ ε |∇ ϕ xt | + c/ε | rot ψ | | rot ψ xt | . Using the above estimates in (3.25) we obtain after integration with respectto time and with the help of (2.27), (2.28) and (3.17) the inequality(3.26) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν D | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t | v x | , , Ω t + cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + |∇ ϕ xt (0) | . Using (3.14) we have(3.27) | v x | , , Ω t ≤ c k v x k , , Ω t ≤ c ( ν k∇ ϕ x k , , Ω t + k rot ψ x k , , Ω t ) ≤ cφ + c ( ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Therefore, (3.26) takes the form(3.28) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + |∇ ϕ xt (0) | . Differentiate (1.7) with respect to x and t , multiply the result by rot ψ xt and integrate over Ω. Then we have(3.29) 12 ddt | rot ψ xt ( t ) | + µ |∇ rot ψ xt | = − Z Ω (rot ψ · ∇ v ) xt · rot ψ xt dx ≡ − J. Performing differentiations with respect to x and t in J yields J = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) · rot ψ xt dx ≡ X i =1 J i . Z132 — −− −−
Differentiate (1.7) with respect to x and t , multiply the result by ∇ ϕ xt and integrate over Ω. Then we have(3.25) 12 ddt |∇ ϕ xt | + µ |∇ ϕ xt | + ν | ∆ ϕ xt | = − Z Ω (rot ψ · ∇ v ) xt · ∇ ϕ xt dx ≡ − I. Performing differentiations with respect to x and t in I implies I = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) ·∇ ϕ xt dx ≡ X i =1 I i . Integrating by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇∇ ϕ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇∇ ϕ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ x | | v t | . Next, we have | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇∇ ϕ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xt | + c/ε | rot ψ t | | v x | . Z132 — −− −− inally, we consider I = Z Ω rot ψ · ∇ ( ∇ ϕ xt + rot ψ xt ) · ∇ ϕ xt dx = Z Ω rot ψ · ∇ rot ψ xt · ∇ ϕ xt dx = − Z Ω (rot ψ ) i (rot ψ xt ) j ∂ x i ∂ x j ϕ xt dx. Then, we have | I | ≤ ε |∇ ϕ xt | + c/ε | rot ψ | | rot ψ xt | . Using the above estimates in (3.25) we obtain after integration with respectto time and with the help of (2.27), (2.28) and (3.17) the inequality(3.26) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν D | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t | v x | , , Ω t + cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + |∇ ϕ xt (0) | . Using (3.14) we have(3.27) | v x | , , Ω t ≤ c k v x k , , Ω t ≤ c ( ν k∇ ϕ x k , , Ω t + k rot ψ x k , , Ω t ) ≤ cφ + c ( ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Therefore, (3.26) takes the form(3.28) |∇ ϕ xt ( t ) | + µ |∇ ϕ xt | , Ω t + ν | ∆ ϕ xt | , Ω t ≤ cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t + cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + |∇ ϕ xt (0) | . Differentiate (1.7) with respect to x and t , multiply the result by rot ψ xt and integrate over Ω. Then we have(3.29) 12 ddt | rot ψ xt ( t ) | + µ |∇ rot ψ xt | = − Z Ω (rot ψ · ∇ v ) xt · rot ψ xt dx ≡ − J. Performing differentiations with respect to x and t in J yields J = Z Ω (rot ψ xt ·∇ v +rot ψ x ·∇ v t +rot ψ t ·∇ v x +rot ψ ·∇ v xt ) · rot ψ xt dx ≡ X i =1 J i . Z132 — −− −− ntegrating by parts in J yields | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xt · ∇ rot ψ xt · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ xt | + c/ε | rot ψ xt | | v | . Similarly, integration by parts in J implies | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇ rot ψ xt · v t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ xt | + cε | rot ψ x | | v t | . Next, we have | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ t · ∇ rot ψ xt · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ xt | + cε | rot ψ t | | v x | . Finally, we examine J = Z Ω rot ψ · ∇ ( ∇ ϕ xt + rot ψ xt ) · rot ψ xt dx = Z Ω rot ψ · ∇∇ ϕ xt · rot ψ xt dx = − Z Ω rot ψ · ∇ rot ψ xt · ∇ ϕ xt dx. Then we have | J | ≤ ε |∇ rot ψ xt | + c/ε | rot ψ | |∇ ϕ xt | . Using the above estimates in (3.29) we obtain after integration with respectto time with the help of (2.27), (2.28), (3.17) and (3.27) the inequality(3.30) | rot ψ xt ( t ) | + µ |∇ rot ψ xt | , Ω t ≤ cD | rot ψ xt | , , Ω t + cD | rot ψ x | , ∞ , Ω t + c | rot ψ t | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + c (cid:18) D + χ ν (cid:19) |∇ ϕ xt | , , Ω t + | rot ψ xt (0) | . Multiplying (3.30) by 1 /ν and summing up with (3.28) we obtain(3.31) |∇ ϕ xt ( t ) | + 1 ν | rot ψ xt ( t ) | + µ (cid:18) |∇ ϕ xt | , Ω t + 1 ν |∇ rot ψ xt | , Ω t (cid:19) + ν | ∆ ϕ xt | , Ω t ≤ cν D | rot ψ x | , ∞ , Ω t + cν | rot ψ t | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + cν (cid:18) D + χ ν (cid:19) ( |∇ ϕ xt | , , Ω t + | rot ψ xt | , , Ω t )+ |∇ ϕ xt (0) | + 1 ν | rot ψ xt (0) | . Z132 — − − ow we shall estimate the unknown quantities appeared on the r.h.s. of(3.31). We need the intepolation | rot ψ xt | , , Ω t ≤ ε |∇ rot ψ xt | , Ω t + c/ε | rot ψ xt | , Ω t , where, in view of (3.8), we have | rot ψ xt | , Ω t ≤ cφ + ν |∇ ϕ t (0) | + | rot ψ t (0) | . Therefore, the following part of the third term on the r.h.s. of (3.31) isbounded by cν (cid:18) D + χ ν (cid:19) | rot ψ xt | , , Ω t ≤ εν |∇ rot ψ xt | , Ω t + cνε ( φ + ν |∇ ϕ t (0) | + | rot ψ t (0) | ) (cid:18) D + χ ν (cid:19) . To estimate the second term on the r.h.s. of (3.31) we use the interpolation | rot ψ t | , ∞ , Ω t ≤ ε |∇ rot ψ t | , ∞ , Ω t + c/ε | rot ψ t | , ∞ , Ω t , where | rot ψ t | , ∞ , Ω t = | rot ψ t + ∇ ϕ t − ∇ ϕ t | , ∞ , Ω t ≤ c ( | v t | , ∞ , Ω t + |∇ ϕ t | , ∞ , Ω t ) ≤ c (cid:18) D + χ ν (cid:19) . Then the second term on the r.h.s. of (3.31) is bounded by εν |∇ rot ψ t | , ∞ , Ω t + cνε (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . The remainintg part of the third term on the r.h.s. of (3.31) is bounded by cν (cid:18) D + χ ν (cid:19) Ψ ν . Using the above estimates in (3.31) and assuming that ε is sufficiently smallwe derive (3.24). This concludes the proof. Lemma 3.5.
Assume that ν ∈ (0 , ∞ ) , A , D , D , χ , Ψ are finite. Let ∇ ϕ (0) , rot ψ (0) ∈ H (Ω) , ∇ ϕ t (0) , rot ψ t (0) ∈ H (Ω) . Then (3.32) |∇ ϕ xx ( t ) | + 1 ν | rot ψ xx ( t ) | + µ (cid:18) |∇ ϕ xx | , Ω t + 1 ν |∇ rot ψ xx | , Ω t (cid:19) + ν | ∆ ϕ xx | , Ω t ≤ cν φ + |∇ ϕ xx (0) | + 1 ν | rot ψ xx (0) | , where φ is defined in (3.39). Z132 — − − roof. Differentiate (1.7) twice with respect to x , multiply by ∇ ϕ xx andintegrate over Ω. Then we obtain(3.33)12 ddt |∇ ϕ xx | + µ |∇ ϕ xx | + ν | ∆ ϕ xx | = − Z Ω (rot ψ · ∇ v ) xx · ∇ ϕ xx dx ≡ − I. Carrying out differentiations in I yields I = Z Ω (rot ψ xx · ∇ v + 2rot ψ x · ∇ v x + rot ψ · ∇ v xx ) · ∇ ϕ xx dx ≡ I + I + I . Integrating by parts in I implies | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xx · ∇∇ ϕ xx · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xx | + cε | rot ψ xx | | v | , where, in view of (2.27), the second term is bounded by cε | rot ψ xx | D . We integrate by parts in I . Then we obtain | I | = 2 (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇ ϕ xx · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xx | + c/ε | rot ψ x | | v x | . Finally, I = Z Ω rot ψ · ∇ ( ∇ ϕ xx + rot ψ xx ) · ∇ ϕ xx dx = Z Ω rot ψ · ∇ rot ψ xx · ∇ ϕ xx dx. Integration by parts in I yields | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ · ∇∇ ϕ xx · rot ψ xx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xx | + c/ε | rot ψ | | rot ψ xx | , where(3.34) | rot ψ | = | rot ψ + ∇ ϕ − ∇ ϕ | ≤ | v | + |∇ ϕ | ≤ D + χ √ ν . Z132 — − − sing the estimates in (3.33) and integrating the result with respect to timeyield |∇ ϕ xx ( t ) | + µ |∇ ϕ xx | , Ω t + ν | ∆ ϕ xx | , Ω t ≤ cν | rot ψ xx | , , Ω t D + cν | rot ψ x | , ∞ , Ω t | v x | , , Ω t + cν | rot ψ xx | , , Ω t (cid:18) D + χ ν (cid:19) + |∇ ϕ xx (0) | . Finally, using (3.27), we get(3.35) |∇ ϕ xx ( t ) | + µ |∇ ϕ xx | , Ω t + ν | ∆ ϕ xx | , Ω t ≤ cν | rot ψ xx | , , Ω t (cid:18) D + χ ν (cid:19) + cν | rot ψ x | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + |∇ ϕ xx (0) | . Differentiate (1.7) twice with respect to x , multiply by rot ψ xx and integrateover Ω. Then we have(3.36) 12 ddt | rot ψ xx | + µ |∇ rot ψ xx | = − Z Ω (rot ψ · ∇ v ) xx · rot ψ xx dx ≡ − J. Performing differentiations in J implies J = Z Ω (rot ψ xx · ∇ v + 2rot ψ x · ∇ v x + rot ψ · ∇ v xx ) · rot ψ xx dx ≡ J + J + J . Integration by parts in J gives | J | = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ xx · ∇ rot ψ xx · vdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ xx | + c/ε | rot ψ xx | | v | . Proceeding, we have | J | = 2 (cid:12)(cid:12)(cid:12)(cid:12) Z Ω rot ψ x · ∇ rot ψ xx · v x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ rot ψ xx | + c/ε | rot ψ x | | v x | . Finally, J = Z Ω rot ψ · ∇ (rot ψ xx + ∇ ϕ xx ) · rot ψ xx dx = Z Ω rot ψ · ∇∇ ϕ xx · rot ψ xx dx = − Z Ω rot ψ · ∇ rot ψ xx · ∇ ϕ xx dx. Z132 — − − ence, we have | J | ≤ ε |∇ rot ψ xx | + cε | rot ψ | |∇ ϕ xx | . Using the above estimates in (3.36) integrating the result with respect totime, exploiting (3.27), (2.27) and (3.34) we obtain the inequality(3.37) | rot ψ xx ( t ) | + µ |∇ rot ψ xx | , Ω t ≤ c | rot ψ xx | , , Ω t D + c | rot ψ x | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | )+ c |∇ ϕ xx | (cid:18) D + χ ν (cid:19) + | rot ψ xx (0) | . Multiplying (3.37) by 1 /ν and adding to (3.35) imply(3.38) |∇ ϕ xx ( t ) | + 1 ν | rot ψ xx ( t ) | + µ (cid:18) |∇ ϕ xx | , Ω t + 1 ν |∇ rot ψ xx | , Ω t (cid:19) + ν | ∆ ϕ xx | , Ω t ≤ cν | rot ψ xx | , , Ω t (cid:18) D + χ ν (cid:19) + cν | rot ψ x | , ∞ , Ω t ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | )+ cν |∇ ϕ xx | , , Ω t (cid:18) D + χ ν (cid:19) + |∇ ϕ xx (0) | + 1 ν | rot ψ xx (0) | . Recall the interpolation | rot ψ xx | , , Ω t (cid:18) D + χ ν (cid:19) ≤ c |∇ rot ψ xx | / , Ω t | rot ψ x | / , Ω t (cid:18) D + χ ν (cid:19) ≤ ε |∇ rot ψ xx | , Ω t + c (1 /ε ) | rot ψ x | , Ω t (cid:18) D + χ ν (cid:19) , where in the second term we use the estimate | rot ψ x | , Ω t = | rot ψ x + ∇ ϕ x − ∇ ϕ x | , Ω t ≤ c ( | v x | , Ω t + |∇ ϕ x | , Ω t ) ≤ c (cid:18) A + Ψ ν (cid:19) . Next we consider |∇ ϕ xx | , , Ω t (cid:18) D + χ ν (cid:19) ≤ c |∇ ϕ xx | / , Ω t |∇ ϕ x | / , Ω t (cid:18) D + χ ν (cid:19) ≤ ε |∇ ϕ xx | , Ω t + c (1 /ε ) Ψ ν (cid:18) D + χ ν (cid:19) . Z132 — − − o estimate the second term on the r.h.s. of (3.38) we use the interpolation | rot ψ x | , ∞ , Ω t ≤ c |∇ rot ψ x | / , ∞ , Ω t | rot ψ | / , ∞ , Ω t ≤ ε |∇ rot ψ x | , ∞ , Ω t + c (1 /ε ) | rot ψ | , ∞ , Ω t where | rot ψ | , ∞ , Ω t = | rot ψ + ∇ ϕ − ∇ ϕ | , ∞ , Ω t ≤ c (cid:18) A + χ √ ν (cid:19) . Therefore the second term on the r.h.s. of (3.38) is bounded by εν |∇ rot ψ x | , ∞ , Ω t + c (1 /ε ) ν (cid:18) A + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Using the above estimates in (3.38) yields(3.39) |∇ ϕ xx ( t ) | + 1 ν | rot ψ xx ( t ) | + µ (cid:18) |∇ ϕ xx | , Ω t + 1 ν |∇ rot ψ xx | , Ω t (cid:19) + ν | ∆ ϕ xx | , Ω t ≤ cν (cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + cν (cid:18) A + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + cν Ψ ν (cid:18) D + χ ν (cid:19) + |∇ ϕ xx (0) | + 1 ν | rot ψ xx (0) | ≡ cν φ (cid:18) A , D , Ψ ν , χ √ ν , φ + X (0) (cid:19) + |∇ ϕ xx (0) | + 1 ν | rot ψ xx (0) | This inequality implies (3.32) and concludes the proof.To prove a global estimate for solutins to problem (1.7) we have to collectall inequalities derived in Lemmas 3.1–3.5. This is a topic of the result
Theorem 3.6.
Assume that ∇ ϕ (0) , rot ψ (0) ∈ Γ (Ω) ( Γ (Ω) is defined atthe beginning of Section 2).Then for ν = ν ∗ sufficiently large and t ≤ T < ν β ∗ , β < − κ ) , κ ∈ (1 / , ,there exists a constant A = A ( ν ∗ , X (0) (sufficiently large, see (3.54)) suchthat (3.40) X ( t ) ≤ A for t ≤ T. We have to emphasize that the bound A in (3.40) can be kept the same if ν > ν ∗ is increasing. Z132 — −−
Assume that ∇ ϕ (0) , rot ψ (0) ∈ Γ (Ω) ( Γ (Ω) is defined atthe beginning of Section 2).Then for ν = ν ∗ sufficiently large and t ≤ T < ν β ∗ , β < − κ ) , κ ∈ (1 / , ,there exists a constant A = A ( ν ∗ , X (0) (sufficiently large, see (3.54)) suchthat (3.40) X ( t ) ≤ A for t ≤ T. We have to emphasize that the bound A in (3.40) can be kept the same if ν > ν ∗ is increasing. Z132 — −− −−
Assume that ∇ ϕ (0) , rot ψ (0) ∈ Γ (Ω) ( Γ (Ω) is defined atthe beginning of Section 2).Then for ν = ν ∗ sufficiently large and t ≤ T < ν β ∗ , β < − κ ) , κ ∈ (1 / , ,there exists a constant A = A ( ν ∗ , X (0) (sufficiently large, see (3.54)) suchthat (3.40) X ( t ) ≤ A for t ≤ T. We have to emphasize that the bound A in (3.40) can be kept the same if ν > ν ∗ is increasing. Z132 — −− −− roof. From Lemma 3.1 we have(3.41) ν |∇ ϕ ( t ) | + | rot ψ ( t ) | + µ ( ν |∇ ϕ | , Ω t + |∇ rot ψ | , Ω t )+ ν | ∆ ϕ | , Ω t ≤ cA D + ν |∇ ϕ (0) | + | rot ψ (0) | . Lemma 3.2 yields(3.42) ν |∇ ϕ t ( t ) | + | rot ψ t ( t ) | + µ ( ν |∇ ϕ t | , Ω t + |∇ rot ψ t | , Ω t )+ ν | ∆ ϕ t | , Ω t ≤ cφ + ν |∇ ϕ t (0) | + | rot ψ t (0) | , where φ = (cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) . Next Lemma 3.3 implies(3.43) ν |∇ ϕ x ( t ) | + | rot ψ x ( t ) | + µ ( ν |∇ ϕ x | , Ω t + |∇ rot ψ x | , Ω t )+ ν | ∆ ϕ x | , Ω t ≤ cφ + ν |∇ ϕ x (0) | + | rot ψ x (0) | , where φ = A (cid:18) D + D + χ ν (cid:19) + D (cid:18) |∇ ϕ (0) | + 1 ν | rot ψ (0) | (cid:19) . Lemma 3.4 gives(3.44) ν |∇ ϕ xt ( t ) | + | rot ψ xt ( t ) | + µ ( ν |∇ ϕ xt | , Ω t + |∇ rot ψ xt | , Ω t )+ ν | ∆ ϕ xt | , Ω t ≤ cD | rot ψ x | , ∞ , Ω t + cφ + ν |∇ ϕ xt (0) | + | rot ψ xt (0) | , where φ = (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ t (0) | + | rot ψ t (0) | )+ (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + (cid:18) D + χ ν (cid:19) Ψ ν . Finally, Lemma 3.5 yields(3.45) ν |∇ ϕ xx ( t ) | + | rot ψ xx ( t ) | + µ ( ν |∇ ϕ xx | , Ω t + |∇ rot ψ xx | , Ω t )+ ν | ∆ ϕ xx | , Ω t ≤ cφ + ν |∇ ϕ xx (0) | + | rot ψ xx (0) | , where φ = (cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + Ψ ν (cid:18) D + χ ν (cid:19) . Z132 — − − sing (3.45) we will be able to estimate the first term on the r.h.s. of (3.44).For this purpose we use the interpolation(3.46) D | rot ψ x | , ∞ , Ω t ≤ c |∇ rot ψ x | , ∞ , Ω t | rot ψ x | , ∞ , Ω t D ≤ ε |∇ rot ψ x | , ∞ , Ω t + c/ε | rot ψ x | , ∞ , Ω t D , where we need to use (3.43) to have the estimate(3.47) | rot ψ x | , ∞ , Ω t ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Summing up (3.44) and (3.45), using (3.46) and (3.47) we obtain the in-equality(3.48) ν ( |∇ ϕ xt ( t ) | + |∇ ϕ xx ( t ) | ) + | rot ψ xt ( t ) | + | rot ψ xx ( t ) | + µ [ ν ( |∇ ϕ xt | , Ω t + |∇ ϕ xx | , Ω t ) + |∇ rot ψ xt | , Ω t + |∇ rot ψ xx | , Ω t ] + ν ( | ∆ ϕ xt | , Ω t + | ∆ ϕ xx | , Ω t ) ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) D + c ( φ + φ )+ ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | ≡ cφ + ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | . Finally, from (3.41), (3.42), (3.43) and (3.48) we derive the inequality(3.49) X ( t ) ≤ c ( A D + φ + φ + φ ) + X (0) . Introducing the notation B (0) = ν |∇ ϕ (0) | , + | rot ψ (0) | , and using the explicit forms of φ , φ , φ , inequality (3.49) takes the form(3.50) X ( t ) ≤ c (cid:20)(cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) D + χ ν (cid:19)(cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) + A (cid:18) D + D + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + A (cid:18) A + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) (1 + D ) B (0)+ (1 + D ) (cid:18) D + D + χ ν + D + χ ν (cid:19) B (0)+ X (0) (cid:21) ≡ φ (cid:18) D , D , Ψ ν , χ √ ν , B (0) (cid:19) + X (0) . Z132 — − − sing (2.25) in (2.27) we have the estimate D ≤ c exp (cid:18) X / (1 − κ ) [ c − √ tν − κ X ] / (1 − κ ) (cid:19)(cid:18) Xν / + X / ν ( κ − / / (cid:19) + c | v (0) | + A ≡ D ( X )Similarly D = | v t (0) | exp[( D A ) / ≤ D ( X ) . Then (3.50) implies(3.51) X ( t ) ≤ φ (cid:18) D ( X ) , D ( X ) , Xν , X √ ν , B (0) (cid:19) + X (0) . Consider the algebraic equation(3.52) A = φ (cid:18) D ( A ) , D ( A ) , Aν , A √ ν , B (0) (cid:19) + X (0) , where φ : R + → R + .If we show the existence of bounded solutions to (3.52) such that A ≤ A ∗ we get the estimate(3.53) X ≤ A. If we show that φ is a contraction then the existence of solutions to (3.52)follows from the method of successive approximations.Since φ is a differentiable function of its arguments we restrict our consid-erations to the case φ = φ ( D ( A )), because the dependence of the otherarguments can be examined similarly. Then we have | φ ( D ( A )) − φ ( D ( A ′ )) | ≤ c | D ( A ) − D ( A ′ ) |≤ c | A − A ′ | [ c − √ tν − κ A ] / (1 − κ ) (cid:18) Aν / + Aν ( κ − / / (cid:19) + other terms . The first term is bounded by cν α | A − A ′ | , α > , where t ≤ ν β , β < − κ ). The other terms can be treated similarly.Hence for ν sufficiently large operator φ is a contraction.34 Z132 — −−
