STABILITY OF DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES

Transcription

1 STABILITY OF DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES JEAN-LUC GUERMOND,, JOSEPH E PASCIAK Abstract Using a general approximation setting aving te generic properties of finite-elements, we prove uniform boundedness and stability estimates on te discrete Stokes operator in Sobolev spaces wit fractional exponents As an application, we construct approximations for te time-dependent Stokes equations wit a source term in L p (, T ; L q (Ω)) and prove uniform estimates on te time derivative and discrete Laplacian of te discrete velocity tat are similar to tose in Sor and von Wal [2] Introduction Scope of te paper Te objective of tis paper is to construct approximations for te time-dependent Stokes equations wit a source term in L p (, T ; L q (Ω)) and to prove uniform estimates on te discrete pressure and te time derivative and discrete Laplacian of te discrete velocity tat are similar to tose proved by Solonnikov [2] and Sor and von Wal [2] To tis purpose we construct a finite-element-like approximate Stokes operator and we prove norm equivalences between te scale of norms wic it generates and te usual fractional order Sobolev norms for 2 < s < 3 2 Te boundary condition under consideration is te omogeneous Diriclet condition By working wit fractional exponents of te discrete Stokes operator and te Fourier tecnique in time we avoid te non-hilbertian L p (L q )- framework, wic we do not know yet ow to andle in finite-element-like discrete settings Tis tecnique yields near optimal counterparts of te estimates of Sor and von Wal on te time derivative and discrete Laplacian of te discrete velocity Te main results summarizing te content of te paper are Teorem 4, and Teorem 5 Te paper is organized as follows Te rest of tis section is devoted to introducing notation and recalling te definitions of te Leray projector and te Stokes operator Te discrete finite-element-like setting alluded to above is introduced in 2 Boundedness and invariance properties of te discrete Date: Draft version: January, 27, Submitted to Journal of Matematical Fluid Mecanics 99 Matematics Subject Classification 35Q3, 65N35, 76M5 Key words and prases Stokes operator, Finite elements, Navier Stokes equations, pressure estimates, negative norms On long leave from LIMSI (CNRS-UPR 325), BP 33, 943, Orsay, France

2 2 J-L GUERMOND AND J PASCIAK Leray projector are stated in 3 Te discrete Stokes operator is analyzed in 4 Te results of 4, in particular Teorem 4, are used to analyze te semi-discrete time-dependent Stokes operator in 5 Discrete counterparts of te estimates of Sor and von Wal using fractional Sobolev spaces are sated in Teorem 5 Te results presented in tis paper are part of a researc program aiming at caracterizing te weak solutions of te tree-dimensional Navier Stokes equations tat are suitable in te sense of Sceffer [8] It as been sown in [] tat, in te tree-dimensional torus, weak solutions tat are constructed as limit of sequences of finite-element-like Galerkin approximations are suitable Te goal we are pursuing is to eventually extend tis result to omogeneous Diriclet conditions One important intermediate step on te way are estimates like (523) and (524) A proof tat finite-element-like Galerkin approximations are indeed suitable is reported in [2] Teorem 5 (wic is a consequence of Teorem 4) is an essential key for proving tis result 2 Notation and conventions Let Ω be a connected, open, bounded domain in R d (d is te space dimension) Te boundary of Ω is assumed to be suc tat te H 2 -regularity property of te Laplace operator olds, ie, tere is c > suc tat () v H (Ω) H 2 (Ω), v H 2 c v L 2 For instance, Ω convex or Ω of class C, are sufficient conditions for tis property to old, cf eg [9] Te boundary of Ω is denoted by Γ We use bold notation to denote te product space wit d components in a given space, eg H (Ω) = (H (Ω)) d, but no notational distinction is made between R-valued and R d -valued functions Wenever E is a normed space, E denotes a norm in E Wenever E is a Hilbert space, (, ) E denotes te scalar product in E Te scalar product in L 2 (Ω) and L 2 (Ω) is simply denoted by (, ) Hencefort c is a generic constant Te symbol c u ( ) denotes a generic positive non decreasing function Te symbol c l ( ) denotes a generic positive non increasing function Bot te generic constant c and te generic functions c u and c l are independent of te mes parameter Te value of c and te exact form of c u and c l may vary at eac occurrence For < s <, te space H s (Ω) is defined by te real metod of interpolation between H (Ω) and L 2 (Ω), ie, te so-called K-metod of Lions and Peetre [7], see also [6] or [3, Appendix A] To define H s (Ω), we interpolate between H (Ω) and H 2 (Ω) if < s < 2 We denote H s (Ω) to be te closure of C (Ω) in Hs (Ω) for < s < and H s (Ω) to be te interpolation space [H (Ω), L2 (Ω)] s for s For s (, 2], Hs (Ω) is defined to be H s (Ω) H (Ω) Note tat te spaces Hs (Ω) and H s (Ω) coincide for s 2 wit uniformly equivalent norms (see [6, Tm ]) Te spaces H s (Ω) and H s(ω) coincide for s < 2 and teir norms are equivalent;

