Numerical verification of regularity in the three-dimensional Navier-Stokes equations 7 0 By James C. Robinson1 & Witold Sadowski2,3 0 2 1Mathematics Institute, University of Warwick, Coventry CV4 7AL. UK. n 2 Faculty of Mathematics, Informatics and Mechanics, a J Warsaw University, Banacha 2 02-097, Warszawa, Poland. 9 3 Institute of Mathematics of the Polish Academy of Sciences, ul. S´niadeckich 8 00-956, Warszawa, Poland. ] P Current theoretical results for the three-dimensional Navier–Stokes equations only A guarantee that solutions remain regular for all time when the initial enstrophy . h (kDu k2 := |curlu |2) is sufficiently small, kDu k2 ≤ χ . In fact, this smallness 0 0 0 0 at conditionissuchthatthe enstrophyisalwaysnon-increasing.Inthis paperwepro- R m vide a numerical procedure that will verify regularity of solutions for any bounded setofinitialconditions,kDu k2 ≤χ .Undertheassumptionthattheequationsare [ 0 1 in fact regularwe show that this procedure canbe guaranteedto terminate after a 1 finite time. v 8 Keywords: Navier-Stokes equations, rigorouscomputation, Galerkin method 6 2 1 1. Introduction 0 7 The globalregularityofsolutions ofthe three-dimensionalNavier-Stokesequations 0 is one of the most intriguing open problems in mathematics. Despite the efforts / h of many eminent mathematicians over the last 150 years, the question whether at the most fundamental model in hydrodynamics is or is not self consistent remains m unanswered. Even when the flow is not disturbed by any external forces and the problem of boundaries is avoided by considering periodic boundary conditions (or : v the whole of R3) the problem is unresolved. i X More precisely, consider the Navier-Stokes equations for the flow of an incom- r pressible fluid a ∂u −ν∆u+(u·∇)u+∇p=0 ∇·u=0 u(0)=u . (1.1) 0 ∂t with periodic boundary conditions on Ω = [0,L]3, where ν > 0 is the kinematic viscosity. It is known that weak solutions of these equations exist that are defined for all t ≥ 0. However, we do not know whether these solutions are unique and regular.On the other hand, we know that if the initial enstrophy is finite then the solutions are regular for t sufficiently small (t<T ) and sufficiently large (t>T ), 1 2 where T and T depend respectively onthe enstrophy andkinetic energy ofinitial 1 2 Article submittedto Royal Society TEXPaper 2 J.C. Robinson & W. Sadowski conditions: T ∼kDu k−4 and T ∼ku k2, (1.2) 1 0 2 0 where ku k2 = |u(x)|2dx and kDu k2 = |curlu(x)|2dx. 0 0 ZΩ ZΩ It is an open problem whether or not weak solutions develop any singularity at some time T <t<T . 1 2 In the light of (1.2) it is natural to conjecture that some restrictions on en- strophy or energy of initial conditions might provide us with regularity of weak solutions.However,nothingisknownaboutregularityofweaksolutionsonlyunder the assumptionthatenergyoftheflowissmall.Inotherwords,thereisnotheoret- ical evidence that there exists an ε>0 such that all initial conditions with kinetic energy ku k2 <ε 0 donotblowup.Ontheotherhand,underthestrongerassumptionthattheenstro- phy of the initial condition u is sufficiently small, 0 kDu k2 <ε 0 1 onecanprovethatu givesrisetoastrongglobalsolution.Thevalueofε forwhich 0 1 the argument holds is very small (see Section 2). This is due to the fact that the proof of regularityof solutions arising from smallinitial data (sketched in the next section)actuallyrulesoutallinitialconditionsforwhichtheenstrophyisincreasing atthe beginning ofthe flow.Thusit isdesirableto extendthis theoreticalresultto a larger class of initial conditions that allow for more complicated and physically realistic behaviour of the flow. Themainaimofthispaperistopresentanumericalmethodwhichmayprovide us withsuchanextension.We