Assume that ∇ ϕ (0) , rot ψ (0) ∈ Γ (Ω) ( Γ (Ω) is defined atthe beginning of Section 2).Then for ν = ν ∗ sufficiently large and t ≤ T < ν β ∗ , β < − κ ) , κ ∈ (1 / , ,there exists a constant A = A ( ν ∗ , X (0) (sufficiently large, see (3.54)) suchthat (3.40) X ( t ) ≤ A for t ≤ T. We have to emphasize that the bound A in (3.40) can be kept the same if ν > ν ∗ is increasing. Z132 — −− −− roof. From Lemma 3.1 we have(3.41) ν |∇ ϕ ( t ) | + | rot ψ ( t ) | + µ ( ν |∇ ϕ | , Ω t + |∇ rot ψ | , Ω t )+ ν | ∆ ϕ | , Ω t ≤ cA D + ν |∇ ϕ (0) | + | rot ψ (0) | . Lemma 3.2 yields(3.42) ν |∇ ϕ t ( t ) | + | rot ψ t ( t ) | + µ ( ν |∇ ϕ t | , Ω t + |∇ rot ψ t | , Ω t )+ ν | ∆ ϕ t | , Ω t ≤ cφ + ν |∇ ϕ t (0) | + | rot ψ t (0) | , where φ = (cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) . Next Lemma 3.3 implies(3.43) ν |∇ ϕ x ( t ) | + | rot ψ x ( t ) | + µ ( ν |∇ ϕ x | , Ω t + |∇ rot ψ x | , Ω t )+ ν | ∆ ϕ x | , Ω t ≤ cφ + ν |∇ ϕ x (0) | + | rot ψ x (0) | , where φ = A (cid:18) D + D + χ ν (cid:19) + D (cid:18) |∇ ϕ (0) | + 1 ν | rot ψ (0) | (cid:19) . Lemma 3.4 gives(3.44) ν |∇ ϕ xt ( t ) | + | rot ψ xt ( t ) | + µ ( ν |∇ ϕ xt | , Ω t + |∇ rot ψ xt | , Ω t )+ ν | ∆ ϕ xt | , Ω t ≤ cD | rot ψ x | , ∞ , Ω t + cφ + ν |∇ ϕ xt (0) | + | rot ψ xt (0) | , where φ = (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ t (0) | + | rot ψ t (0) | )+ (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + (cid:18) D + χ ν (cid:19) Ψ ν . Finally, Lemma 3.5 yields(3.45) ν |∇ ϕ xx ( t ) | + | rot ψ xx ( t ) | + µ ( ν |∇ ϕ xx | , Ω t + |∇ rot ψ xx | , Ω t )+ ν | ∆ ϕ xx | , Ω t ≤ cφ + ν |∇ ϕ xx (0) | + | rot ψ xx (0) | , where φ = (cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + Ψ ν (cid:18) D + χ ν (cid:19) . Z132 — − − sing (3.45) we will be able to estimate the first term on the r.h.s. of (3.44).For this purpose we use the interpolation(3.46) D | rot ψ x | , ∞ , Ω t ≤ c |∇ rot ψ x | , ∞ , Ω t | rot ψ x | , ∞ , Ω t D ≤ ε |∇ rot ψ x | , ∞ , Ω t + c/ε | rot ψ x | , ∞ , Ω t D , where we need to use (3.43) to have the estimate(3.47) | rot ψ x | , ∞ , Ω t ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Summing up (3.44) and (3.45), using (3.46) and (3.47) we obtain the in-equality(3.48) ν ( |∇ ϕ xt ( t ) | + |∇ ϕ xx ( t ) | ) + | rot ψ xt ( t ) | + | rot ψ xx ( t ) | + µ [ ν ( |∇ ϕ xt | , Ω t + |∇ ϕ xx | , Ω t ) + |∇ rot ψ xt | , Ω t + |∇ rot ψ xx | , Ω t ] + ν ( | ∆ ϕ xt | , Ω t + | ∆ ϕ xx | , Ω t ) ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) D + c ( φ + φ )+ ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | ≡ cφ + ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | . Finally, from (3.41), (3.42), (3.43) and (3.48) we derive the inequality(3.49) X ( t ) ≤ c ( A D + φ + φ + φ ) + X (0) . Introducing the notation B (0) = ν |∇ ϕ (0) | , + | rot ψ (0) | , and using the explicit forms of φ , φ , φ , inequality (3.49) takes the form(3.50) X ( t ) ≤ c (cid:20)(cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) D + χ ν (cid:19)(cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) + A (cid:18) D + D + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + A (cid:18) A + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) (1 + D ) B (0)+ (1 + D ) (cid:18) D + D + χ ν + D + χ ν (cid:19) B (0)+ X (0) (cid:21) ≡ φ (cid:18) D , D , Ψ ν , χ √ ν , B (0) (cid:19) + X (0) . Z132 — − − sing (2.25) in (2.27) we have the estimate D ≤ c exp (cid:18) X / (1 − κ ) [ c − √ tν − κ X ] / (1 − κ ) (cid:19)(cid:18) Xν / + X / ν ( κ − / / (cid:19) + c | v (0) | + A ≡ D ( X )Similarly D = | v t (0) | exp[( D A ) / ≤ D ( X ) . Then (3.50) implies(3.51) X ( t ) ≤ φ (cid:18) D ( X ) , D ( X ) , Xν , X √ ν , B (0) (cid:19) + X (0) . Consider the algebraic equation(3.52) A = φ (cid:18) D ( A ) , D ( A ) , Aν , A √ ν , B (0) (cid:19) + X (0) , where φ : R + → R + .If we show the existence of bounded solutions to (3.52) such that A ≤ A ∗ we get the estimate(3.53) X ≤ A. If we show that φ is a contraction then the existence of solutions to (3.52)follows from the method of successive approximations.Since φ is a differentiable function of its arguments we restrict our consid-erations to the case φ = φ ( D ( A )), because the dependence of the otherarguments can be examined similarly. Then we have | φ ( D ( A )) − φ ( D ( A ′ )) | ≤ c | D ( A ) − D ( A ′ ) |≤ c | A − A ′ | [ c − √ tν − κ A ] / (1 − κ ) (cid:18) Aν / + Aν ( κ − / / (cid:19) + other terms . The first term is bounded by cν α | A − A ′ | , α > , where t ≤ ν β , β < − κ ). The other terms can be treated similarly.Hence for ν sufficiently large operator φ is a contraction.34 Z132 — −− −−
Assume that ∇ ϕ (0) , rot ψ (0) ∈ Γ (Ω) ( Γ (Ω) is defined atthe beginning of Section 2).Then for ν = ν ∗ sufficiently large and t ≤ T < ν β ∗ , β < − κ ) , κ ∈ (1 / , ,there exists a constant A = A ( ν ∗ , X (0) (sufficiently large, see (3.54)) suchthat (3.40) X ( t ) ≤ A for t ≤ T. We have to emphasize that the bound A in (3.40) can be kept the same if ν > ν ∗ is increasing. Z132 — −− −− roof. From Lemma 3.1 we have(3.41) ν |∇ ϕ ( t ) | + | rot ψ ( t ) | + µ ( ν |∇ ϕ | , Ω t + |∇ rot ψ | , Ω t )+ ν | ∆ ϕ | , Ω t ≤ cA D + ν |∇ ϕ (0) | + | rot ψ (0) | . Lemma 3.2 yields(3.42) ν |∇ ϕ t ( t ) | + | rot ψ t ( t ) | + µ ( ν |∇ ϕ t | , Ω t + |∇ rot ψ t | , Ω t )+ ν | ∆ ϕ t | , Ω t ≤ cφ + ν |∇ ϕ t (0) | + | rot ψ t (0) | , where φ = (cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) . Next Lemma 3.3 implies(3.43) ν |∇ ϕ x ( t ) | + | rot ψ x ( t ) | + µ ( ν |∇ ϕ x | , Ω t + |∇ rot ψ x | , Ω t )+ ν | ∆ ϕ x | , Ω t ≤ cφ + ν |∇ ϕ x (0) | + | rot ψ x (0) | , where φ = A (cid:18) D + D + χ ν (cid:19) + D (cid:18) |∇ ϕ (0) | + 1 ν | rot ψ (0) | (cid:19) . Lemma 3.4 gives(3.44) ν |∇ ϕ xt ( t ) | + | rot ψ xt ( t ) | + µ ( ν |∇ ϕ xt | , Ω t + |∇ rot ψ xt | , Ω t )+ ν | ∆ ϕ xt | , Ω t ≤ cD | rot ψ x | , ∞ , Ω t + cφ + ν |∇ ϕ xt (0) | + | rot ψ xt (0) | , where φ = (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ t (0) | + | rot ψ t (0) | )+ (cid:18) D + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + (cid:18) D + χ ν (cid:19) Ψ ν . Finally, Lemma 3.5 yields(3.45) ν |∇ ϕ xx ( t ) | + | rot ψ xx ( t ) | + µ ( ν |∇ ϕ xx | , Ω t + |∇ rot ψ xx | , Ω t )+ ν | ∆ ϕ xx | , Ω t ≤ cφ + ν |∇ ϕ xx (0) | + | rot ψ xx (0) | , where φ = (cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) + Ψ ν (cid:18) D + χ ν (cid:19) . Z132 — − − sing (3.45) we will be able to estimate the first term on the r.h.s. of (3.44).For this purpose we use the interpolation(3.46) D | rot ψ x | , ∞ , Ω t ≤ c |∇ rot ψ x | , ∞ , Ω t | rot ψ x | , ∞ , Ω t D ≤ ε |∇ rot ψ x | , ∞ , Ω t + c/ε | rot ψ x | , ∞ , Ω t D , where we need to use (3.43) to have the estimate(3.47) | rot ψ x | , ∞ , Ω t ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) . Summing up (3.44) and (3.45), using (3.46) and (3.47) we obtain the in-equality(3.48) ν ( |∇ ϕ xt ( t ) | + |∇ ϕ xx ( t ) | ) + | rot ψ xt ( t ) | + | rot ψ xx ( t ) | + µ [ ν ( |∇ ϕ xt | , Ω t + |∇ ϕ xx | , Ω t ) + |∇ rot ψ xt | , Ω t + |∇ rot ψ xx | , Ω t ] + ν ( | ∆ ϕ xt | , Ω t + | ∆ ϕ xx | , Ω t ) ≤ c ( φ + ν |∇ ϕ x (0) | + | rot ψ x (0) | ) D + c ( φ + φ )+ ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | ≡ cφ + ν ( |∇ ϕ xt (0) | + |∇ ϕ xx (0) | ) + | rot ψ xt (0) | + | rot ψ xx (0) | . Finally, from (3.41), (3.42), (3.43) and (3.48) we derive the inequality(3.49) X ( t ) ≤ c ( A D + φ + φ + φ ) + X (0) . Introducing the notation B (0) = ν |∇ ϕ (0) | , + | rot ψ (0) | , and using the explicit forms of φ , φ , φ , inequality (3.49) takes the form(3.50) X ( t ) ≤ c (cid:20)(cid:18) A + Ψ ν (cid:19)(cid:18) D + χ ν (cid:19) + (cid:18) D + χ ν (cid:19)(cid:18) D + χ ν (cid:19)(cid:18) D + Ψ ν (cid:19) + A (cid:18) D + D + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + A (cid:18) A + χ ν (cid:19)(cid:18) D + D + χ ν (cid:19) + (cid:18) A + χ ν (cid:19) (1 + D ) B (0)+ (1 + D ) (cid:18) D + D + χ ν + D + χ ν (cid:19) B (0)+ X (0) (cid:21) ≡ φ (cid:18) D , D , Ψ ν , χ √ ν , B (0) (cid:19) + X (0) . Z132 — − − sing (2.25) in (2.27) we have the estimate D ≤ c exp (cid:18) X / (1 − κ ) [ c − √ tν − κ X ] / (1 − κ ) (cid:19)(cid:18) Xν / + X / ν ( κ − / / (cid:19) + c | v (0) | + A ≡ D ( X )Similarly D = | v t (0) | exp[( D A ) / ≤ D ( X ) . Then (3.50) implies(3.51) X ( t ) ≤ φ (cid:18) D ( X ) , D ( X ) , Xν , X √ ν , B (0) (cid:19) + X (0) . Consider the algebraic equation(3.52) A = φ (cid:18) D ( A ) , D ( A ) , Aν , A √ ν , B (0) (cid:19) + X (0) , where φ : R + → R + .If we show the existence of bounded solutions to (3.52) such that A ≤ A ∗ we get the estimate(3.53) X ≤ A. If we show that φ is a contraction then the existence of solutions to (3.52)follows from the method of successive approximations.Since φ is a differentiable function of its arguments we restrict our consid-erations to the case φ = φ ( D ( A )), because the dependence of the otherarguments can be examined similarly. Then we have | φ ( D ( A )) − φ ( D ( A ′ )) | ≤ c | D ( A ) − D ( A ′ ) |≤ c | A − A ′ | [ c − √ tν − κ A ] / (1 − κ ) (cid:18) Aν / + Aν ( κ − / / (cid:19) + other terms . The first term is bounded by cν α | A − A ′ | , α > , where t ≤ ν β , β < − κ ). The other terms can be treated similarly.Hence for ν sufficiently large operator φ is a contraction.34 Z132 — −− −− o perform the method of successive approximations we have to find a lowerbound for A . The lower bound for A must be greater than the r.h.s. of (3.52)at ν = ∞ . We have D | ν = ∞ = c | v (0) | + A ,D | ν = ∞ = | v t (0) | exp[( c | v (0) | + A ) A / A > φ ( c | v (0) | + A , | v t (0) | exp[( c | v (0) | + A ) A / , , , B (0)) + X (0) . This concludes the proof.