3 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 3 ie, tere is c > and a non-decreasing function c u suc tat (2) c v H s v eh s c u (s) v H s, v H s, s [, 2 ), wit lim s c u (s) =, see [6, Tm 7] 2 For negative s, H s s (Ω) is te dual of H (Ω) Te space H s (Ω) for s > is defined by duality, ie, v H s = (v, w) w C (Ω) w H s For s [, 2 ) ( 2, 3 2 ), H s coincides wit (Ω) We define HR s = (Ω), s [, ], to be composed of tose functions in H s (Ω) tat are of zero mean It can be sown tat HR s = (Ω) = [LR 2 = (Ω), HR = (Ω)] s, for all s [, ], cf eg [] H s 3 Te Leray projector Following [4, 22] we define (3) V = {v C (Ω); v = } to account for solenoidal vector fields, and we set (4) V = V L2, V = V H, V 2 = V H 2 (Ω) Te following caracterizations of V and V old, cf [22], (5) (6) V = {v L 2 (Ω); v = ; v n Γ = }, V = {v H (Ω); v = ; v Γ = } V is a closed subspace of L 2 (Ω) and te following well known Helmoltz decomposition olds, see eg [4, 22] (7) L 2 (Ω) = V H R = (Ω) We denote by P : L 2 (Ω) V te L 2 -projection onto V (ie, te socalled Leray projection) Lemma Tere is c > suc tat for all s [, ], (8) v H s (Ω), P v H s c v eh s Proof Let v L 2 (Ω) Define q HR = (Ω) be suc tat ( q, r) = (v, r) for all r in HR = (Ω) It is clear tat q H c v L 2 Assume moreover tat v H (Ω), ten q solves te omogeneous Neumann problem q = v, n q Γ = Owing to te regularity ypotesis on Ω, te regularity teory of elliptic operators implies q H 2 c v H

4 4 J-L GUERMOND AND J PASCIAK Ten, by interpolation, we obtain q H +s c v eh s, v H s (Ω) Ten we define P v = v q Te triangle inequality yields P v H s v H s + q H s c v eh s 4 Te Stokes operator Let us define te unbounded vector-valued Laplace operator : D( ):=H (Ω) H2 (Ω) L 2 (Ω) We introduce te so-called Stokes operator A : D(A):=V 2 V by setting A = P V 2 We assume tat te domain Ω is suc tat tere is c > (9) v V 2, v H 2 c Av L 2 Tis property olds in two and tree dimensions (d = 2, 3) wenever Ω is convex or of class C,, see [6, Tm 63] It follows from (9) tat A is closed Moreover, it is positive and selfadjoint and its inverse is compact We denote by (φ k, λ k ) k te eigenpairs of A so tat te family (φ k ) k forms an ortonormal basis for V Following [7] we define { () E = v = } N k= v kφ k ; N N; (v,, v N ) R N, and for all s R we denote by E s te completion of E in te norm ( ) () j= λs k v k 2 2 = (A s v, v) 2 It is clear tat E s = V s, for s =,, 2 We encefort introduce te notation V s := E s for all s R and we set v V s := (A s v, v) 2 Using te K-interpolation metod, it can be sown also tat {V s } s R forms an Hilbert scale 2 Te discrete setting and preliminaries We introduce in tis section a discrete approximation setting and we prepare te ground for te main result of 4, ie, Teorem 4 2 Te discrete setting We assume tat we ave at and two families of finite-dimensional spaces, {X } > and {M } > suc tat X H (Ω) and M LR 2 = (Ω) To avoid irrelevant tecnicalities we assume M HR = (Ω) Te spaces {X } > and {M } > ave approximating properties in te sense tat tere is a constant c uniform in suc tat for all l, s, l min(, s), l s 2, (2) (22) v H s (Ω), q H s R = (Ω), inf v v eh l c s l v eh s v X inf q q L 2 c s q H s q M

5 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 5 We moreover assume tat te following inverse inequality olds: Tere is a positive non-decreasing function c u, uniform in, suc tat for all s [, 3 2 ) (23) v X, v eh s c u (s) s v L 2 We assume also tat te L 2 -projection π : L 2 (Ω) X onto X is stable on H s (Ω), for s < 3 2, ie, tere is a positive non-decreasing function c u, uniform in, so tat (24) v H s (Ω), π v eh s c u (s) v eh s, for all s [, 3 2 ) Remark 2 Te ypoteses (23) is usually satisfied wen X, and M are constructed by using finite elements on quasi-uniform meses By redoing carefully te computation in [2, Appendix] we can sow tat c u (s) ( 2s) 2 Te ypotesis (24) can be proved to old on quasi-uniform meses also by using Lemma A3 wit ρ = π and T = π 22 Discrete projections and Laplace operator Let E : H (Ω) X be te so-called elliptic projection defined by (25) ( E x, x ) = ( x, x ), x H (Ω), x X Lemma 2 Tere is c independent of suc tat l, s, l min(, s), l s 2, (26) E x x eh l c s l x eh s, x H s (Ω) Proof Tis is a standard result wen l is integer; see eg [8] Te result follows by interpolation for non-integer l in (, ) We also assume tat te family (X ) > is suc tat E is uniformly H s -stable for s [, 3 2 ), ie, tere is a positive non-decreasing function c u, independent of, suc tat (27) E v eh s c u (s) v eh s, for all v in H s (Ω) Wen te spaces (X ) > are finite-element-based, tis assumption is known to old under quite weak regularity requirements on te underlying mes family, see [2] or Lemma A3 wit ρ = π and T = E We define te discrete Laplace operator : X X as follows: ( x, y ) = ( x, y ), x, y X Clearly te four operators, E, π, and are related by (28) E x = π x, x D( )