showthat,under the assumptionthatthe equations are well-posed, it is possible to verify numerically the regularityof solutions for all initial conditions with kDu k ≤ R for any choice of R > 0. In Section 2 we set 0 up the problemmore formally,and recallsome classicalresults on the regularityof solutions. Section 3 reproduces results from Dashti & Robinson (2006)that can be used to verify that a single initial condition u ∈ V gives rise to a strong solution 0 of some bounded interval [0,T]. Section 4.2 shows that for initial conditions in any bounded ball in V it is sufficient to guarantee that solutions are regular on somefinite time interval[0,T],since allsolutionsareregularaftersomefinitetime. Section 5 is the core of the paper, and shows that one can verify that all initial conditions in some bounded ball in H2 give rise to regularsolutions – this requires no additional assumptions. The final section shows his this can be used to verify the same property for all initial conditions in a bounded ball in H1, although we note here thatthis requiresthe assumptionofregularityforallinitialconditions in H1. 2. Setting of the problem As enstrophy and kinetic energy play key role in the problem of existence and regularity of solutions it is worth recalling two basic facts concerning them. First Article submitted to Royal Society Regularity via numerics 3 of all, it follows from the energy inequality d kuk2+νkDuk2 ≤0 (2.1) dt that the kinetic energy of the flow decreases to zero as time goes to infinity (the flow is dissipative). Moreover,we have T kDu(s)k2ds≤ν−1ku(0)k2 (2.2) Z0 which shows that mean value of kDuk2 over the interval [0,T]is bounded in terms of the initial kinetic energy and decreasing at least as fast as T−1. Throughout the paper we use standard notation for the Lebesgue and Sobolev spacesL2, H1 andH2.The set ofallsmooth,periodic anddivergentfree functions onΩwith u(x)dx=0wedenotebyV.TheclosureofV intheL2 andH1 norms Ω we denote by H and V, respectively. The eigenfunctions of the Stokes operator A R defined by A(u)=−P∆(u), whereP isthe Lerayprojector(the orthogonalprojectionofthe spaceL2 ontoH), we denote by w and their corresponding eigenvalues by λ . It follows from the k k theory of compact self-adjoint operators in Hilbert spaces that we can order the eigenvalues so that 0<λ ≤λ ≤λ ≤...→∞. 1 2 3 Since we considerperiodic boundary conditions we canin factspecify the valuesof λ exactly, and in particular we have k 4π2 λ = . 1 L2 In what follows we will concentrate on a case L = 2π for which λ = 1. This is 1 convenient in the light of the following Poincar´e inequalities kuk2 ≤λ−1kDuk2, kDuk2 ≤λ−1kAuk2 1 1 which we use frequently in what follows. As mentioned above, solutions to (1.1) with sufficiently small initial data are regularforalltime.Withzeroforcingtheconditionguaranteeingregularityisgiven by following theorem. Theorem 2.1. If the initial condition u for problem (1.1) satisfies 0 kDu k2 ≤c−1/2ν2λ1/2, (2.3) 0 1 where c is some absolute constant†, then u gives rise to a global strong solution. 0 † For later use it is worthwhile to note that c = 27k4/16, where k is the constant in the inequality [(u·∇)v]·∆wdx ≤kkDukkDvk1/2kAvk1/2kAwk. ˛˛ZΩ ˛˛ Inthecubicgeometry˛˛ herewecantakek˛˛=9×215/4,seeDashti&Robinson(2006). Article submittedto Royal Society 4 J.C. Robinson & W. Sadowski The proof is based on the inequality d c kDuk2≤ kDuk6−νλ kDuk2 (2.4) dt ν3 1 which may be obtained from the enstrophy inequality with some help from the Sobolevembeddingtheorem.Whatiscrucialisthatforinitialconditionssatisfying (2.3) the right hand side of (2.4) is non-positive. In consequence kDuk2 will never exceed the value c−1/2ν2λ1/2. (For details of the proof see, for example, Theorem 