Remark 3.7.
To eliminate the dependence of t in the r.h.s. of (3.52) weset t = ν − κ . Since the dependence appears in D ( A ) only we will eliminatethe explicit dependence on time in (3.51). Then D ( A ) = c exp (cid:18) A / (1 − κ ) [ c − A/ν (1 − κ ) / ] / (1 − κ ) (cid:19)(cid:18) Aν / + A / ν ( κ − / / (cid:19) + c | v (0) | r + A . Assume that for ν = ν ∗ we found a bound A ∗ for A . We can increase A by kA , k > ν by k ∗ ν , k ∗ >
1, in such a way that the r.h.s. of (3.52)does not increase. Then we obtain the same bound A ∗ but time of existenceincreases to t = ( k ∗ ν ∗ ) − κ . Since such k and k ∗ can be chosen as large aswe want the same bound A ∗ for A holds for t and ν arbitrary large. In Theorem 3.6 we proved the existence of long time solutions to problem(1.7). However, to prove global existence of solutions to the Navier-Stokesequations (1.1) we need to have global existence of solutions to (1.7). Forthis purpose we use the two steps in time technique. We have to empha-size that in this paper any problem of existence is restricted to derive anappropriate estimate. Hence we need the result
Lemma 4.1.
Let the assumptions of Theorem 3.6 hold. Let A and T bedefined in Theorem 3.6. Let cA < µ T Let Y ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , Z132 — − − hen (4.1) Y ( T ) ≤ exp (cid:18) − µ T (cid:19) Y (0) . This means that the initial data for solutions to problem (1.7) for timeinterval [ T, ∞ ) are small for large T . This suggests the existence of globalsolutions to (1.7) in [ T, ∞ ). In Lemma 4.2 we derive a necessary globalestimate. Proof.
From (3.3) and (3.5) we have(4.2) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . Next, (3.9) and (3.11) imply(4.3) ddt ( ν |∇ ϕ t | + | rot ψ t | ) + µ ( ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v t | + | rot ψ t | | v | ) . From (3.15) and (3.19) it follows(4.4) ddt ( ν |∇ ϕ x | + | rot ψ x | ) + µ ( ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v x | + | rot ψ x | | v | ) . Next, (3.25) and (3.29) imply(4.5) ddt ( ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ | | v xt | ) . Finally, (3.33) and (3.36) yield(4.6) ddt ( ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v xx | ) . We use the following interpolation(4.7) | u | d ≤ c |∇ u | | u | d ≤ ε |∇ u | + c/ε | u | d , Z132 — −−
From (3.3) and (3.5) we have(4.2) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . Next, (3.9) and (3.11) imply(4.3) ddt ( ν |∇ ϕ t | + | rot ψ t | ) + µ ( ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v t | + | rot ψ t | | v | ) . From (3.15) and (3.19) it follows(4.4) ddt ( ν |∇ ϕ x | + | rot ψ x | ) + µ ( ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v x | + | rot ψ x | | v | ) . Next, (3.25) and (3.29) imply(4.5) ddt ( ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ | | v xt | ) . Finally, (3.33) and (3.36) yield(4.6) ddt ( ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v xx | ) . We use the following interpolation(4.7) | u | d ≤ c |∇ u | | u | d ≤ ε |∇ u | + c/ε | u | d , Z132 — −− −−
From (3.3) and (3.5) we have(4.2) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . Next, (3.9) and (3.11) imply(4.3) ddt ( ν |∇ ϕ t | + | rot ψ t | ) + µ ( ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v t | + | rot ψ t | | v | ) . From (3.15) and (3.19) it follows(4.4) ddt ( ν |∇ ϕ x | + | rot ψ x | ) + µ ( ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v x | + | rot ψ x | | v | ) . Next, (3.25) and (3.29) imply(4.5) ddt ( ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ | | v xt | ) . Finally, (3.33) and (3.36) yield(4.6) ddt ( ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v xx | ) . We use the following interpolation(4.7) | u | d ≤ c |∇ u | | u | d ≤ ε |∇ u | + c/ε | u | d , Z132 — −− −− here d is a constant. Using (4.7) in (4.2) yields(4.8) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . In view of (4.7) we obtain from (4.3) and (4.8) the inequality(4.9) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ t | | v | ) . In view of (4.7) inequalities (4.4) and (4.8) imply(4.10) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | rot ψ x | | v | + | rot ψ t | | v | ) . Using (4.7) inequalities (4.5), (4.9) and (4.10) give(4.11) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ c ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ t | | v | + | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | v xt | | rot ψ | ) . Finally, (4.2), (4.4) and (4.6) imply(4.12) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v | + | rot ψ | | v x | + | rot ψ x | | v | + | v xx | | rot ψ | ) . Introduce the quantity Y ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , . Z132 — −−
From (3.3) and (3.5) we have(4.2) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . Next, (3.9) and (3.11) imply(4.3) ddt ( ν |∇ ϕ t | + | rot ψ t | ) + µ ( ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v t | + | rot ψ t | | v | ) . From (3.15) and (3.19) it follows(4.4) ddt ( ν |∇ ϕ x | + | rot ψ x | ) + µ ( ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v x | + | rot ψ x | | v | ) . Next, (3.25) and (3.29) imply(4.5) ddt ( ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ | | v xt | ) . Finally, (3.33) and (3.36) yield(4.6) ddt ( ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v xx | ) . We use the following interpolation(4.7) | u | d ≤ c |∇ u | | u | d ≤ ε |∇ u | + c/ε | u | d , Z132 — −− −− here d is a constant. Using (4.7) in (4.2) yields(4.8) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . In view of (4.7) we obtain from (4.3) and (4.8) the inequality(4.9) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ t | | v | ) . In view of (4.7) inequalities (4.4) and (4.8) imply(4.10) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | rot ψ x | | v | + | rot ψ t | | v | ) . Using (4.7) inequalities (4.5), (4.9) and (4.10) give(4.11) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ c ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ t | | v | + | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | v xt | | rot ψ | ) . Finally, (4.2), (4.4) and (4.6) imply(4.12) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v | + | rot ψ | | v x | + | rot ψ x | | v | + | v xx | | rot ψ | ) . Introduce the quantity Y ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , . Z132 — −− −−
From (3.3) and (3.5) we have(4.2) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . Next, (3.9) and (3.11) imply(4.3) ddt ( ν |∇ ϕ t | + | rot ψ t | ) + µ ( ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v t | + | rot ψ t | | v | ) . From (3.15) and (3.19) it follows(4.4) ddt ( ν |∇ ϕ x | + | rot ψ x | ) + µ ( ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v x | + | rot ψ x | | v | ) . Next, (3.25) and (3.29) imply(4.5) ddt ( ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ | | v xt | ) . Finally, (3.33) and (3.36) yield(4.6) ddt ( ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v xx | ) . We use the following interpolation(4.7) | u | d ≤ c |∇ u | | u | d ≤ ε |∇ u | + c/ε | u | d , Z132 — −− −− here d is a constant. Using (4.7) in (4.2) yields(4.8) ddt ( ν |∇ ϕ | + | rot ψ | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | ) ≤ c | rot ψ | | v | . In view of (4.7) we obtain from (4.3) and (4.8) the inequality(4.9) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ t | | v | ) . In view of (4.7) inequalities (4.4) and (4.8) imply(4.10) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | )+ µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | ) ≤ c ( | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | rot ψ x | | v | + | rot ψ t | | v | ) . Using (4.7) inequalities (4.5), (4.9) and (4.10) give(4.11) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ t | + | rot ψ t | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xt | + | rot ψ xt | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ t | + |∇ rot ψ t | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xt | + |∇ rot ψ xt | ) ≤ c ( | rot ψ xt | | v | + | rot ψ x | | v t | + | rot ψ t | | v x | + | rot ψ t | | v | + | rot ψ | | v | + | rot ψ | | v t | + | rot ψ | | v x | + | v xt | | rot ψ | ) . Finally, (4.2), (4.4) and (4.6) imply(4.12) ddt ( ν |∇ ϕ | + | rot ψ | + ν |∇ ϕ x | + | rot ψ x | + ν |∇ ϕ xx | + | rot ψ xx | ) + µ ( ν |∇ ϕ | + |∇ rot ψ | + ν |∇ ϕ x | + |∇ rot ψ x | + ν |∇ ϕ xx | + |∇ rot ψ xx | ) ≤ c ( | rot ψ xx | | v | + | rot ψ x | | v x | + | rot ψ | | v | + | rot ψ | | v x | + | rot ψ x | | v | + | v xx | | rot ψ | ) . Introduce the quantity Y ( t ) = ν |∇ ϕ ( t ) | , + | rot ψ ( t ) | , . Z132 — −− −− hen inequalities (4.8)–(4.12) imply the inequality(4.13) ddt Y + µY ≤ cY ( | v | + | v x | + | v t | + | rot ψ | ) . From (4.13) we have(4.14) ddt (cid:18) Y exp (cid:18) µt − c | v | , , ∞ , Ω t t Z | v ( t ′ ) | , dt ′ (cid:19)(cid:19) ≤ . Hence the decay follows(4.15) Y ( t ) ≤ exp( − µt + cA ) Y (0) . For t = T , the restriction cA ≤ µ T and Remark 3.7 we derive (4.1) andconclude the proof.In Theorem 3.6 estimate (3.40) is proved for t ≤ T , where T < ν − κ ) .To prove Theorem 3.6 we need that ν is large so T can also be large. Toshow (3.40) we need that the norm X (0) = Y (0) of initial data for solutionsto (1.7) must be finite. Lemma 4.1 implies that the norm Y ( T ) for T sufficiently large satisfies(4.16) Y ( T ) ≤ Y (0) . It suggests that the data at time T to problem (1.7) satisfying (4.16) can betreated as the initial data for [ T, T ] so also the proof of Theorem 3.6 can berepeated for this interval. However, to prove Theorem 3.6 for interval [ T, T ]we need Lemma 2.4. Therefore the following assumptions are necessary(4.17) c ν κ ≤ | ϕ ( T ) | p ≤ c ν κ , | ϕ ( T ) | ≤ c ν κ . But we do not know how satisfy (4.17). Therefore we have to derive anestimate of type (3.40) for t > T in a different way. This is the topic of thenext lemma
Lemma 4.2.