6 6 J-L GUERMOND AND J PASCIAK In oter words te following diagram commutes: D( ) L 2 (Ω) (29) E π X X Te operator is self-adjoint and positive definite so we can define ( ) s for all s R and te following norm makes sense (2) v X s := (( ) s v, v ) 2, v X We denote by X s te vector space X equipped wit tis norm X s is clearly a Hilbert space Te family {X s } s R is a Hilbert scale in te sense of Lions and Peetre [7], [6], [3, Appendix A] Lemma 22 Under te above assumptions, tere is a positive non-increasing function c l and tere is a positive non-decreasing function c u, bot uniform in, suc tat for all s ( 3 2, 3 2 ), (2) v X, c l ( s ) v eh s v X s c u ( s ) v eh s For completeness, we sketc a proof of tis result in Appendix A 23 Compatibility between X and M We assume tat X and M are compatible in te sense tat tere is a constant c > independent of suc tat (22) q M, π q L 2 c q L 2 Tis inequality can also be equivalently rewritten as ( q, v ) (23) q M, c q v X v L 2 L 2 A first consequence of tis ypotesis is tat X and M satisfy te socalled LBB condition, see eg [8] Lemma 23 Assume tat (2) olds wit l =, s =, and (23) olds wit s = Ten (22) implies tat tere is a constant c independent of suc tat (q, v ) (24) inf c q M v X q L 2 v H Proof See te proof of Lemma 2 in [] Te operator C can be eg te Clément interpolation operator [5] or te Scott-Zang operator [9] Lemma 24 Hypotesis (22) olds in eiter one of te following situations: (i) X is composed of P Bubble H -conforming finite elements and M is composed of P H -conforming finite elements (ie, te so-called MINI element)

7 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 7 (ii) X is composed of P 2 H -conforming finite elements and M is composed of P H -conforming finite elements (ie, te so-called Hood- Taylor element), and no tetraedron as more tan 3 edges on Ω Proof See te proof of Lemma 22 in [] 24 Te Discrete Leray projection and Stokes operator We now define te space V to be te set of discretely divergence free vectors, ie, (25) V = {v X ; (v, q ) =, q M } Ten let P : L 2 (Ω) V be te L 2 -projection onto V P is a discrete version of te Leray projection We also introduce te mapping R : H (Ω) V defined by (26) ( R v, v ) = ( v, v ), v V Lemma 25 Under te ypoteses of Lemma 23, tere is a constant c independent of suc tat (27) v V 2, R v v H c v H 2 Proof (27) is a standard result; see eg [8, 4] We now define te discrete Stokes operator A : V V as follows: For all u V, A u is te element of V suc tat (28) (A u, v ) = ( u, v ), v V Note tat A = P V Observe tat te four operators A, R, P, and are related by (29) A R v = P v, v D( ) An identical argument sows tat (22) A R x = P x, x X In oter words te following diagram commutes (22) D( ) E X R V P A V R P L 2 (Ω) π X Since A is self-adjoint and positive definite, te operator A s is well defined for all s R We equip te vector space V wit te norm (222) v V s = (A s v, v ) 2, and we denote by V s te corresponding normed (Hilbert) space Using te so-called K-interpolation metod of Lions and Peetre [7], [6], [3, Appendix A], it is clear tat {V s} s R is a scale

8 8 J-L GUERMOND AND J PASCIAK 3 Properties of te discrete Leray projection Te goal of tis section is to provide a preliminary result concerning P tat will be used in te proof of Teorem 4 Te main result of tis section is Lemma 3 Lemma 3 Tere is a positive non-decreasing function c u, independent of, suc tat (3) v H s (Ω), P v eh s c u (s) v eh s, s [, 2 ) Proof Let v be a member of H s (Ω) = H s (Ω) for s < 2 Let P v be te L 2 -projection of v onto V Tere is q HR = (Ω) suc tat (P v, l) + ( q, l) = (v, l), l L 2 (Ω), ( r, P v) =, r H R = (Ω) Te above problem is clearly a well posed mixed problem Let (v, q ) X M solve (v, l ) + ( q, l ) = (v, l ), l X, ( r, v ) =, r M Tis is a stable mixed problem by (22) Clearly, v = P v Tus P v and q are te mixed approximations of P v and q, respectively Owing to (23) te approximation teory of mixed problems yields (see eg [8, 4]) P v P v L 2 + q q H c( inf P v w L 2 + inf q r H ) w X r M Since v is in H s (Ω), Lemma implies P v H s (Ω) = H s (Ω), s < 2 Te approximation ypoteses (2) (22) togeter wit te norm equivalence (2) ten yield P v P v L 2 c s ( P v eh s + q H s+) c s (c u (s) P v H s + q H s+) c u (s) s v eh s Ten using te above approximation result togeter wit te approximation and stability properties of π and te inverse inequality (23), we infer P v eh s P v π P v eh s + π P v eh s (32) c s P v π P v L 2 + c P v eh s c s ( P v P v L 2 + P v π P v L 2) + c P v eh s c u (s) v eh s Tis completes te proof Remark 3 Observe tat te above result does not old for s 2 Even if v is in H s, P v is not in H s in general if s 2, ie, te boundary conditions are lost on P v (te normal component of P v is zero, but te tangential