1 * in Constantin & Foias (1988).) Below we treat the question of regularity for periodic boundary conditions on [0,2π]3 andforviscosityν =1.Wenowshow,usingasimplescalingargument,that a proof of regularity for this particular case would imply regularity for all [0,L]3 and all ν >0. Indeed, starting with the equation ∂u −ν∆u+(u·∇)u+∇p=0 ∇·u=0, x∈[0,L]3, ∂t we set u˜=Lu/2πν, x˜=2πx/L, t˜=4π2νt/L2, and p˜=L2p/4π2ν2, and then obtain ∂u˜ −∆ u˜+(u˜·∇ )u˜+∇ p˜=0 (2.5) ∂t˜ x˜ x˜ x˜ for x˜∈[0,2π]3. For this non-dimensionalized form of the Navier-Stokes equations we can get an explicit value R for the radius of the ball in V in which standard results V guaranteethat everyinitial condition givesrise to a globalstrong solution.Indeed, since c=27(9×215/4)4/16 we get RV =c−1/4νλ11/2 =c−41 =2/(9×215/4×271/4)≈0.007. Nothingisknownaboutregularityofsolutionsstartingfrominitialconditionslying in a ball in V with R > R , and so the theoretical results currently available V really are about “small data”. In this paper we present a method of numerical verification of regularity for more general, physically reasonable initial conditions. More precisely, denote by B (R) the ball of radius R in V. V We prove below under the assumption of regularity that for any fixed value of R>0 the following statement can be verified numerically in finite time. Statement 2.2. Every initial condition u ∈ B (R) gives rise to a global strong 0 V solution of problem (1.1). Of course, regularity of solutions with initial conditions in some fixed ball in V for the non-dimensionalizedformof the Navier-Stokesequations implies parallel results for other cubic domains anddifferentvalues of the viscosity.Indeed, scaling initial condition changes its enstrophy as we have ∂u˜ L2 ∂u = . ∂x˜ 4π2ν∂x Article submitted to Royal Society Regularity via numerics 5 So L |Du˜(x)|2dx˜= |Du(x)|2dx Z[0,2π]3 2πν2 Z[0,L]3 and since λ = 4π2 we see that 1 L2 1 ||Du˜||2 = ||Du||2 ν2λ1/2 1 which is in agreement with theorem 2.1. Thusacompleteresultontheregularityofsolutionsofthenon-dimensionalized Navier-Stokes equations would immediately imply the same result for any L and any ν, while regularity only for kDu˜k2 ≤ R in the non-dimensionalized equation would provide regularity for all kDuk2 ≤2πRν2/L. 3. Numerical verification of regularity for a single u ∈ V 0 Beforepresentingamethodofverifyingregularityforallinitialconditionsinbounded subsets of H2 (and then H1) we need a result that allows us to check numerically the regularity of a particular solution arising from some given initial condition. In other words,we have to be able to make a judgement based only on numericalcal- culationswhetheragiveninitialconditionu givesriseto aregularsolutionu.The 0 first step to this end is a standard one: We construct the Galerkin approximations u of u by solving the equation n du n +Au +P B(u ,u )=0 n n n n dt with u (0) = P u , where P is orthogonal projection on the space spanned by n n 0 n first n eigenfunctions of the Stokes operator. We can deduce solely from the properties of these Galerkin approximations whether they approximate a regular solution on a finite time interval. A method for this was provided by Chernyshenko et al. (2006) under the assumption that u ∈ Hm with m ≥ 3. Dashti & Robinson (2006) adapt these arguments to show 0 that the same is possible given a regular solution arising from an initial condition in V, and we now briefly outline the main idea of this method. The first ingredient is a robustness result for strong solutions: Theorem 3.1. There exists a constant c > 0 such that if u ∈ V gives rise to a 0 strong solution u(t) of (2.5) on [0,T∗] then so does any initial condition v and 0 forcing g(t) with T∗ T∗ kD(v −u )k+ kDg(s)kds<c(T∗)−1/4exp −c kDu(s)k4+kDu(s)kkAu(s)kds . 