AN A POSTERIORI CONDITION ON THE NUMERICAL APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS FOR THE EXISTENCE OF A STRONG SOLUTION∗ MASOUMEH DASHTI AND JAMES C. ROBINSON † 7 Abstract. In their 2006 paper, Chernyshenko et al prove that a sufficiently smooth strong 0 solutionofthe 3dNavier-Stokes equations isrobustwithrespect tosmallenough changes ininitial 0 conditionsandforcingfunction. Theyalsoshowthatifaregularenoughstrongsolutionexiststhen 2 Galerkinapproximationsconvergetoit. Theythenusetheseresultstoconcludethattheexistenceof asufficientlyregularstrongsolutioncanbeverifiedusingsufficientlyrefinednumericalcomputations. n In this paper we study the solutions with minimal required regularity to be strong, which are less a regular than those considered in Chernyshenko et al (2006). We prove a similar robustness result J andshowthevalidityoftheresultsrelatingconvergentnumericalcomputationsandtheexistenceof 2 thestrongsolutions. 1 Key words. Navier-Stokesequations, Galerkinmethod ] P A 1. Introduction. For the three-dimensional Navier-Stokes equations the existence of strong solutions over an arbitrary time interval is not known. Therefore the . h validity of the results of the numerical solutions of 3d Navier-Stokes equations is not t a obvious. This problem is addressed in Chernyshenko et al (2006) where a rigorous m relationshipbetweennumericalandsufficiently regularexactsolutions is given. They [ show for sufficiently smooth initial conditions and forcing functions (data) that al- though a priori there is no guaranteeof the validity of the numericalsolutions, there 1 is an a posteriori condition that if satisfied by the numerical results guarantees the v 1 existenceofastrongsolution. Inthispaperweshowthevalidityoftheresultsproved 4 in Chernyshenko et al (2006) for the less regular strong solutions which are not cov- 3 ered there. 1 WewillstudytheNavier-Stokesequationsintheirfunctionalform. Forabounded 0 domainΩ we let be the space ofdivergence-freesmoothvector-valuedfunctions on 7 H 0 Ω with compact support and zero average and define / h H =closure of in[L2(Ω)]3, t H a V =closure of in[H1(Ω)]3. m H : We use the same notation H and V for the similar spaces of periodic functions over v i the periodic domain Q. Then the Navier-Stokes equations in their functional form X are written as (Constantin and Foias 1988, Robinson 2001) r a du +νAu+B(u,u)=f, with u(0)=u (1.1) 0 dt whereAu= Π∆,B(u,u)=Π(u. )uwithΠtheorthogonalprojectionfromL2 into − ∇ H. We will consider this equation with the following cases for the data (a) ‘minimal regularity’when u V and f L2(0,T;H) L1(0,T;V) 0 ∈ ∈ ∩ (b) ‘second order regularity’when u V2 and f L2(0,T;V) L1(0,T;V2), 0 ∈ ∈ ∩ ∗SUBMITTEDTOSIAMJOURNALONNUMERICALANALYSIS †Mathematics Institute, University of Wawick, Coventry CV4 7AL. UK (m.dashti@warwick.ac.uk), (j.c.robinson@warwick.ac.uk). 1 where Vm = Hm V. For the periodic case we know that D(Am/2) = Hm V ∩ ∩ for all m and therefore we define the normon Vm as u = Am/2u. We denote by m k k | | (, ) and the inner product and norm on H. · · |·| We will show that the strong solution of (1.1) with the data introduced in (a) or (b), ifitexists,remainsstrongifthe changesinthe initial conditionandforcingfunc- tion are small enough. The exact conditions required for these changes to be ‘small’ aregivenintheorems4.1and4.2. Forexampleinthecaseofminimallyregularstrong solutions we require T D(u v ) + Df(s) Dg(s) ds 0 0 | − | | − | Z0 1 ν3 1/4 k2 T 27k2 1 1 < exp Du(s)4+ Du(s) Au(s) ds . k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | ! We then, in theorems 6.1 and 6.2, use these robustness results to find an a posteriori condition that if satisfied by sufficiently refined numerical approximations, implies the existence of a strong solution. We also show that if a strong solution exists the Galerkin approximations convergeto it and then use this to prove that the existence ofastrongsolutioncanbeverifiedbytheGalerkinapproximations. Inthelastsection weconsiderachannelflowasaphysicalexamplethatcanbedescribedbytheNavier- Stokesequationswiththeconditionsintroducedabove. Forthisexamplewewillshow how the results of this paper canbe applied to the Galerkinapproximationsto verify the existence of a strong solution. The results we prove here for a strong solution with lowest regularity hold in a generalbounded domainas wellas inthe absence ofboundaries unlike the results for more regular solutions which are proved only for the equations in a periodic domain or the whole space. 2. General ODE lemma. WefirstproveanODElemma whichwillbeusedin dealing with the differential inequalities that appear in the proofs. We consider the differential inequality dy δ(t)+αyn with y(0)=y >0 0 dt ≤ and find the conditions on y , δ(t) and α that ensure that y(t) exists on a finite time 0 interval [0,T]. This lemma is a generalization of the result obtained for n = 2 in Chernyshenko et al (2006). Lemma 2.1. Let T > 0, α > 0 and n > 1 be constants and let δ(t) be a non- negative continuous function on [0,T]. Let y satisfy the differential inequality dy δ(t)+αyn with y(0)=y >0 (2.1) 0 dt ≤ and define T η =y + δ(s)ds. 0 Z0 (i) If 1 η < (2.2) [(n 1)αT]1/(n−1) − then y(t) remains bounded on [0,T] 2 (ii) y(t) 0 uniformly on [0,T] as η 0. → → Proof. We first consider the following differential inequality z˙ αzn with z(0)=η (2.3) ≤ and show that sup y(T) sup z(T). (2.4) ≤ S1 S2 where S and S are the sets of all possible solutions of inequalities (2.1) and (2.3) 1 2 respectively. Since y˙, z˙, y(0)andz(0)arenon-negative,the suprema ofy(T)and z(T)happen when y˙=δ(t)+αyn, z˙=αzn for all t [0,T]. In this case, for the difference w =y z we have ∈ − n−1 n 1 t w˙ =δ(t)+αw − yn−1−k zk with w(0)= δ(s)ds. k − k=0(cid:18) (cid:19) Z0 X Since y(t) and z(t) are greater than zero and assuming they remain finite over [0,T], there exists some M >0 such that w˙ αMw+δ(t) ≤ and therefore by Gronwall’s inequality t w(t) w(0)eαMt+ δ(s)ds ≤ Z0 t =(1 eαMt) δ(s)ds. − Z0 This implies w(t) 0 for any t [0,T] and (2.4) follows. ≤ ∈ The inequality (2.3) can be written as z˙ α zn ≤ Integrating both sides from 0 to t yields η z(T) . (2.5) ≤ ( αT(n 1)(η)n−1+1)1/n−1 − − Since y(t) y(T) z(T), y(t) remains finite on [0,T] provided that ≤ ≤ αT(n 1)(η)n−1 <1 − which yields (2.2). Furthermore, it is clear from (2.5) that z(T) 0 as η 0, from which it follows → → that y(t) 0 uniformly on [0,T]. → 3 3. The inequalities for the nonlinear term. For the nonlinear operator,we will use the following inequalities in this paper (B(u,v),Aw) k Du Dv 1/2 Av 1/2 Aw (3.1) | |≤ | || | | | | | (B(w,u),A2w) c u w 2 (3.2) | |≤ k k3k k2 (B(u,w),A2w) c′ u w 2, (3.3) | |≤ k k3k k2 The first inequality holds for both Dirichlet and periodic boundary conditions (Con- stantinandFoias1988)whiletheothertwoarevalidonlyintheabsenceofboundaries with c and c′ independent of the size of the domain (Kato 1972). The constant k for the periodic and no-slip domain and also the whole of R3, is independent of the domain Ω. In