Assume that T and ν are large. Assume that ϕ ( T ) ∈ H (Ω) , v ( T ) ∈ H (Ω) . Then there exists a constant B such that k ϕ ( T ) k + k v ( T ) k < cB, where B = (cid:0) e − µ T k v (0) k + A √ ν (cid:1) and A appears in (3.40). Moreover, for T and ν sufficiently large the following inequality holds (4.18) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT + k∇ ϕ k , , Ω tT < cB for any t ∈ ( T, ∞ ) . Z132 — −−
Assume that T and ν are large. Assume that ϕ ( T ) ∈ H (Ω) , v ( T ) ∈ H (Ω) . Then there exists a constant B such that k ϕ ( T ) k + k v ( T ) k < cB, where B = (cid:0) e − µ T k v (0) k + A √ ν (cid:1) and A appears in (3.40). Moreover, for T and ν sufficiently large the following inequality holds (4.18) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT + k∇ ϕ k , , Ω tT < cB for any t ∈ ( T, ∞ ) . Z132 — −− −−
Assume that T and ν are large. Assume that ϕ ( T ) ∈ H (Ω) , v ( T ) ∈ H (Ω) . Then there exists a constant B such that k ϕ ( T ) k + k v ( T ) k < cB, where B = (cid:0) e − µ T k v (0) k + A √ ν (cid:1) and A appears in (3.40). Moreover, for T and ν sufficiently large the following inequality holds (4.18) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT + k∇ ϕ k , , Ω tT < cB for any t ∈ ( T, ∞ ) . Z132 — −− −− roof. The first statement follows directly from (4.15). To derive the esti-mate we first need to show that data for solutions of (1.7) at time T aresufficiently small for sufficiently large T . Then we get problem (1.7) withsmall initial data so it is clear that we would be able to show (3.40) for any t > T .Multiply (1.7) by v and integrate over Ω. Then we have(4.19) ddt | v | + µ |∇ v | + ν | div v | = 0 . Since R Ω vdx = 0, (4.19) yields ddt ( | v | e µt ) ≤ | v ( t ) | ≤ e − µt | v (0) | . Let T be large. Then the quantity | v ( T ) | (4.21) | v ( T ) | ≤ e − µT | v (0) | is small. Integrating (4.19) with respect to time from t = T to t > T we get(4.22) | v ( t ) | + µ |∇ v | , Ω tT + ν | ∆ ϕ | , Ω tT = | v ( T ) | . Since(4.23) Z Ω vdx = 0 , Z Ω ϕdx = 0we obtain from (4.22) the inequality(4.24) | v ( t ) | + µ k v k , , Ω tT + ν k ϕ k , , Ω tT ≤ c | v ( T ) | . Differentiate (1 . with respect to x , multiply by v x and integrate over Ω.Then we have ddt | v x | + µ |∇ v x | + ν | div v x | = − Z Ω (rot ψ ) x · ∇ v · v x dx = Z Ω (rot ψ ) x · ∇ v x · vdx. Z132 — − − pplying the H¨older and Young inequalities to the r.h.s., integrating withrespect to time from t = T to t > T and using (4.24), we obtain(4.25) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ Ω tT | v ( T ) | + c k v ( T ) k . Differentiate (1 . twice with respect to x , multiply by v xx and integrateover Ω. Then we derive(4.26) ddt | v xx | + µ |∇ v xx | + ν | div v xx | = − Z Ω (rot ψ ) xx · ∇ v · v xx dx − Z Ω (rot ψ ) x · ∇ v x · v xx dx − Z Ω rot ψ · ∇ v xx · v xx dx ≡ I + I + I , where by the H¨older and Young inequalities we get | I | ≤ ε | v xx | + c/ε |∇ v | | rot ψ xx | , | I | ≤ ε | v xx | + c/ε | rot ψ x | | v xx | ,I = 0 . Using the above estimates in (4.26), summing up the result integrated withrespect to time with (4.25) and assuming that ε is sufficiently small wederive the inequality(4.27) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ , Ω tT | v ( T ) | + c |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT + c | rot ψ x | , ∞ , Ω tT | v xx | , , Ω tT + c k v ( T ) k . Using the interpolations | v ( T ) | | rot ψ x | , ∞ , Ω tT ≤ c | v ( T ) | ( ε / | rot ψ xx | , ∞ , Ω tT + cε − / | rot ψ | , ∞ , Ω tT ) = ε / | rot ψ xx | , ∞ , Ω tT + cε − / | v ( T ) | | rot ψ | , ∞ , Ω tT , |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT ≤ |∇ v | , ∞ , Ω tT ( ε / | rot ψ xxx | , Ω tT + cε − / | rot ψ | , Ω tT ) = ε / | rot ψ xxx | , Ω tT + cε − / |∇ v | , ∞ , Ω tT | rot ψ | , Ω tT Z132 — −−
Assume that T and ν are large. Assume that ϕ ( T ) ∈ H (Ω) , v ( T ) ∈ H (Ω) . Then there exists a constant B such that k ϕ ( T ) k + k v ( T ) k < cB, where B = (cid:0) e − µ T k v (0) k + A √ ν (cid:1) and A appears in (3.40). Moreover, for T and ν sufficiently large the following inequality holds (4.18) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT + k∇ ϕ k , , Ω tT < cB for any t ∈ ( T, ∞ ) . Z132 — −− −− roof. The first statement follows directly from (4.15). To derive the esti-mate we first need to show that data for solutions of (1.7) at time T aresufficiently small for sufficiently large T . Then we get problem (1.7) withsmall initial data so it is clear that we would be able to show (3.40) for any t > T .Multiply (1.7) by v and integrate over Ω. Then we have(4.19) ddt | v | + µ |∇ v | + ν | div v | = 0 . Since R Ω vdx = 0, (4.19) yields ddt ( | v | e µt ) ≤ | v ( t ) | ≤ e − µt | v (0) | . Let T be large. Then the quantity | v ( T ) | (4.21) | v ( T ) | ≤ e − µT | v (0) | is small. Integrating (4.19) with respect to time from t = T to t > T we get(4.22) | v ( t ) | + µ |∇ v | , Ω tT + ν | ∆ ϕ | , Ω tT = | v ( T ) | . Since(4.23) Z Ω vdx = 0 , Z Ω ϕdx = 0we obtain from (4.22) the inequality(4.24) | v ( t ) | + µ k v k , , Ω tT + ν k ϕ k , , Ω tT ≤ c | v ( T ) | . Differentiate (1 . with respect to x , multiply by v x and integrate over Ω.Then we have ddt | v x | + µ |∇ v x | + ν | div v x | = − Z Ω (rot ψ ) x · ∇ v · v x dx = Z Ω (rot ψ ) x · ∇ v x · vdx. Z132 — − − pplying the H¨older and Young inequalities to the r.h.s., integrating withrespect to time from t = T to t > T and using (4.24), we obtain(4.25) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ Ω tT | v ( T ) | + c k v ( T ) k . Differentiate (1 . twice with respect to x , multiply by v xx and integrateover Ω. Then we derive(4.26) ddt | v xx | + µ |∇ v xx | + ν | div v xx | = − Z Ω (rot ψ ) xx · ∇ v · v xx dx − Z Ω (rot ψ ) x · ∇ v x · v xx dx − Z Ω rot ψ · ∇ v xx · v xx dx ≡ I + I + I , where by the H¨older and Young inequalities we get | I | ≤ ε | v xx | + c/ε |∇ v | | rot ψ xx | , | I | ≤ ε | v xx | + c/ε | rot ψ x | | v xx | ,I = 0 . Using the above estimates in (4.26), summing up the result integrated withrespect to time with (4.25) and assuming that ε is sufficiently small wederive the inequality(4.27) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ , Ω tT | v ( T ) | + c |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT + c | rot ψ x | , ∞ , Ω tT | v xx | , , Ω tT + c k v ( T ) k . Using the interpolations | v ( T ) | | rot ψ x | , ∞ , Ω tT ≤ c | v ( T ) | ( ε / | rot ψ xx | , ∞ , Ω tT + cε − / | rot ψ | , ∞ , Ω tT ) = ε / | rot ψ xx | , ∞ , Ω tT + cε − / | v ( T ) | | rot ψ | , ∞ , Ω tT , |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT ≤ |∇ v | , ∞ , Ω tT ( ε / | rot ψ xxx | , Ω tT + cε − / | rot ψ | , Ω tT ) = ε / | rot ψ xxx | , Ω tT + cε − / |∇ v | , ∞ , Ω tT | rot ψ | , Ω tT Z132 — −− −−