0 0 Z0 Z0 ! (3.1) (The constant c is related to the constants arising in certain Sobolev embedding results.) This has the following corollary,key to all that follows: Article submittedto Royal Society 6 J.C. Robinson & W. Sadowski Corollary 3.2. Suppose that v ∈ L∞(0,T∗;V)∩L2(0,T∗;D(A)) is a numerical approximation of u such that dv +Av+B(v,v)∈L1(0,T∗;V)∩L2(0,T∗;H) dt and T∗ dv |Dv(0)−Du |+ (s)+Av(s)+B(v(s),v(s)) ds 0 dt Z0 (cid:13) (cid:13)1 (cid:13)(cid:13) T∗ (cid:13)(cid:13) <c(T∗)−1/4e(cid:13)xp −c |Dv(s)|4+|Dv(s)(cid:13)||Av(s)|ds . (3.2) Z0 ! Then u is a regular solution of (2.5) with u∈L∞(0,T;V)∩L2(0,T;D(A)). Thus to verify that u ∈ V gives rise to a strong solution on [0,T∗], it suffices 0 to show that for sufficiently large n the Galerkin solution u will satisfy (3.2) in n Corollary 3.2. If this is the case then on can compute ever more accurate Galerkin approximations until (3.2) is eventually satisfied. If the approximated solution is regular then this method has to be successful since it is the case that the Galerkin approximations of regular solutions must converge in a sufficiently strong sense: Theorem 3.3. If u ∈ V gives rise to a strong solution u(t) on [0,T∗] then the 0 Galerkin approximations u convergetoustronglyin L∞(0,T;V)∩L2(0,T;D(A)). n Of course, this neglects two issues: first, that the numerical solution will not be a continuous function of t, but given at discrete time points (cf. comments in Chernyshenkoetal.(2006)).However,sinceCorollary3.2saysnothingoftheprove- nance of v, one can construct v via linear interpolation between these time points. A further complication is that the integrals in (3.2) cannot be computed exactly and have to be approximated; but this is not a major obstruction. 4. Finite time of calculation Ourgoalinthefollowingtwosectionsistoprove,undertheassumptionofregularity, that the following statement may be verified numerically in a finite time. Statement 4.1. Consider the non-dimensional 3d Navier-Stokes equations on [0,2π]3 with zero forcing, ∂u −ν∆u+(u·∇)u+∇p=0 ∇·u=0 u(0)=u . (4.1) 0 ∂t Then every initial condition in a ball in V B (R)={u : kDu k≤R} V 0 0 gives rise to a strong solution that exists for all t≥0. Since the method in the previous section only works to verify the regularity of a given solution on some finite time interval, it is vital to show that this is all that is required. Article submitted to Royal Society Regularity via numerics 7 Lemma 4.2. Any (weak) solution of (4.1) becomes regular after a finite time T∗ =c1/2ku k2 ≤c1/2kDu k2, 0 0 where c is the absolute constant from theorem 2.1. Proof. Since T kDu(s)k2ds≤ku k2, 0 Z0 it follows that onany time interval [0,T]with T >T∗ as above,there must exist a t such that 0 kDu(t )k2 ≤c−1/2, 0 andhencefromtheorem2.1itfollowsthatthesolutionisregularforallt≥T∗. Thus if we are interested in regularity of a particular ball B (R) we have to V checkregularityofsolutionsonlyonsomefixedintervaloftime:nowwecanproceed with the proof of the main theorem of this paper. Themoststraightforwardapproachwouldbetoverifytheregularityofsolutions arising from some finite sequence of initial conditions u , where 1 ≤ n ≤ N 0n and N is large enough. Then Corollary 3.2 would imply regularity of all initial conditions in balls with radii δ ,δ ,...,δ centred at u ,u ,...,u respectively. 