a general domain with nonzero boundary conditions however, it depends on the regularity properties of the boundary of Ω. To see this we use the Ho¨lder inequality to write 3 B(u,v,Aw) u D v Aw . i L6 i j L3 j | |≤ k k k k | | i,j=1 X By the Sobolev inequality u c Du (3.4) L6 s k k ≤ | | we have Dv Dv 1/2 Dv 1/2 c1/2 Av 1/2 Dv 1/2. k kL3 ≤k kL6 | | ≤ s | | | | Therefore we can write 3 B(u,v,Aw) u D v Aw i L6 i j L3 j | |≤ k k k k | | i,j=1 X 9 u Dv Aw L6 L3 ≤ k k k k | | 9c3/2 Du Dv 1/2 Av 1/2 Aw ≤ s | || | | | | | and so k = 9c3/2. For no-slip boundary conditions and the whole of R3, c , the s s constant of the Sobolev inequality (3.4), does not depend on Ω (Ziemer 1989). For a bounded domain with non-zero boundary conditions, c depends on the regularity s properties of ∂Ω but is independent of the size of Ω (Adams 1975). Therefore for the cubicdomainofthe periodicboundaryconditionsalso,itdoesnotdependonthesize of the domain. In fact, the cubic domain has the strong Lipschitz property and for suchadomainAdams(1975,Lemma5.10)hasshownthatc =4√2. Similarly,cand s c′ depend on the constant of the Sobolev inequality kukL6/(3−2k) ≤ cs,kkukHk which is again independent of the size of the domain (Adams 1975 and Ziemer 1989). In obtaining the robustness conditions in the next section, we keep track of the constants k, c and c′ when they appear, so that the value of the constant coefficients in the robustness conditions could be computed if desired. The inequalities (3.1)–(3.3) are not as elegant as the inequalities available for the more regular solutions considered in Chernyshenko et al (2006) and this is why different proofs are needed in the less regular cases we are studying here. We note here that assuming a strong solution u L∞(0,T;V) L2(0,T;V2) ∈ ∩ exists, from (3.1) we have B(u,u) L2(0,T;H) and therefore (1.1) implies that ∈ 4 du/dt L2(0,T;H). Having u V2 and f L2(0,T;V) in case (b), from the 0 ∈ ∈ ∈ regularityresultforperiodicdomainsprovedinConstantinandFoias(1988)weknow that the strong solution u is in fact more regular and an element of L∞(0,T;V2) ∩ L2(0,T;V3). 4. Robustness of strong solutions. Here we show that as long as a strong solution exists for some specific initial data and forcing function, the equations with sufficiently close data also have a strong solution. A similar result about the robustness of strong solutions with respect to changes in the forcing function is proved by Fursikov in his 1980 paper. For the three-dimensional Navier-Stokes equations with initial condition u Vm where m 1/2, he shows that the set of forcing functions 0 ∈ ≥ for which a strong solution exists is open in Lq(0,T;Vm−1) with q 2. ≥ Forbothminimallyregularsolutions(a)andthe moreregularcase(b)thecondi- tion we obtain here for the data depends explicitly on the viscosity coefficient unlike themoreregularcasesconsideredinChernyshenkoetal(2006). Therobustnessresult we prove for the minimally regular strong solutions holds in a general bounded domain. For the secondorderregularsolutionshowever,we need to restrictthe domain to be periodic or the whole of R3 since the inequalities (3.2) and (3.3) only hold in a periodic domain (Constantin and Foias 1988) or the whole of R3 (Kato 1972). (a) Strong solutionswithminimalregularity. Toprovetherobustnesswith respect to the data we write the governing equation for the difference between the solution of the equations with nearby data and the strong solution and find a bound on the norm of this difference. Theorem 4.1. Let f L1(0,T,V) L2(0,T;H), u V and u L∞(0,T;V) 0 ∈ ∩ ∈ ∈ ∩ L2(0,T;V2) be a strong solution of du +Au+B(u,u)=f with u(0)=u . 