9 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 9 component is not zero) On te oter and, observe tat P v X H (Ω) Hence, in general P v H s but P v H s wen s 2 Tis boundary value incompatibility implies tat for all s 2, P v P v L 2 c(ɛ) 2 ɛ v eh s, ɛ >, is te best estimate tat can be obtained in general 4 Properties of te discrete Stokes operator Te main result of tis section is embodied in Teorem 4 4 Stability properties of R on X We first derive a discrete counterpart of (9) Lemma 4 Tere is c independent of suc tat (4) v L 2 c A v L 2, v V Proof Let v be a member of V Let (v, p) H (Ω) L2 R = (Ω) be te solution of te Stokes problem wit data A v, ie, ( v, l) + (p, l) = (A v, l), l H (Ω), ( v, q) =, q L 2 R = (Ω) Let (w, r ) X M be te solution to ( w, l ) + (r, l ) = (A v, l ), l X, ( w, q ) =, q M Clearly w V and actually w = v Tis means tat v is te Galerkin approximation to v Te teory of mixed problems togeter wit (24) implies v v H c ( v H 2 + p H ) c A v L 2 We ten ave for x X, Tus, ( v, x ) ( (v v), x ) + ( v, x ) c( x H + x L 2) A v L 2 c x L 2 A v L 2 v L 2 = wic completes te proof of te lemma ( v, x ) c A v x X x L 2, L 2 We now turn our attention to te discrete operator R It is obvious tat R is stable on H (Ω) Te following lemma sows tat it is also stable in X in te L 2 (Ω)-norm Lemma 42 Tere is c independent of suc tat (42) R x L 2 c x L 2, x X

10 J-L GUERMOND AND J PASCIAK Proof By (22), A R = P wen R and P are restricted to X It follows tat R = A P and so (42) will follow if we sow R v L 2 c v L 2, v V Here R : V X is te adjoint of R and is given by R = A Te above inequality is equivalent to proving wic is exactly (4) in Lemma 4 v L 2 c A v L 2, v V, Remark 4 Somewat similar forms of Lemmas 4, 42 can also be found in Heywood and Rannacer [3, 4] 42 Comparing H s - and Vs -norms Te following teorem is te major result of tis section Teorem 4 Tere is is positive function c l >, non-decreasing for negative arguments and non-increasing for positive arguments, and a positive non-decreasing function c u >, bot independent of, suc tat te following olds for all v in V : (43) c l (s) v eh s v V s c u ( s ) v eh s, { 2 < s < 3 2, 3 2 < s < 3 2, lower bound, upper bound Proof Step () Clearly, for v V, (4) means v X 2 c v V 2 and v X v V is evident Te lower bound in (43) for s < 3 2 follows by interpolation and (2) Step (2) Applying tis bound, we observe tat for 3 2 < s, v V s = (v, θ) θ V θ V s (v, θ) c l ( s ) θ V θ eh s c u ( s ) v eh s Tis is te upper bound for 3 2 < s Step (3) To prove te upper bound for s 3 2, we observe tat for all x X R x V 2 = A R x L 2 = P x L 2 x L 2 = x X 2 Moreover R x V c x X owing to Lemma 42 By interpolation tis gives R x V s c x X s for s [, 2] By applying tis result to v V, we infer v V s c x X s Ten we conclude using te upper bound in (2) for s [, 3 2 ) Step (4) Finally, we prove te lower bound for 2 < s Let v be in V, ten v eh s = (v, x) x H e s (Ω) x eh s = x e H s (v, P x) x eh s

11 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES Te key estimate in Lemma 3 ten implies (v, P x) (v, x ) v eh s c u ( s ) c u ( s ) x H s P x eh s x V x eh s x V c u ( s ) v V s c u ( s ) v V s x V x eh s were we used te upper estimate in (43) for te last inequality completes te proof of te teorem Similarly we ave te following s Tis Corollary 4 Tere is positive non-increasing function c l > and a positive non-decreasing function c u >, bot independent of, suc tat for all s ( 3 2, ] (44) c l ( s ) v eh s A v V s c u ( s ) v eh s, v V Proof Let v be a member of V By reasoning as in step () of te proof of Teorem 4, we infer v X s+2 c v V s+2 for s [ 2, ], ie, v X s c A v V s Using te lower bound in (2) yields te desired result for 3 2 s < For te upper bound we reason as in step (3) of te proof of Teorem 4 and we ave R x V s+2 c x X s+2 for all x X and s [ 2, ] By applying tis result to v V, we infer A v V s c v X s Ten we conclude using te upper bound in (2) for s ( 3 2, ] 5 Te semi-discrete time-dependent Stokes problem We sow in tis section an application of Teorem 4 5 Formulation of te problem Let (, T ) be a time interval (T is arbitrary) Let u H, let p [, 2], q [, 2], and f L p (, T ; L q (Ω)), and consider te following non stationary Stokes problem in weak form t u u + p = f, in Ω T (5) u =, in Ω T u Γ =, u t= = u were Ω T = Ω (, T ) It is well known tat tis classical problem as a unique solution In particular, if u = and p = q (, ), it is proved in Solonnikov [2] tat te following bound olds (52) p L p (Ω T ) + t u L p (Ω T ) + u L p (Ω T ) c f L p (Ω T ) Tis estimate as been significantly generalized by Sor and von Wal [2] to account for different exponents p (, ), q (, ), (53) p L p (,T ;L q ) + t u L p (,T ;L q ) + u L p (,T ;L q ) c f L p (,T ;L q )