Assume that T and ν are large. Assume that ϕ ( T ) ∈ H (Ω) , v ( T ) ∈ H (Ω) . Then there exists a constant B such that k ϕ ( T ) k + k v ( T ) k < cB, where B = (cid:0) e − µ T k v (0) k + A √ ν (cid:1) and A appears in (3.40). Moreover, for T and ν sufficiently large the following inequality holds (4.18) k v k , ∞ , Ω tT + k v k , , Ω tT + k ϕ k , ∞ , Ω tT + k∇ ϕ k , , Ω tT < cB for any t ∈ ( T, ∞ ) . Z132 — −− −− roof. The first statement follows directly from (4.15). To derive the esti-mate we first need to show that data for solutions of (1.7) at time T aresufficiently small for sufficiently large T . Then we get problem (1.7) withsmall initial data so it is clear that we would be able to show (3.40) for any t > T .Multiply (1.7) by v and integrate over Ω. Then we have(4.19) ddt | v | + µ |∇ v | + ν | div v | = 0 . Since R Ω vdx = 0, (4.19) yields ddt ( | v | e µt ) ≤ | v ( t ) | ≤ e − µt | v (0) | . Let T be large. Then the quantity | v ( T ) | (4.21) | v ( T ) | ≤ e − µT | v (0) | is small. Integrating (4.19) with respect to time from t = T to t > T we get(4.22) | v ( t ) | + µ |∇ v | , Ω tT + ν | ∆ ϕ | , Ω tT = | v ( T ) | . Since(4.23) Z Ω vdx = 0 , Z Ω ϕdx = 0we obtain from (4.22) the inequality(4.24) | v ( t ) | + µ k v k , , Ω tT + ν k ϕ k , , Ω tT ≤ c | v ( T ) | . Differentiate (1 . with respect to x , multiply by v x and integrate over Ω.Then we have ddt | v x | + µ |∇ v x | + ν | div v x | = − Z Ω (rot ψ ) x · ∇ v · v x dx = Z Ω (rot ψ ) x · ∇ v x · vdx. Z132 — − − pplying the H¨older and Young inequalities to the r.h.s., integrating withrespect to time from t = T to t > T and using (4.24), we obtain(4.25) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ Ω tT | v ( T ) | + c k v ( T ) k . Differentiate (1 . twice with respect to x , multiply by v xx and integrateover Ω. Then we derive(4.26) ddt | v xx | + µ |∇ v xx | + ν | div v xx | = − Z Ω (rot ψ ) xx · ∇ v · v xx dx − Z Ω (rot ψ ) x · ∇ v x · v xx dx − Z Ω rot ψ · ∇ v xx · v xx dx ≡ I + I + I , where by the H¨older and Young inequalities we get | I | ≤ ε | v xx | + c/ε |∇ v | | rot ψ xx | , | I | ≤ ε | v xx | + c/ε | rot ψ x | | v xx | ,I = 0 . Using the above estimates in (4.26), summing up the result integrated withrespect to time with (4.25) and assuming that ε is sufficiently small wederive the inequality(4.27) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c | rot ψ x | , ∞ , Ω tT | v ( T ) | + c |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT + c | rot ψ x | , ∞ , Ω tT | v xx | , , Ω tT + c k v ( T ) k . Using the interpolations | v ( T ) | | rot ψ x | , ∞ , Ω tT ≤ c | v ( T ) | ( ε / | rot ψ xx | , ∞ , Ω tT + cε − / | rot ψ | , ∞ , Ω tT ) = ε / | rot ψ xx | , ∞ , Ω tT + cε − / | v ( T ) | | rot ψ | , ∞ , Ω tT , |∇ v | , ∞ , Ω tT | rot ψ xx | , , Ω tT ≤ |∇ v | , ∞ , Ω tT ( ε / | rot ψ xxx | , Ω tT + cε − / | rot ψ | , Ω tT ) = ε / | rot ψ xxx | , Ω tT + cε − / |∇ v | , ∞ , Ω tT | rot ψ | , Ω tT Z132 — −− −− nd | rot ψ | , ∞ , Ω tT ≤ | v | , ∞ , Ω tT ≤ | v ( T ) | , | rot ψ | , Ω tT ≤ | v | , Ω tT ≤ | v ( T ) | , and similarly we have | rot ψ x | , ∞ , Ω tT | v xx | , , Ω tT ≤ ε / | v xxx | , Ω tT + cε − / | rot ψ x | , ∞ , Ω tT | v | , Ω tT . Using rot ψ = v − ∇ ϕ and | v ( t ) | = e − µ ( t − T ) | v ( T ) | so(4.28) t Z T | v ( t ′ ) | ≤ | v ( T ) | we obtain from (4.27) the following inequality for sufficiently small ε − ε (4.29) k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ c |∇ v | , ∞ , Ω tT | v ( T ) | + c | v ( T ) | | v ( T ) | + c | rot ψ x | , ∞ , Ω tT | v ( T ) | + c k v ( T ) k . Introduce the quantity X ( t ) = k v ( t ) k + µ k v k , , Ω tT + ν k ∆ ϕ k , , Ω tT . Then (4.29) implies for t > T (4.30) X ( t ) ≤ cX ( t ) | v ( T ) | + c k v ( T ) k . Then for T sufficiently large and a fixed point argument we obtain theestimate(4.31) X ( t ) ≤ c k v ( T ) k , where c is a correspondingly large constant.To derive more delicate estimate for ϕ we consider problem (1.7). From(1.7) we have(4.32) ϕ t − ( µ + ν )∆ ϕ = ∆ − ∂ x i ∂ x j ( v i v j ) − ∆ − ∂ x i ∂ x j ( ϕ x i v j ) . Multiplying (4.32) by ϕ and integrating over Ω yields(4.33) ddt | ϕ | + ( µ + ν ) |∇ ϕ | ≤ cν | vv | / + cν |∇ ϕv | / ≤ cν ( | v | | v | + | v | |∇ ϕ | ) . Z132 — − − ntegrating (4.33) with respect to time implies(4.34) | ϕ ( t ) | + ( µ + ν ) |∇ ϕ | , Ω tT ≤ cν | v ( T ) | ( | v | , , Ω tT + |∇ ϕ | , , Ω tT ) + | ϕ ( T ) | . Multiplying (1 . by ∇ ϕ and integrating over Ω gives ddt |∇ ϕ | + µ |∇ ϕ | + ν | ∆ ϕ | ≤ cν | rot ψ | | v | . Integrating with respect to time implies(4.35) |∇ ϕ ( t ) | + µ |∇ ϕ | , Ω tT + ν | ∆ ϕ | , Ω tT ≤ cν | rot ψ | , ∞ , Ω tT | v | , , Ω tT + |∇ ϕ ( T ) | . Differentiate (1 . with respect to x , multiply by ∇ ϕ x and integrate overΩ. Then we obtain(4.36) ddt |∇ ϕ x | + µ |∇ ϕ x | + ν | ∆ ϕ | = − Z Ω rot ψ · ∇ v x · ∇ ϕ x dx − Z Ω rot ψ x · ∇ v · ∇ ϕ x dx = Z Ω rot ψ · ∇∇ ϕ x · v x dx + Z Ω rot ψ x · ∇∇ ϕ x · vdx ≡ I + I , where | I | ≤ ε |∇ ϕ x | + c/ε | rot ψ | | v x | , | I | ≤ ε |∇ ϕ x | + c/ε | rot ψ x | | v | . Using the above estimates in (4.36), assuming that ε is sufficiently smalland integrating the result with respect to time we have(4.37) |∇ ϕ x ( t ) | + µ |∇ ϕ x | , Ω tT + ν | ∆ ϕ x | , Ω tT ≤ cν | rot ψ | , ∞ , Ω tT | v x | , , Ω tT + cν | rot ψ x | , , Ω tT | v | , ∞ , Ω tT + |∇ ϕ x ( T ) | . Differentiate (1 . twice with respect to x , multiply by ∇ ϕ xx and integrateover Ω. Then we obtain(4.38) ddt |∇ ϕ xx | + µ |∇ ϕ xx | + ν | ∆ ϕ xx | ≤ (cid:12)(cid:12)(cid:12)(cid:12) − Z Ω (rot ψ · ∇ v ) xx · ∇ ϕ xx dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω (rot ψ · ∇ v ) x · ∇ ϕ xxx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε |∇ ϕ xxx | + cε | rot ψ | |∇ v x | + cε | rot ψ x | |∇ v | . Z132 — − − ntegrating (4.38) with respect to time yields(4.39) |∇ ϕ xx ( t ) | + µ |∇ ϕ xx | , Ω tT + ν | ∆ ϕ xx | , Ω tT ≤ cν | rot ψ | , ∞ , Ω tT |∇ v x | , , Ω tT + cν | rot ψ x | , ∞ , Ω tT |∇ v | , , Ω tT + |∇ ϕ xx ( T ) | . From (4.34), (4.35), (4.37) and (4.39) it follows(4.40) k ϕ ( t ) k + µ k∇ ϕ k , , Ω tT + ν k ∆ ϕ k , , Ω tT ≤ cν | v ( T ) | ( | v | , , Ω tT + |∇ ϕ | , , Ω tT )+ cν ( k rot ψ k , ∞ , Ω tT + k rot ψ k , , Ω tT ) ·· ( k v k , , Ω tT + | v | , ∞ , Ω tT ) + k ϕ ( T ) k . ≤ cν k v ( T ) k + cν k ϕ ( T ) k , where the last inequality follows from (4.31). Hence (4.31) and (4.40) imply(4.18). This concludes the proof. u Lemma 5.1.
Let the assumptions of Theorem 3.6 hold. Let u ( t ) ∈ H (Ω) , u t ∈ L (Ω) . Then (5.1) ddt | u | + µ k u k ≤ c | u | ( |∇ v | + |∇ ϕ | )+ c ( k v k |∇ ϕ | + k∇ ϕ k + |∇ ϕ t | ) . Proof.