1 2 N 01 02 0N If these balls cover all of B then the Statement 4.1 would be verified. However, R several obstacles stand in the way of such a happy ending. First of all, a ball in infinite dimensional space is never compact andone cannotcoverit with any finite number of smaller balls. Even if we were interested only in checking regularity in some finite-dimensional subspace of V we would still encounter a serious problem: without a proof of some generaluniform lower bound onthe sequence δ we might k not be able to construct a proper covering (e.g. it is not possible to cover the unit interval with a sequence of intervals of length δ =3−n). n To circumvent these difficulties we will prove in the next section an auxiliary result under a stronger assumption on initial condition. 5. Numerical verification of regularity for all initial conditions in a ball in H2 In this section we show that it is possible to verify numerically the following state- ment, which requires more regularity of the initial condition. Statement 5.1. Consider the 3d Navier-Stokes equations on [0,2π]3 with zero forcing, ∂u −∆u+(u·∇)u+∇p=0 ∇·u=0 u(0)=u . (5.1) 0 ∂t Then every initial condition in a ball BH2(S)={u0 :|Au0|≤S} gives rise to a strong solution that exists for all t≥0. Article submittedto Royal Society 8 J.C. Robinson & W. Sadowski Theorem 5.2. Statement 5.1 can be verified numerically in a finite time. This theorem is proved via a series of subsidiary results. From lemma 4.2 and thePoincar´einequalityitfollowsthatinordertoverifyStatement5.1weneedonly proveregularityonthefinite timeinterval[0,T∗].Itisthereforeveryusefultohave the followingresult,whichgivesuniformbounds overtheH1 andH2 normsofany solution with u0 ∈BH2(S). Proposition 5.3. Suppose that Statement 5.1 is true. Then for any T > 0 there exist constants D (T) and E (T) such that for any solution u arising from an S S initial condition u0 ∈BH2(S) we have T sup kDu(s)k≤D and kAu(s)k2ds≤E . (5.2) S S 0≤t≤T Z0 Proof. This result follows from the proof, but not the statement of Theorem * in Foias & Temam (19**), see also Theorem 12.10 in Robinson (2001). Here we give a sketch of the relevant part of the argument. If the first bound in (5.2) does not hold then there must be sequences u with kAu k≤S and t ∈[0,T] such that 0n 0n n kDu (t )k→∞asn→∞.Byextractingappropriatesubsequenceswecanassume n n that t → T ∈ [0,T] and that u ⇀ u in D(A): in particular, kAu k ≤ S and n 0n 0 0 u → u in H. As in the standard proof of the existence of weak solutions, one 0n 0 can show that the solutions u (t) of n du /dt+Au +B(u ,u )=0 u(0)=u n n n n 0n are uniformly bounded in L∞(0,T;H)∩ L2(0,T;V); taking the limit as n → 0 produces a weak solution v of dv/dt+Av+B(v,v)=0 with v(0)=v . 0 However, by assumption, this equation possesses a strong solution which is unique in the class of weak solutions, and so in fact v ∈ L∞(0,T;V)∩L2(0,T;D(A)). Onecannowusethisregularityofv alongwithstandardestimatesonthe equation for the difference v −u to show that u → v strongly in L2(0,T;V). Since L2 n n convergenceonanintervalimpliesthatthereexistsasubsequenceconvergingalmost everywhere, at each such point of convergence,s, we must have kDu (s)k≤S := n 0 1+kvkL∞(0,T;V) for n sufficiently large. Standard estimates can be used to show that there is a time τ such that kDu (s+t)k≤2(1+S ) for all 0≤t≤τ n 0 for all suchs and n.By covering[0,T]with a finite number ofintervals [s ,s +τ], j j where u (s ) →u(s ) in V, one can deduce that in fact kDu (t)k ≤ 2(1+S ) for n j j n 0 all t ∈ [0,T] and obtain a contradiction. The second bound in (5.2) then follows from the inequality d kDuk2+kAuk2 ≤ckDuk6 dt which can be obtained by standard methods. Article submitted to Royal Society Regularity via numerics 9 Now we are ready to prove that if an initial condition u0 ∈ BH2(S) gives rise to a regular solution then there exist a ball in V of some radius δ depending on S but not on u such that all initial conditions in that ball give rise to regular 0 solutions. In other words, we may say that if Statement 5.1 is true, then all points of BH2(S) are separated