12 2 J-L GUERMOND AND J PASCIAK Tese estimates are important to construct weak solutions to te Navier Stokes equations tat are suitable in te sense of Sceffer [8] Te goal we ave in mind now is to derive similar estimates using te discrete (finite-element-like) setting introduced above under te assumption p [, 2], q [, 2] Te long term program we are pursuing is to eventually extend te results of [] to omogeneous Diriclet conditions Te results of [] old in te tree-dimensional torus only, ie, for periodic boundary conditions Proving a discrete counterpart of (53) wit Diriclet conditions is a key step in tis program However, since we ave not yet been able to andle te discrete setting associated wit Diriclet boundary conditions using te non-hilbertian L p -framework, we are going to reformulate (53) using fractional Sobolev spaces Te idea is to use te Fourier transform in time as done in Lions [5, p 77] Let H be a Hilbert space wit norm H Let δ, δ <, and define L δ + (R; H) = {ψ : R t ψ(t) H; ψ(t) δ Hdt < } For all ψ L (R; H), denote by ˆψ(k) = + ψ(t)e 2iπkt dt for all k R Te Fourier transform is extended to te space of tempered distributions wit values in H, say S (R; H) We sall make use of te following Lemma 5 (Hausdorff-Young Inequality) Tere is c > suc tat for all p, δ 2, and for all ψ L δ (R; H) L (R; H), (54) ˆψ L δ (R;H) c ψ L δ (R;H), δ + δ = Following [6, p 2], we now define, (55) H γ (R; H) = {v S (R; H); tat we equip wit te norm + ( + k ) 2γ ˆv 2 Hdk < + }, + (56) v 2 H γ (R;H) := ( + k ) 2γ ˆv 2 Hdk Te space H γ ((, T ); H) is composed of tose tempered distributions in S ((, T ); H) tat can be extended to S (R; H) and wose extension is in H γ (R; H) Te norm in H γ ((, T ); H) is te quotient norm, ie, (57) v H γ ((,T );H) = inf ṽ H γ (R;H) ṽ=u ae on (,T ) Hencefort we set s := s(q) := d( q 2 ), r := p 2 Tis definition of s implies tat te embedding H s (Ω) L q (Ω) olds, were q + q = Note tat te embedding H s (Ω) Hs (Ω) (if s [, 2]) being continuous implies tat te embedding L q (Ω) Te Hausdorff Young inequality ten implies (58) f L p ((, T ); L q (Ω)) H r ((, T ); H s (Ω) is also continuous s H (Ω)), r > r

13 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 3 Our goal now is to derive estimates in spaces like H r ((, T ); H s (Ω)) 52 Te a priori estimates In addition to f L p ((, T + ); L q (Ω)), we also assume u = and f L ((, T + ); H (Ω)) Tese two ypoteses could be avoided at te price of additional irrelevant tecnicalities Te approximate counterpart of (5) is as follows: { t u + A u = P f, ae t (, T + ) (59) u t= = We start by proving a series of key estimates Lemma 52 Assume s(q) [, 3 2 ) Tere is c independent of so tat (5) t u H r ((,T );V s ) + A u H r ((,T );V s ) c, r > r Moreover, if u L 2 ((,T );H (Ω)) is uniformly bounded, te following uniform estimates also old: (5) t u H τ ((,T );V α ) + u H τ ((,T );V α ) c, for all α, α s + 2α, and for all τ < τ := +α +s ( 3 2 p ); and (52) A u H ρ ((,T );V α ) c, for all α, 2α s α, for all ρ > ρ := α s ( p 2 ) Proof () By taking te scalar product of (5) wit A u we infer (using P f V c f H ) 2 d t u 2 V + u 2 L P 2 f V u V c f H u V Ten, since f L (H ) is bounded, te Gronwall Lemma yields Tis bound clearly implies u L (V ) + u L 2 (L 2 ) c (53) u L p (L q ) c (2) Extension We extend u and f by zero on (, ] and (T +, + ), and we sligtly abuse te notation by still denoting tese extensions by u and f, respectively Let ϕ C (R) be an infinitely smoot function compactly ported on (, T + ) and equal to on [, T ] We now set ũ = ϕu and f = ϕf + ϕ u It is clear tat ũ and f are well defined on te time interval (, + ), and (53) implies tat f L p ((,T ); H e s ) is uniformly bounded Te approximate problem takes te following form in S (R; V ): d + A ũ = P f dtũ

14 4 J-L GUERMOND AND J PASCIAK Ten, denoting by û and ˆf te Fourier transform of ũ and f, respectively, and upon taking te Fourier transform of te above equation, we obtain (54) 2iπkû + A û = P ˆf (3) Bound (5) Let α R + Testing te above equation wit te complex conjugate of A α û and taking te imaginary part of te result yields 2π k û 2 V α ˆf eh s A α u eh s Using te lower bound in (43) for s [, 3 2 ), we obtain k û 2 V α c ˆf eh s c ˆf eh s A α û eh s c ˆf eh s û V s 2α Assume α s + 2α, ten by interpolation we obtain û V s 2α û γ V α û γ, V A α û V s were γ = 2α+ s +α Inserting tis inequality in te previous estimate yields Tis in turn implies k 2 were µ [, to k, + k 2 2 k û 2 γ V α 2 γ µ û 2 V α 2 γ 2 γ µ û 2 V α c ˆf eh s c( + k ) µ ˆf û γ V 2 2 γ eh s 2( γ) û 2 γ, ] is still arbitrary We now integrate over R wit respect dk c (+ k ) µ L l ˆf 2 2 γ L 2m 2 γ ( e H s V ) û 2( γ) 2 γ 2n( γ), L 2 γ (V ) were l + m + n = Te first integral in te rigt-and side is bounded provided µl > Furtermore, we set m and n so tat p = 2m 2 γ and 2 = 2n( γ) Te Hausdorff Young Inequality togeter wit te embedding L q (Ω) 2 γ s H + (Ω) ten implies k 2τ û 2 V α dk c f 2 γ L p ((,+ );L q ) ũ ( γ) 2 γ L 2 ((,+ );H ), were τ = 2 γ µ 2 < 2 ( 2 2 γ l ) Owing to te definition of ũ and f, tis and u being uniformly bounded in L 2 ((, T ); H (Ω)) L 2 ((, T ); L 2 (Ω)) imply (55) t u H τ ((,T );V α ) + u H τ ((,T );V α ) c Te bound on t u is obtained by using t u = 2iπkû