Since u is not divergence free we multiply (1.9) by u − ∇ ϕ . Integrat-ing the result over Ω we obtain(5.2) Z Ω u t · ( u − ∇ ϕ ) dx + Z Ω rot ψ · ∇ u · ( u − ∇ ϕ ) dx + Z Ω ( u − ∇ ϕ ) · ∇ ( v − u ) · ( u − ∇ ϕ ) dx + µ Z Ω ∇ u · ∇ ( u − ∇ ϕ ) dx = 0 . Z132 — −−
Since u is not divergence free we multiply (1.9) by u − ∇ ϕ . Integrat-ing the result over Ω we obtain(5.2) Z Ω u t · ( u − ∇ ϕ ) dx + Z Ω rot ψ · ∇ u · ( u − ∇ ϕ ) dx + Z Ω ( u − ∇ ϕ ) · ∇ ( v − u ) · ( u − ∇ ϕ ) dx + µ Z Ω ∇ u · ∇ ( u − ∇ ϕ ) dx = 0 . Z132 — −− −−
Since u is not divergence free we multiply (1.9) by u − ∇ ϕ . Integrat-ing the result over Ω we obtain(5.2) Z Ω u t · ( u − ∇ ϕ ) dx + Z Ω rot ψ · ∇ u · ( u − ∇ ϕ ) dx + Z Ω ( u − ∇ ϕ ) · ∇ ( v − u ) · ( u − ∇ ϕ ) dx + µ Z Ω ∇ u · ∇ ( u − ∇ ϕ ) dx = 0 . Z132 — −− −− ontinuing, we have(5.3) 12 ddt | u | − Z Ω u t · ∇ ϕdx + Z Ω rot ψ · ∇ u · udx − Z Ω rot ψ · ∇ u · ∇ ϕdx − Z Ω ( u − ∇ ϕ ) · ∇ u · udx + Z Ω ( u − ∇ ϕ ) · ∇ u · ∇ ϕdx + Z Ω ( u − ∇ ϕ ) · ∇ v · ( u − ∇ ϕ ) dx + µ |∇ u | − µ Z Ω ∇ u · ∇ ϕdx = 0 . The second term on the l.h.s. of (5.3) equals Z Ω div u t ϕdx = Z Ω ∆ ϕ t ϕdx = − Z Ω ∇ ϕ t · ∇ ϕdx = − ddt |∇ ϕ | . The third term vanishes. We estimate the fourth term by ε |∇ u | + c/ε | rot ψ | |∇ ϕ | . The fifth term vanishes. We estimate the sixth term by ε |∇ u | + c/ε ( | u | + |∇ ϕ | ) |∇ ϕ | . The seventh term is written in the form Z Ω u · ∇ vudx − Z Ω ∇ ϕ · ∇ v · udx − Z Ω u · ∇ v · ∇ ϕdx + Z Ω ∇ ϕ · ∇ v · ∇ ϕdx ≡ J + J + J + J , where | J | ≤ ε | u | + c/ε | u | |∇ v | , | J | ≤ ε | u | + c/ε |∇ v | |∇ ϕ | , | J | ≤ ε | u | + c/ε |∇ v | |∇ ϕ | , | J | ≤ |∇ ϕ | + |∇ v | |∇ ϕ | . Finally, the last term is bounded by ε |∇ u | + c/ε |∇ ϕ | . Z132 — − − mploying the above estimates in (5.3) and assuming that ε is sufficientlysmall yield(5.4) ddt | u | + µ k u k ≤ ddt |∇ ϕ | + c ( | rot ψ | |∇ ϕ | + | u | |∇ ϕ | + |∇ ϕ | |∇ ϕ | + | u | |∇ v | + |∇ v | |∇ ϕ | + |∇ ϕ | + |∇ ϕ | + |∇ ϕ | ) . By some interpolation inequality we have | u | |∇ ϕ | ≤ ε |∇ u | + c/ε | u | |∇ ϕ | . Then we write (5.4) in the form(5.5) ddt | u | + µ k u k ≤ c | u | ( |∇ v | + |∇ ϕ | )+ c [( | rot ψ | + |∇ ϕ | ) |∇ ϕ | + |∇ v | |∇ ϕ | + |∇ ϕ | + |∇ ϕ | ] + ddt |∇ ϕ | . Using that | rot ψ | + |∇ ϕ | ≤ c ( k rot ψ k + k∇ ϕ k ) ≤ c k v k we simplify (5.5) to the inequality(5.6) ddt | u | + µ k u k ≤ c | u | ( |∇ v | + |∇ ϕ | )+ c ( k v k |∇ ϕ | + |∇ ϕ | + |∇ ϕ | ) + ddt |∇ ϕ | . This implies (5.1) and concludes the proof.Next we have
Lemma 5.2.
Let the assumptions of Theorem 3.6 hold, let u ( t ) ∈ H (Ω) , u t ∈ H (Ω) . Then (5.7) ddt | u x | + µ k∇ u k ≤ c | u x | + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | )+ |∇ ϕ x | |∇ ϕ x | ) . Z132 — − − roof. Differentiate (1.9) with respect to x , multiply by ( u − ∇ ϕ ) x andintegrate the result over Ω. Then we have(5.8) Z Ω u xt · ( u x − ∇ ϕ x ) dx + Z Ω (rot ψ x · ∇ u + rot ψ · ∇ u x )( u x − ∇ ϕ x ) dx + Z Ω [( u − ∇ ϕ ) · ∇ ( v − u )] ,x · ( u x − ∇ ϕ x ) dx + µ Z Ω ∇ u x · ( ∇ u x − ∇ ϕ x ) dx = 0 . The first term on the l.h.s. of (5.8) equals12 ddt | u x | − Z Ω u xt · ∇ ϕ x dx, where the second term reads − Z Ω ∇ ϕ xt · ∇ ϕ x dx = − ddt |∇ ϕ x | . We write the second term on the l.h.s. of (5.8) in the form Z Ω rot ψ x · ∇ u · u x dx − Z Ω rot ψ x · ∇ u · ∇ ϕ x dx + Z Ω rot ψ · ∇ u x · u x dx − Z Ω rot ψ · ∇ u x · ∇ ϕ x dx ≡ X i =1 I i , where | I | ≤ ε | u x | + c/ε | rot ψ x | |∇ u | , | I | ≤ ε |∇ u | + c/ε | rot ψ x | |∇ ϕ x | ,I = 0 and | I | ≤ ε |∇ u x | + c/ε | rot ψ | ∞ |∇ ϕ x | . Next we examine the third term on the l.h.s. of (5.8). We write it in theform Z Ω ( u x − ∇ ϕ x ) · ∇ ( v − u ) · ( u x − ∇ ϕ x ) dx + Z Ω ( u − ∇ ϕ ) · ∇ ( v x − u x ) · ( u x − ∇ ϕ x ) dx ≡ J + L. Z132 — − − irst we consider J . It can be expressed in the form − Z Ω ( u x − ∇ ϕ x ) · ∇ u · ( u x − ∇ ϕ x ) dx + Z Ω ( u x − ∇ ϕ x ) · ∇ v · ( u x − ∇ ϕ x ) dx = − Z Ω u x · ∇ u · u x dx + Z Ω u x · ∇ u · ∇ ϕ x dx + Z Ω ∇ ϕ x · ∇ u · u x dx − Z Ω ∇ ϕ x · ∇ u · ∇ ϕ x dx + Z Ω u x · ∇ v · u x dx − Z Ω u x · ∇ v · ∇ ϕ x dx − Z Ω ∇ ϕ x · ∇ v · u x dx + Z Ω ∇ ϕ x · ∇ v · ∇ ϕ x dx ≡ X i =1 J i , where | J | ≤ | u x | ≤ c | u xx | / | u x | / ≤ ε | u xx | + c (1 /ε ) | u x | , | J | ≤ ε | u x | + c/ε |∇ u | |∇ ϕ x | , | J | ≤ ε | u x | + c/ε |∇ u | |∇ ϕ x | , | J | ≤ ε |∇ u | + c/ε |∇ ϕ x | |∇ ϕ x | , | J | ≤ ε | u x | + c/ε | u x | |∇ v | , | J | ≤ ε | u x | + c/ε |∇ v | |∇ ϕ x | , | J | ≤ ε | u x | + c/ε |∇ v | |∇ ϕ x | , | J | ≤ |∇ ϕ x | + |∇ v | |∇ ϕ x | . Next we examine L . We express it in the form − Z Ω ( u − ∇ ϕ ) · ∇ u x · ( u x − ∇ ϕ x ) dx + Z Ω ( u − ∇ ϕ ) · ∇ v x · ( u x − ∇ ϕ x ) dx = − Z Ω ( u − ∇ ϕ ) · ∇ u x · u x dx + Z Ω u · ∇ u x · ∇ ϕ x dx − Z Ω ∇ ϕ · ∇ u x · ∇ ϕ x dx + Z Ω u · ∇ v x · u x dx − Z Ω u · ∇ v x · ∇ ϕ x dx − Z Ω ∇ ϕ · ∇ v x · u x dx + Z Ω ∇ ϕ · ∇ v x · ∇ ϕ x dx ≡ X i =1 L i , Z132 — − − here L = 0, | L | ≤ ε |∇ u x | + c/ε | u | |∇ ϕ x | , | L | ≤ ε |∇ u x | + c/ε |∇ ϕ | |∇ ϕ x | , | L | ≤ ε | u x | + c/ε | u | |∇ v x | , | L | ≤ |∇ ϕ x | + | u | |∇ v x | , | L | ≤ ε | u x | + c/ε |∇ ϕ | |∇ v x | , | L | ≤ |∇ ϕ x | + |∇ ϕ | |∇ v x | . Finally, the last term on the l.h.s. of (5.8) equals µ |∇ u x | − µ Z Ω ∇ u x · ∇ ϕ x dx, where the second integral is bounded by ε |∇ u x | + c/ε |∇ ϕ x | . Using the above estimates in (5.8) yields(5.9) ddt | u x | + µ k∇ u k ≤ c | u x | + c k u k ( | rot ψ x | + |∇ ϕ x | + |∇ v | + |∇ v x | ) + c ( | rot ψ x | |∇ ϕ x | + | rot ψ | ∞ |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | + |∇ ϕ x | + |∇ ϕ x | + |∇ ϕ | |∇ ϕ x | + |∇ v | |∇ ϕ x | + |∇ ϕ | |∇ v x | + |∇ ϕ xt | ) . Simplifying (5.9) we obtain(5.10) ddt | u x | + µ k∇ u k ≤ c | u x | + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . This implies (5.7) and concludes the proof.
Lemma 5.3.
Assume that Lemma 4.1 and Lemma 4.2 hold for any t ∈ R + .Assume that cA ≤ µT / , u (0) ∈ L (Ω) . Then (5.11) | u ( kT ) | ≤ c exp( cA )( A + 1) A ν / (1 − exp( − µT / − µkT / | u (0) | , k ∈ N Z132 — −−
Assume that Lemma 4.1 and Lemma 4.2 hold for any t ∈ R + .Assume that cA ≤ µT / , u (0) ∈ L (Ω) . Then (5.11) | u ( kT ) | ≤ c exp( cA )( A + 1) A ν / (1 − exp( − µT / − µkT / | u (0) | , k ∈ N Z132 — −− −−
Assume that Lemma 4.1 and Lemma 4.2 hold for any t ∈ R + .Assume that cA ≤ µT / , u (0) ∈ L (Ω) . Then (5.11) | u ( kT ) | ≤ c exp( cA )( A + 1) A ν / (1 − exp( − µT / − µkT / | u (0) | , k ∈ N Z132 — −− −− nd for t ∈ [ kT, ( k + 1) T ] , (5.12) | u ( t ) | ≤ c exp( cA )( A + 1) A ν + exp( − µ ( t − kT ) + cA ) ·· (cid:20) c exp( cA )( A + 1) A ν / (1 − exp( − µT / − µkT / | u (0) | (cid:21) . Proof.
In view of Lemma 4.1 it follows that Lemma 4.2 holds for interval[ kT, ( k + 1) T ], k ∈ N . Considering (5.1) in the interval [ kT, ( k + 1) T ] wehave(5.13) | u ( t ) | ≤ exp( − µt + cA ) t Z kT ( k v k |∇ ϕ | + k∇ ϕ k + |∇ ϕ t | ) exp( µt ′ ) dt ′ + exp[ − µ ( t − kT ) + cA ] | u ( kT ) | . Continuing, we have(5.14) | u ( t ) | ≤ c exp( cA )( k v k , ∞ , Ω tkT |∇ ϕ | , , Ω tkT + k∇ ϕ k , , Ω tk + |∇ ϕ t | , Ω tkT ) + exp( − µ ( t − kT ) + cA ) | u ( kT ) | , where Ω tkT = Ω × ( kT, t ). Setting t = ( k + 1) T and using the properties ofsolutions described by Lemma 4.2 we have(5.15) | u (( k + 1) T ) | ≤ c exp( cA )( A + 1) A ν + exp( − µT + cA ) | u ( kT ) | . Using − µ T + cA ≤ | u ( t ) | ≤ c exp( cA )( A + 1) A ν + exp[ − µ ( t − kT ) + cA ] | u ( kT ) | . From (5.16) and (5.11) we obtain (5.12). This concludes the proof.