0 dt If T D(u v ) + Df(s) Dg(s) ds 0 0 | − | | − | Z0 1 ν3 1/4 k2 T 27k2 1 1 < exp Du(s)4+ Du(s) Au(s) ds (4.1) k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | ! then the solution of dv +νAv+B(v,v)=g with v(0)=v 0 dt is also a strong solution on [0,T] with the same regularity as u. Proof. By the local existence of strong solutions (Constantin and Foias 1988, Temam 1995) we know that there exists T∗ > 0 such that v L∞(0,T′;V) ∈ ∩ L2(0,T′;V2) for any T′ < T∗. We consider T∗ the maximal time of existence of the strongsolutionv meaning that limsup Dv = . In the followingargument t→T∗| | ∞ we assume T∗ T and deduce a contradiction. ≤ We consider w =u v which satisfies − dw +νAw+B(u,w)+B(w,u) B(w,w)=f g. dt − − 5 Over the time interval t [0,T′) for any T′ < T∗ we have dv/dt L2(0,T′;H) and ∈ ∈ since T∗ T, du/dt L2(0,T′;H). Therefore taking the inner product of the above ≤ ∈ equation with Aw and using (3.1) we can write 1 d Dw2+ν Aw2 k Du Dw1/2 Aw3/2+k Dw Du1/2 Au1/2 Aw 2dt| | | | ≤ | || | | | | || | | | | | +k Dw3/2 Aw3/2+ D(f g) Dw. | | | | | − || | We then use Young’s inequality to remove Aw (which causes the appearence of ν in | | the coefficients) and obtain 1 d 27k4 k2 27k4 Dw2 Du4+ Du Au Dw2+ Dw6+ D(f g) Dw. 2dt| | ≤ 4ν3 | | 2ν| || | | | 4ν3 | | | − || | (cid:18) (cid:19) Dividing by Dw (assuming Dw is nonzero. If not, replacing Dw by Dw +e 0 | | | | | | | | with e >0 will leadus to the same result after tending e to zeroat the end) we get 0 0 d 27k4 k2 27k4 Dw Du4+ Du Au Dw + Dw5+ D(f g). dt| |≤ 4ν3 | | 2ν| || | | | 4ν3 | | | − | (cid:18) (cid:19) Now letting k2 t 27k2 1 1 β(t)= Du4+ Du Au ds 2 2 ν3| | ν| || | Z0 (cid:18) (cid:19) and setting y(t)= Dw(t)e−β(t) the above inequality can be written as | | dy αy5+δ(t), dt ≤ where α= 27k4e4β(T) and δ(t)= Df(t) Dg(t). 4ν3 | − | By lemma 2.1, if the condition(4.1) is satisfied, Dw(t) is uniformly bounded on | | [0,T∗). This implies that Dv(T∗) < which contradicts the maximality of T∗. It | | ∞ follows that v(t) is a strong solution on [0,T]. (b) Strong solutions with second order regularity. We follow the same approachasintheminimallyregularcasetoshowtherobustnessforastrongsolution withsecondorderregularity. Herealsotheinequalitiesweneedtouseforthenonlinear operator to find a bound on the V2-norm of the difference between nearby solutions depend on the V3-norm of this difference and therefore the robustness condition on the data depends explicitly on the viscosity coefficient. However, its dependence is weaker in this case. Theorem4.2. Letf L1(0,T,V2) L2(0,T;V),u V2 andu L∞(0,T;V2) 0 ∈ ∩ ∈ ∈ ∩ L2(0,T;V3) be a strong solution of du +Au+B(u,u)=f with u(0)=u . 0 dt If T 1 2ν T A(u v ) + A(f g) dt < exp (c+c′) u dt , (4.2) 0 0 3 | − | Z0 | − | crT −Z0 k k ! 6 then the solution of dv +Av+B(v,v)=g with v(0)=v 0 dt is also a strong solution on [0,T] with the same regularity as u. Proof. As before, the difference w=u v satisfies − dw +νAw+B(u,w)+B(w,u)+B(w,w)=f g. dt − Taking the inner product of the above equation with A2w and using (3.2) and (3.3) we obtain 1 d w +ν w 2 (c+c′) u 2 w 2+c w w 2+ f g w . 2dtk k2 k k3 ≤ k k3k k2 k k3k k2 k − k2k k2 We apply Young’s inequality to the second term on the right hand side and get 1 d c2 w 2 (c+c′) u w 2+ w 4+ f g w . 