15 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 5 By collecting te definitions of γ, l, m, and n, we deduce tat te above inequality olds for all τ and α suc tat τ < τ := +α +s ( 3 2 p ), and α s + 2α (4) Bound (52) Multiply (54) by A α û and take te real part to obtain A û 2 V α ˆf eh s A α û eh s ˆf eh s û V 2+s 2α Note again tat we used te lower bound in (43) for s [, 3 2 ) Assume now tat 2α s α, ten by interpolation we obtain û V 2+s 2α û δ V û δ, V 2 α were δ = +s 2α α Inserting tis inequality in te previous estimate yields Tis in turn implies Âu 2 δ V α ( + k ) ν û 2 V α c ˆf H s û δ V c ( + k ) ν ˆf 2 2 δ û H s 2( δ) 2 δ V were ν is still arbitrary By proceeding as in step (3) we finally infer (56) A u H ρ ((,T );V α ) c, for all α and ρ suc tat ρ > ρ := α s ( p 2 ), and 2α s α (5) Te estimate A u in (5) is obtained by using α = s in (56), ie, δ = Te estimate on t u in (5) is obtained by using α = s in (55) We are now in measure to conclude by stating te discrete counterpart of te Sor and von Wal estimates (53) Te following Teorem is te main result of tis section Teorem 5 Assume s(q) [, 3 2 ) Tere is c independent of so tat for all r > r (57) u H r ((,T ); e H s ) c If q is suc tat s(q) < 2, ten (58) t u H r ((,T ); e H s ) c Moreover, if u L 2 ((,T );H (Ω)) is uniformly bounded, te following uniform estimates also old: (59) t u H τ ((,T ); e H α ) + u H τ ((,T ); e H α ) c,,

16 6 J-L GUERMOND AND J PASCIAK for all τ < τ = +α +s ( r) and all α [, 2 ) suc tat s [α, + 2α]; and (52) u H ρ ((,T ); e H α ) c, for all ρ > ρ = α s r and all α [, 3 2 ) suc tat s [2α, α] Proof Te inequality (57) is a consequence of te lower bound in (44) togeter wit (5) Te inequality (58) is a consequence of te lower bound in (43) in Teorem 4 togeter wit (5) Te rest of te proof follows along te same lines Remark 5 Te ypotesis f L ((, T + ); H (Ω)) is not really necessary It is just meant to deduce an easy bound on u in L 2 ((, T +); L 2 (Ω)) to guaranty tat te extension f is bounded in L p (R; H s (Ω)), see (53) Tis type of bound could be deduced witout tis ypotesis by invoking more involved arguments Modulo more tecnicalities, te ypotesis u = can be removed by assuming u D(A 2 s ) Remark 52 Working wit fractional exponents of te Stokes operator is not te most elegant way to treat te above problem It would be more satisfactory to directly deduce L p (L q ) estimates, but tis necessitate a L p (L q ) teory of te resolvent of te finite-element-based Stokes operator tat seems unavailable (or of wic we are unaware) at te present time 53 Application to te 3D Navier Stokes equations Note tat wen applied to te Navier-Stokes equations in tree space dimensions, te restriction s(q) < 2 in Teorem 5 makes te bound (58) somewat useless Actually, in tis case te above analysis applies wit f = g u u were g is a smoot source and u u is te nonlinear advection term Since a standard uniform estimate in L ((, T ); L 2 (Ω)) L 2 ((, T ); H (Ω)) olds on u, it comes tat f L p (, T ; L q (Ω)) were p and q satisfy te equality 2 p + 3 q = 4 and p 2, q 3 2 Te restriction on q yields 2 s = 3( q 2 ) 3 2, wic is contradictory wit te assumption s < 2 wic is needed for te bound (58) to old Tis remark is te reason wy we ave been led to account for te additional uniform estimate u L 2 ((, T ); H (Ω)) in Teorem 5 wic gives te more sopisticated estimate (59) Let us illustrate te use of (59) in te tree-dimensional Navier-Stokes situation Assume now tat α [, 2 ), ten (52) t u H τ (H α ) + u H τ (H α ) c, for all τ < τ provided α s + 2α and s < 3 2 Note tat τ = +α +s ( s ) owing to te relation 2 p + 3 q = 4 and te definition s = 3( q 2 ) Observe tat +α +s ( s ) is maximum at s = 3 2 ; as a result, (52) olds for all τ < 2 5 ( + α) for all α [ 4, 2 )