in V from any initial conditions giving rise to a solution with singularities (if such solutions exist) by at least some fixed distance δ. Most importantly for us it means that there is a uniform lower bound on the radius of the “balls of regularity”whose existence is guaranteedby theorem 3.1, centered at points of BH2(S) . Proposition5.4. Supposethat Statement5.1 is true.Then thereexistsaδ =δ(S) such that if u0 ∈ BH2(S) then the value of δ required by the regularity check of Corollary 3.2 is bounded below by some δ: any initial condition v ∈V with 0 kDu −Dv k<δ 0 0 also gives rise to a strong solution on [0,T∗]. Proof. Given the result of Proposition 5.3, the integrand on the right-hand side of (3.1) is bounded above: T∗ I := kDu(s)k4+kDu(s)kkAu(s)kds≤I :=T∗D4 +(T∗D2)1/2E , S S S S Z0 and so the right-hand side of (3.1) is bounded below, cexp(−cI)≥δ :=cexp(−cI ). S It follows that there exist a lower bound on the radius of “ball of regularity” pro- vided by theorem 3.1 Nowweuseacompactnessargument.Looselyspeaking,wecouldcoverBH2(S) by open balls of radius δ centred at every point of BH2(S) and then choose finite sub-covering (in topology of V). This argument, based on the Rellich-Kondrachov theorem on the compact embedding of H2 in H1, is formulated more precisely in thefollowinglemmawhichalsogivesthe explicitformofpointsinBH2(S)thatwe need to test for regularity numerically. Lemma 5.5. Given δ > 0 there exist N, M such that every v0 ∈ BH2(S) can be approximated to within δ (in the V norm) by elements of the set N U ={u :u = α w , with α =a /2M, a ∈Z; kAu k≤S} M,N 0 0 j j j j j 0 j=1 X where {w } are the Stokes eigenfunctions. j Note that the proof shows that one can take N such that λ ≥ 2S2/δ and M N+1 such that 2−M <δ/2. Article submittedto Royal Society 10 J.C. Robinson & W. Sadowski Proof. Take any v ∈BH2(S). Since kAvk≤S, we have ∞ kD(P v−v)k2 = λ |(v,w )|2 n k k k=n+1 X ∞ ≤ λ−1 λ2|(v,w )|2 n+1 k k k=n+1 X ≤ λ−1 kAvk2 n+1 ≤ λ−1 S2. n+1 Thus we can choose an N that for any n≥N we have δ kD(P v−v)k2 ≤λ−1 S2 ≤ n n+1 2 and we can cover the finite dimensional ball PN(BH2(S)) with a finite number of balls of radius r ≤ δ with centres in U , where 2−M < δ/2. It follows that for 2 M,N any v0 ∈BH2(S) we can find u˜0 ∈UM,N such that kDv −Du˜ k≤kD(v −P v )k+kP v −Du˜ k≤δ. 0 0 0 N 0 N 0 0 Finally we assemble all these ingredients to provide a proof of theorem 5.2. Proof of thereom 5.2. Given S > 0, let δ be the lower bound on the radius of the “ball of regularity” provided by proposition 5.4. Then to prove regularity for all initial conditions in BH2(S) it suffices to check that the finite set of initial conditions in U (the set defined in lemma 5.5) all give rise to strong solutions M,N on [0,T∗]. This can be done for each initial condition in turn using the procedure outlined in Section 3. Observethattheproofisbasedonlyonanassumptionofregularityofsolutions arising from initial conditions in BH2(S). Whether or not singularities may occur in solutions arising from initial conditions u with kAu k>S has no effect on the 0 0 possibility of numerical verification of regularity in the ball BH2(S). 6. Numerical verification of regularity for all u ∈ B (R) 0 V Now we will use the results of previous section to prove the main theorem of this paper Theorem 6.1. If the Navier-Stokes equations are regular, then statement 4.1 can be verified numerically in a finite time. Proof. We findaT >0suchthateverysolutionwithinitialconditionkDu k≤R R 0 is regular on the interval [0,T ] and has an explicit bound on kAu(T )k, using a R R result due to Foias & Temam (1988) on the Gevrey regularity of solutions. More explicitly, if we define etA1/2 = ∞ tnAn/2 (6.1) n! n=0 X Article submitted to Royal Society