2dtk k2 ≤ k k3k k2 4νk k2 k − k2k k2 Dividing the above inequality by w yields 2 k k d c2 w (c+c′) u w + w 3+ f g . dtk k2 ≤ k k3k k2 4νk k2 k − k2 Now we consider y(t)= w e−β(t) 2 k k with β(t)= t(c+c′) u dt. The equation with this new variable becomes 0 k k3 R dy δ(t)+αy3, dt ≤ where α = c2e2β(T) and δ = f g . Using lemma 2.1 w is bounded if the 4ν k − k2 k k2 condition (4.2) is satisfied. 5. Convergence of Galerkin approximations. Hereweshowthatifastrong solutionwithminimalorsecondorderregularityexists,Galerkinapproximationscon- verge to it. Similar results about convergence of various numerical method assuming the existence of a strong solution, are given for finite element methods by Heywood andRannacher(1982),foraFouriercollocationmethodbyE(1993)andforanonlin- ear Galerkinmethod by Devulder, Marion and Titi (1993). Here as in Chernyshenko etal(2006),wemakenoassumptionontheregularityoftheGalerkinapproximations. For the minimally regular case the result we prove here holds for the solution of the equations overa generalbounded domain. For the second orderregular solutions we need to use inequality (3.3) which is valid only in a periodic domain or the whole of R3. In the following theorems we let P be the orthogonal projection in H onto the n space spanned by the first n eigenfunctions of the Stokes operator A, w n , or- { j}j=1 dered so that their corresponding eigenvalues satisfy 0 < λ λ ... . Since the 1 2 ≤ ≤ eigenfunctions of A are smooth (Constantin and Foias 1988), for any u Vm, m 0 (with V0 =H) we have u =P u= n (u,w )w Vm. ∈ ≥ n n j=1 j j ∈ We notethatwhatweobtainhereaboutthe convergenceofGalerkinapproxima- P tions would be useful even if the existence of regular solutions was known. 7 (a) Minimal regularity. Thefollowingtheorem,liketherobustnesstheoremin the minimal regularity case, holds in sufficiently smooth bounded domains as well as in the absence of boundaries. Theorem 5.1. Let u V, f L2(0,T;H) and u L∞(0,T;V) L2(0,T;V2) 0 ∈ ∈ ∈ ∩ be a strong solution of the Navier-Stokes equations du +νAu+B(u,u)=f(t) with u(0)=u . 0 dt Then u , the solution of Galerkin system n du n +νAu +P B(u ,u )=P f(t) with u (0)=P u , (5.1) n n n n n n n 0 dt converges strongly to u in both L∞(0,T;V) and L2(0,T;V2) as n . →∞ Proof. We consider w =u u which satisfies n n − dw n +νAw +P B(u,w )+P B(w ,u) P B(w ,w )=Q f(t) Q B(u,u). n n n n n n n n n n dt − − Letting q =u P u and taking the inner product of the above equation with Aw n n n − while noting that (P B(u,v),Aw )=b(u,v,P Aw ) we obtain n n n n 1 d Dw 2+ν Aw 2 k Du Dw 1/2 Aw 3/2 n n n n 2dt| | | | ≤ | || | | | +(cid:16)Dw Du1/2 Au1/2 Aw + Dw 3/2 Aw 3/2 n n n n | || | | | | | | | | | +Du3/2 Au1/2 Aq + f Aq . n n | | | | | | | || | (cid:17) Applying Young’s inequality and redefining the constant k gives d 1 1 1 Dw 2 k Du4+ Du Au Dw 2+ Dw 6 dt| n| ≤ ν3| | ν| || | | n| ν3| n| (cid:26)(cid:18) (cid:19) +Du3/2 Au1/2 Aq + f Aq . n n | | | | | | | || | o Now letting t 1 1 β(t)=k Du4+ Du Au ds ν3| | ν| || | Z0 (cid:18) (cid:19) and y (t)=e−β(t) Dw 2 n n | | the above inequality can be written as dy n αy3 +δ (t), dt ≤ n n with α= k e2β(T) and δ (t)= f Aq +k Du3/2 Au1/2 Aq . ν3 n | || n| | | | | | n| Using the Ho¨lder inequality we have 1/2 1/2 1/2 T T T T δ (s)ds f 2ds +k Du3 Auds Aq 2ds n n Z0 ≤ Z0 | | ! Z0 | | | | !  Z0 | | !  