17 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 7 Let ɛ > and ɛ > be two small positive numbers, and set α = 2 ɛ, s = 3 2 ɛ Assume tat ɛ and ɛ are small enoug so tat (522) α s + 2α, 2 5 ɛ < 2 ɛ 5 2ɛ < ɛ Ten 2 5 +α + 3ɛ > +s ( s ), and te bound (52) can be rewritten as (523) u H 3 5 3ɛ ((,T );H 2 +ɛ ) + tu H 2 5 3ɛ ((,T );H 2 +ɛ ) c Note tat (523) is sligtly better tan wat te Sor and von Wal estimate gives by embedding Actually, taking q = 3 2, we conjecture tat a discrete version of te inequality (53) togeter wit (58) would give t u H 2 ɛ ((,T );H c, for all ɛ >, wic is clearly weaker tan 2 ) (523) wen ɛ is close to zero since 2 < 2 5 Tis is not a surprise since more information on u as been used to deduce (523) Te estimate (523) is a key to extend to omogeneous Diriclet conditions te results of [], wic for te time being olds only in te treedimensional torus An important link still missing in tis program is an estimate on te pressure tat allows for te convergence of te product p u in some reasonable sense To derive suc an estimate, set s = α = 2 Ten p =, and (57) yields (524) u H 2 ɛ ((,T ); H e 2 c, ) for all ɛ > Tis yield a uniform bound on p H 2 2 ɛ ((,T );H (note in 2 ) passing tat tis is coerent wit te estimate p L +ɛ ((, T ); L 3+3ɛ +3ɛ (Ω)) given in [2, Tm 33]) Tis sows tat it sould be possible to pass to te limit on te product p u, tus implying tat te result of [] sould old for Diriclet boundary conditions Tese developments are reported in [2] Appendix A Proof of Lemma 22 Lemma 22 Tere is a non-increasing function c l > and a non-decreasing function c u >, bot uniform in, suc tat for all s ( 3 2, 3 2 ), (A) c l ( s ) v eh s v X s c u ( s ) v eh s, v X To prove te above lemma, we sall use te following two lemmas Lemma A Tere is constant c > suc tat for all s ( 2, 3 2 ), (A2) v H s v, x, v eh s c x H 2 s (Ω) H (Ω) x H 2 s Here te brackets represent te duality paring between H s (Ω) H +s (Ω) and

18 8 J-L GUERMOND AND J PASCIAK Lemma A2 Tere is a non-decreasing function c u >, uniform in, suc tat for all s [, 2 ) (A3) E (v) eh s c u (s) v eh s, v H s (Ω) Proof of Lemma 22 Step (): Te case s [, ] is given in Bank and Dupont [, Lemma ] (Te upper bound is a consequence of π : L 2 (Ω) X and π : H (Ω) X being stable (see (24)) Te lower bound is a consequence of te natural injection I : X L2 (Ω), and I : X H (Ω) being stable) Step (2): Te lower bound for + s, s (, 2 ) is verified as follows Let v X Owing to (A2), (A3), and step (), we get v eh +s c c u (s) But we also ave w e H s (Ω) w e H s (Ω) v, w w eh s ( v, E w) E w eh s ( v, w ) c u (s) w X w X s = c w e H s (Ω) ( v, E w) w eh s ( v, w ) c u (s) w X w eh s ( v, w ) ( v, w ) = w X w X s w X ( ) s 2 w L2 +s (( ) 2 v, w ) = = ( ) +s 2 v L2 = v w X w X +s, L2 wic combined wit te previous bound yields te desired result Step (3): We next prove te upper bound for + s, s (, 2 ) We use Step () to conclude v X +s ( v, w ) = w X w X s v eh +s c w X w eh s w eh s v ehs w X w X s w eh s c v eh +s Note tat tis also sows tat c u in (A) is uniformly bounded on [, 3 2 ] Step (4): We now consider te case 3 2 < s < Using te lower bound just proved for < s < 3 2 yields v X s = (v, w ) w X w X s (v, w ) c u (s) c u (s) v eh s w X w eh s

19 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 9 Finally, applying (24) and te upper bound just proved for < s < 3 2 gives v eh s = (v, w) w H e s w eh s c u (s) w e H s (v, π w) π w eh s (v, w ) (v, w ) c u (s) c u (s) w X w eh s w X w X s Tis completes te proof c u (s) v X s Proof of Lemma A Step (): Let s [, +s 2 ) and let v H (Ω) and set +s f = v Clearly f H (Ω) and elliptic regularity implies tat tere is a constant c >, independent of s, suc tat c v H +s f eh +s Hence c v H +s f eh +s = φ e H s (Ω) = φ e H s (Ω) v, φ φ eh s Note tat te last equality olds because C s [, 2 ) Step (2): Let s [, 2 ) solving f, φ φ eh s = φ e H s (Ω) v, φ φ eh s (Ω) is dense in H s (Ω) for s For all w H (Ω), define x(w) H (Ω) x(w), y = (w, y) eh s, for all y H (Ω) s were (, ) eh s is te scalar product in H (Ω) Elliptic regularity implies tat tere is c, independent of s, suc tat Let v C x(w) eh +s (Ω) Ten c w eh s, w H s (Ω) v eh s = w e H s (Ω) c c w e H s (Ω) x e H +s (Ω) (v, w) eh s w eh s v, x(w) x(w) eh +s v, x x eh +s = w e H s (Ω) v, x(w) w eh s Te desired inequality follows by density since s [, 2 ) Tis completes te proof of te lemma