Lemma 5.4.
Let the assumptions of Theorem 3.6, Lemmas 4.1, 4.2 hold.Let k u (0) k ≤ γ , where γ ≤ γ ∗ and γ ∗ is so small that µ − c exp(2 cA ) γ ∗ ≥ µ/ .Then for ν and T sufficiently large we have (5.17) k u ( kT ) k ≤ γ, k ∈ N , and (5.18) k u ( t ) k ≤ (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) , t ∈ [ kT, ( k + 1) T ] . Z132 — − − roof. From (5.1) and (5.7) we have(5.19) ddt k u k + µ k∇ u k ≤ c k u k + c k u k k v k + ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We write (5.19) in the form(5.20) ddt k u k ≤ − ( µ − c k u k ) k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We consider (5.20) in the interval [ kT, ( k + 1) T ], k ∈ N . We know thatLemma 4.2 holds in this interval. Assume that(5.21) k u ( kT ) k ≤ γ, where γ will be chosen sufficiently small. Introduce the quantity(5.22) X ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) k u ( t ) k . Assume that estimate (3.53) holds for interval [ kT, ( k + 1) T ]. Then(5.23) t Z kT k v ( t ′ ) k dt ′ ≤ A , t ∈ [ kT, ( k + 1) T ] . Therefore(5.24) X ( t ) ≥ exp( − cA ) k u ( t ) k . Introduce the quantity(5.25) G ( t ) = c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Let(5.26) K ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) G ( t ) . Z132 — − − hen (5.20) takes the form(5.27) ddt X ≤ − (cid:20) µ − c exp (cid:18) c t Z kT k v ( t ′ ) k dt ′ (cid:19) X (cid:21) X + K . In view of (3.53) we have(5.28) K ( t ) ≤ c ( A + 1) A ν and (5.22) implies(5.29) X ( kT ) = k u ( kT ) k ≤ γ. Suppose that t ∗ = inf { t ∈ [ kT, ( k + 1) T ] : X ( t ) > γ } . Let γ ∈ (0 , γ ∗ ], where γ ∗ is so small that(5.30) µ − c exp(2 cA ) γ ∗ ≥ µ/ . Hence, for t ≤ t ∗ we derive from (5.27) the inequality(5.31) ddt X ≤ − µ X + K . Assume that ν is so large that K ( t ) ≤ c ( A + 1) A ν ≤ µ γ for t ∈ [ kT, ( k + 1) T ] . Then ddt X | t = t ∗ ≤ − (cid:18) µ − µ (cid:19) γ < t ∗ does not exist in [ kT, ( k + 1) T ]. Hence(5.32) k u ( t ) k ≤ γ exp( cA ) , t ∈ [ kT, ( k + 1) T ] . In view of (5.30), (5.32) and for γ ≤ γ ∗ we obtain from (5.20) the inequality(5.33) ddt k u k ≤ − µ k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Z132 — −−
Let the assumptions of Theorem 3.6, Lemmas 4.1, 4.2 hold.Let k u (0) k ≤ γ , where γ ≤ γ ∗ and γ ∗ is so small that µ − c exp(2 cA ) γ ∗ ≥ µ/ .Then for ν and T sufficiently large we have (5.17) k u ( kT ) k ≤ γ, k ∈ N , and (5.18) k u ( t ) k ≤ (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) , t ∈ [ kT, ( k + 1) T ] . Z132 — − − roof. From (5.1) and (5.7) we have(5.19) ddt k u k + µ k∇ u k ≤ c k u k + c k u k k v k + ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We write (5.19) in the form(5.20) ddt k u k ≤ − ( µ − c k u k ) k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We consider (5.20) in the interval [ kT, ( k + 1) T ], k ∈ N . We know thatLemma 4.2 holds in this interval. Assume that(5.21) k u ( kT ) k ≤ γ, where γ will be chosen sufficiently small. Introduce the quantity(5.22) X ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) k u ( t ) k . Assume that estimate (3.53) holds for interval [ kT, ( k + 1) T ]. Then(5.23) t Z kT k v ( t ′ ) k dt ′ ≤ A , t ∈ [ kT, ( k + 1) T ] . Therefore(5.24) X ( t ) ≥ exp( − cA ) k u ( t ) k . Introduce the quantity(5.25) G ( t ) = c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Let(5.26) K ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) G ( t ) . Z132 — − − hen (5.20) takes the form(5.27) ddt X ≤ − (cid:20) µ − c exp (cid:18) c t Z kT k v ( t ′ ) k dt ′ (cid:19) X (cid:21) X + K . In view of (3.53) we have(5.28) K ( t ) ≤ c ( A + 1) A ν and (5.22) implies(5.29) X ( kT ) = k u ( kT ) k ≤ γ. Suppose that t ∗ = inf { t ∈ [ kT, ( k + 1) T ] : X ( t ) > γ } . Let γ ∈ (0 , γ ∗ ], where γ ∗ is so small that(5.30) µ − c exp(2 cA ) γ ∗ ≥ µ/ . Hence, for t ≤ t ∗ we derive from (5.27) the inequality(5.31) ddt X ≤ − µ X + K . Assume that ν is so large that K ( t ) ≤ c ( A + 1) A ν ≤ µ γ for t ∈ [ kT, ( k + 1) T ] . Then ddt X | t = t ∗ ≤ − (cid:18) µ − µ (cid:19) γ < t ∗ does not exist in [ kT, ( k + 1) T ]. Hence(5.32) k u ( t ) k ≤ γ exp( cA ) , t ∈ [ kT, ( k + 1) T ] . In view of (5.30), (5.32) and for γ ≤ γ ∗ we obtain from (5.20) the inequality(5.33) ddt k u k ≤ − µ k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Z132 — −− −−
Let the assumptions of Theorem 3.6, Lemmas 4.1, 4.2 hold.Let k u (0) k ≤ γ , where γ ≤ γ ∗ and γ ∗ is so small that µ − c exp(2 cA ) γ ∗ ≥ µ/ .Then for ν and T sufficiently large we have (5.17) k u ( kT ) k ≤ γ, k ∈ N , and (5.18) k u ( t ) k ≤ (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) , t ∈ [ kT, ( k + 1) T ] . Z132 — − − roof. From (5.1) and (5.7) we have(5.19) ddt k u k + µ k∇ u k ≤ c k u k + c k u k k v k + ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We write (5.19) in the form(5.20) ddt k u k ≤ − ( µ − c k u k ) k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . We consider (5.20) in the interval [ kT, ( k + 1) T ], k ∈ N . We know thatLemma 4.2 holds in this interval. Assume that(5.21) k u ( kT ) k ≤ γ, where γ will be chosen sufficiently small. Introduce the quantity(5.22) X ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) k u ( t ) k . Assume that estimate (3.53) holds for interval [ kT, ( k + 1) T ]. Then(5.23) t Z kT k v ( t ′ ) k dt ′ ≤ A , t ∈ [ kT, ( k + 1) T ] . Therefore(5.24) X ( t ) ≥ exp( − cA ) k u ( t ) k . Introduce the quantity(5.25) G ( t ) = c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Let(5.26) K ( t ) = exp (cid:18) − c t Z kT k v ( t ′ ) k dt ′ (cid:19) G ( t ) . Z132 — − − hen (5.20) takes the form(5.27) ddt X ≤ − (cid:20) µ − c exp (cid:18) c t Z kT k v ( t ′ ) k dt ′ (cid:19) X (cid:21) X + K . In view of (3.53) we have(5.28) K ( t ) ≤ c ( A + 1) A ν and (5.22) implies(5.29) X ( kT ) = k u ( kT ) k ≤ γ. Suppose that t ∗ = inf { t ∈ [ kT, ( k + 1) T ] : X ( t ) > γ } . Let γ ∈ (0 , γ ∗ ], where γ ∗ is so small that(5.30) µ − c exp(2 cA ) γ ∗ ≥ µ/ . Hence, for t ≤ t ∗ we derive from (5.27) the inequality(5.31) ddt X ≤ − µ X + K . Assume that ν is so large that K ( t ) ≤ c ( A + 1) A ν ≤ µ γ for t ∈ [ kT, ( k + 1) T ] . Then ddt X | t = t ∗ ≤ − (cid:18) µ − µ (cid:19) γ < t ∗ does not exist in [ kT, ( k + 1) T ]. Hence(5.32) k u ( t ) k ≤ γ exp( cA ) , t ∈ [ kT, ( k + 1) T ] . In view of (5.30), (5.32) and for γ ≤ γ ∗ we obtain from (5.20) the inequality(5.33) ddt k u k ≤ − µ k u k + c k u k k v k + c ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) . Z132 — −− −− rom (5.33) we have(5.34) ddt (cid:20) k u k exp (cid:18) µ t − c t Z kT k v ( t ′ ) k dt ′ (cid:19)(cid:21) ≤ c (cid:0) k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | (cid:1) exp (cid:18) µ t − c t Z kT || v ( t ′ k dt ′ (cid:19)(cid:19) . Integrating (5.34) with respect to time yields(5.35) k u ( t ) k ≤ c exp (cid:18) − µ t + c t Z kT k v ( t ′ ) k dt ′ (cid:19) t Z kT ( k v k k∇ ϕ k + |∇ ϕ xt | + k∇ ϕ k + |∇ ϕ | |∇ ϕ x | + |∇ ϕ x | |∇ ϕ x | ) exp (cid:18) µ t ′ (cid:19) dt ′ + exp (cid:18) − µ t − kT ) + c t Z kT k v ( t ′ ) k dt ′ (cid:19) k u ( kT ) k . Setting t = ( k + 1) T we derive(5.36) k u (( k + 1) T ) k ≤ c exp( cA )( A + 1) A ν + exp (cid:18) − µ T + cA (cid:19) k u ( kT ) k . Assuming that − µ T + cA ≤ k u ( kT ) k ≤ γ we obtain k u (( k + 1) T ) k ≤ c exp( cA )( A + 1) A ν + exp (cid:18) − µ T (cid:19) γ ≤ γ which holds for sufficiently large ν and T . Hence(5.37) k u ( kT ) k ≤ γ for any k ∈ N . Using (5.37) in (5.35) yields(5.38) k u ( t ) k ≤ (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) , where t ∈ [ kT, ( k + 1) T ]. Estimates (5.37) and (5.38) imply (5.17) and(5.18). This concludes the proof. 52 Z132 — − − Estimate for solutions to the Navier--Stokes equations
Theorem 6.1.
Let the assumptions of Theorem 3.6, Lemmas 4.1, 4.2, 5.3,5.4 hold. Then (6.1) k V ( t ) k ≤ A + (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) . Proof.
Recall that V = v − u Hence using estimates (3.40), (5.18) yields k V ( t ) k ≤ k v k + k u k ≤ A + (cid:20) c ( A + 1) A ν + γ (cid:21) exp( cA ) . This concludes the proof.
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