8  Since u L2(0,T;V2), u(s) V2 for almost every s [0,T] and therefore q (s) = n ∈ ∈ ∈ Q u(s) 0 in V2 as n and Aq (s)2 Au(s)2 for a.e. s [0,T]. Hence the n n → →∞ | | ≤| | ∈ Lebesgue dominated convergence theorem for each s implies that T Aq 2ds tends 0 | n| to zero as n and therefore →∞ R T δ (s)ds 0 n → Z0 as n . Since y (0) 0 as n , lemma 2.1 shows that y , and hence Dw 2, n n n →∞ → →∞ | | converges to zero uniformly on [0,T] as n . →∞ (b) Second order regularity. Here also,in a similar way to the previous case, we try to find a bound on the V2-norm of the difference between a second order regularstrong solutionand Galerkinapproximations. Howeverhere the more regular solution space makes the proof easier. Theorem5.2. Letu V2,f L2(0,T;V) L1(0,T;V2)andu L∞(0,T;V2) 0 ∈ ∈ ∩ ∈ ∩ L2(0,T;V3) be a strong solution of the Navier-Stokes equations du +νAu+B(u,u)=f(t) with u(0)=u . 0 dt Then u , the solution of Galerkin system n du n +νAu +P B(u ,u )=P f(t) with u (0)=P u , (5.2) n n n n n n n 0 dt converges strongly to u in both L∞(0,T;V2) and L2(0,T;V3) as n . →∞ Proof. Let w =u u . Then w satisfies n n n − dw n +νAw +P B(u,w )+P B(w ,u)+P B(w ,w )=Q f Q B(u,u) n n n n n n n n n n dt − Taking the inner product of the above equation with A2w and using (3.2) and (3.3) n we obtain 1 d w 2+ν w 2 (c+c′) u w 2+c w w 2 2dtk nk2 k nk3 ≤ k k3k nk2 k nk3k nk2 + Q f Q B(u,u) w n n 2 n 2 k − k k k Now we use Young’s inequality to remove the dependence on w and then divide n 3 k k by w to obtain 2 k k d c2 w (c+c′) u w + w 3+ Q f Q B(u,u) , dtk nk2 ≤ k k3k nk2 4νk nk2 k n − n k2 in which the coefficient of w does not depend on n. n 2 k k Letting y = w e−β(t), n n 2 k k with β(t)=exp t(c+c′) u(s) ds yields 0 k k3 (cid:16)R (cid:17) y˙ δ(t)+αy3, n ≤ n 9 where α= c2e2β(T) and δ(t)= Q f Q B(u,u) . So by lemma 2.1 if 4ν k n − n k2 T η = Q u + Q f Q B(u,u) ds 0 as n , n 0 m n n 2 k k k − k → →∞ Z0 then y (t) 0 uniformly on [0,T] as n . n → →∞ By (5.2) Q u 0 as n . We know that n 0 m k k → →∞ B(u,u) c u u (5.3) 2 2 3 k k ≤ k k k k (Kato 1972, Constantin and Foias 1988) and therefore f(s) B(u(s),u(s)) V2 − ∈ for almost every s [0,T]. So since w n form a basis for V2 as well as H, ∈ { j}j=1 Q (f(s) B(u(s),u(s))) 0 and n 2 k − k → Q (f(s) B(u(s),u(s))) f(s) B(u(s),u(s)) n 2 2 k − k ≤k − k foralmosteverys [0,T]. ThereforebytheLebesguedominatedconvergencetheorem ∈ it follows that T Q (f(s) B(u(s),u(s))) 0 n 2 k − k → Z0 and the result follows. 6. Numerical verification of the existence of a strong solution. Here we show that the existence of minimal and second order regular strong solutions can be verified via computations using sufficiently refined Galerkin approximations. Theorem 6.1. i) Consider the Navier-Stokes equations du +νAu+B(u,u)=f with u(0)=u (6.1) 0 dt with u V, f L2(0,T;H) L1(0,T;V), that hold in a bounded domain Ω with 0 ∈ ∈ ∩ sufficiently smooth boundary or in a periodic domain. Let v L∞(0,T;V) L2(0,T;V2) be a numerical approximation of u satisfying ∈ ∩ dv +νAv+B(v,v) L1(0,T;V) L2(0,T;H) dt ∈ ∩ and T dv(s) Dv(0) Du + +νAv(s)+B(v(s),v(s)) f(s) ds 0 1 | − | k ds − k Z0 1 ν3 1/4 k2 T 27k2 1 1 < exp Dv(s)4+ Dv(s) Av(s) ds . (6.2) k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | ! Then the Navier-Stokes equations (6.1) have a strong solution u L∞(0,T;V) ∈ ∩ L2(0,T;V2). ii) Let u be a strong solution of (6.1). Then there exists an N such that the Galerkin approximations u satisfy the inequality (6.2) for all n > N. Therefore in n viewofpart(i), upasses the aposterioritestasasolutionapproximated byu i.e. the n 10

An a posteriori condition on the numerical approximations of the Navier-Stokes equations for the existence of a strong solution

Similar An a posteriori condition on the numerical approximations of the Navier-Stokes equations for the existence of a strong solution