20 2 J-L GUERMOND AND J PASCIAK Proof of Lemma A2 Let v C (Ω) and let s [, 2 ) Clearly E (v) H s (Ω) H (Ω) Owing to (A2) and (27), we ave E v H s c c x e H +s (Ω) x e H +s c v eh s Ten use te fact tat C ( E v, x) x eh +s ( v, E x) x eh +s x e H +s (Ω) E x eh +s x eh +s c u (s) v eh s (Ω) is dense in H s (Ω) to conclude Lemma A3 Assume tat te family {X } > is suc tat tere is a non decreasing function c u (s) >, s [, 2 ), so tat v eh s c u (s) s v L 2, v X + x X + + xd X, s [, 2 ), and tere is a linear operator ρ : H s (Ω) X and a constant c, independent of and s, suc tat ρ v eh s c v eh s, (ρ )v L 2 c s v eh s, v H s (Ω) Let T : Z H +s (Ω) X be a linear operator, were Z is a closed subspace of H +s (Ω) Assume tat te family {T } > is suc tat tere is a constant c 2, uniform in and s, so tat T u u H c 2 s u H +s, u Z, s [, 2 ) Ten tere is a non decreasing function c u(s) >, s [, 2 ), so tat (A4) T u H +s c u(s) u H +s, u Z, s [, 2 ) Proof We proceed as in [2, Appendix] Te norm in H +s (Ω) can be defined by d v H +s = v H + xi v H s Ten T u H +s T u H + d i= x i T u H s and for all i {, d}, i= xi T u H s c( π xi u eh s + π xi u xi T u eh s ) c(c xi u eh s + c u (s) s π xi u xi T u L 2) c u (s)( xi u H s + s ( (π ) xi u L 2 + xi (u T u) L 2)) c u (s)( xi u H s + s ( s xi u eh s + s u H +s)) c u (s) u H +s Tis concludes te proof

21 DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES 2 Remark A Te inverse inequality ypotesis is reasonable if X is a finite element constructed on a quasi-uniform mes In tis case, by redoing carefully te computation in [2, Appendix] we infer tat tere is c, uniform in bot and s suc tat v H s c ( 2s) 2 s v L 2, v X + x X + + xd X, s [, 2 ) Tat is to say te inverse inequality ypotesis olds wit c u (s) ( 2s) 2 References [] Randolp E Bank and Todd Dupont An optimal order process for solving finite element equations Mat Comp, 36(53):35 5, 98 [2] James H Bramble, Josep E Pasciak, and Jincao Xu Te analysis of multigrid algoritms wit nonnested spaces or noninerited quadratic forms Mat Comp, 56(93): 34, 99 [3] James H Bramble and Xuejun Zang Te analysis of multigrid metods In Handbook of numerical analysis, Vol VII, Handb Numer Anal, VII, pages Nort-Holland, Amsterdam, 2 [4] F Brezzi and M Fortin Mixed and Hybrid Finite Element Metods Springer-Verlag, New York, NY, 99 [5] P Clément Approximation by finite element functions using local regularization RAIRO, Anal Num, 9:77 84, 975 [6] Monique Dauge Stationary Stokes and Navier-Stokes systems on two- or treedimensional domains wit corners I Linearized equations SIAM J Mat Anal, 2():74 97, 989 [7] A V Fursikov Optimal control of distributed systems Teory and applications, volume 87 of Translations of Matematical Monograps American Matematical Society, Providence, RI, 2 [8] V Girault and P-A Raviart Finite Element Metods for Navier-Stokes Equations Teory and Algoritms Springer Series in Computational Matematics Springer- Verlag, Berlin, Germany, 986 [9] P Grisvard Elliptic problems in nonsmoot domains, volume 24 of Monograps and Studies in Matematics Pitman (Advanced Publising Program), Boston, MA, 985 [] J-L Guermond Finite-element-based Faedo-Galerkin weak solutions to te Navier- Stokes equations in te tree-dimensional torus are suitable J Mat Pures Appl (9), 85(3):45 464, 26 [] J-L Guermond Te LBB condition in fractional sobolev spaces and applications IMA, Numer Anal, 26, Submitted [2] J-L Guermond Faedo-galerkin weak solutions of te navier stokes equations wit diriclet boundary conditions are suitable J Mat Pures Appl (9), 27, Submitted [3] Jon G Heywood and Rolf Rannacer Finite element approximation of te nonstationary Navier-Stokes problem I Regularity of solutions and second-order error estimates for spatial discretization SIAM J Numer Anal, 9(2):275 3, 982 [4] O A Ladyzenskaya Te matematical teory of viscous incompressible flow Second Englis edition, revised and enlarged Translated from te Russian by Ricard A Silverman and Jon Cu Matematics and its Applications, Vol 2 Gordon and Breac Science Publisers, New York, 969 [5] J-L Lions Quelques métodes de résolution des problèmes aux limites non linéaires, volume Dunod, Paris, France, 969 [6] J-L Lions and E Magenes Problèmes aux limites non omogènes et applications, volume Dunod, Paris, France, 968

22 22 J-L GUERMOND AND J PASCIAK [7] J-L Lions and J Peetre Sur une classe d espaces d interpolation Inst Hautes Études Sci Publ Mat, 9:5 68, 964 [8] V Sceffer Hausdorff measure and te Navier-Stokes equations Comm Mat Pys, 55(2):97 2, 977 [9] RL Scott and S Zang Finite element interpolation of nonsmoot functions satisfying boundary conditions Mat Comp, 54(9): , 99 [2] Hermann Sor and Wolf von Wal On te regularity of te pressure of weak solutions of Navier-Stokes equations Arc Mat (Basel), 46(5): , 986 [2] V A Solonnikov Estimates of te solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations Zap Naučn Sem Leningrad Otdel Mat Inst Steklov (LOMI), 59:78 254, 257, 976 [22] R Temam Navier Stokes Equations, volume 2 of Studies in Matematics and its Applications Nort-Holland, 977 Department of Matematics, Texas A&M University 3368 TAMU, College Station, TX , USA address: guermond@mattamuedu, pasciak@mattamuedu