Struwe Partial Regularity

8/3/2019 Struwe Partial Regularity

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On PartialRegularity Results for theNavier-Stokes Equations

MICHAEL STRUWEETH, Ziirich

AbsbartLooking at the reg ularity results of Scheffer, respectively, CaKarelli,Kohn and Nirenberg from anew point of view indicates that es timates for the pressure do not play an essential role in partialregularity results for the Navier-Stokes equations.

1. IntroductionLet P be a domain in R " with smooth boundary aP; P may be unbounded.is modelled byhe motion of an incompressible fluid confined in the regionthe Navier-Stokes equations

(1.1) 8 , u - A u + ( u . v ) u + ~ p = f in P X O , T ] ,(1.2) div u = 0,

for the velocity u. Of course, the initial data uo should satisfy the compatibilityconditionsdiv uo = 0, uOla,= 0.

For smooth forces f, global weak solutions u E L2,"(P) to (1.1)-(1.3) withVu E L2(P X [0, TI) were constructed by Leray [9] and Hopf [7].Serrin [15] hasshown that local weak solutions u to (1.1)-(1.2) will be locally bounded (andhence locally regular in the spatial variables) if in addition

with(1 4)Compare also Okyama [U].

n 2- + - c 1r s

Communications on Pure and Applied Mathematics, Vol. XLI 437-458 (1988)0 988JohnWiley & Sons, Inc. CCC 0010-3640/88/040437-22$04.00


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438 M. STRUWEFabes, Jones and Riviere [3], respectivelySohr and v. Wahl (cp. [20], p. 190mhave extended these results to cases where equality holds in (1.4); see also GigaHowever, in dimensions n 2 a large gap opens between the regularityavailable in the existence theorems and the additional regularity required forSerrins result. In n = 3 dimensions, this gap has recently been narrowed byScheffer [13], respectively Caffarelli, Kohn and Nirenberg [2]. In particular, basedon Scheffers ideas the latter authors prove the following local partial regularityresult for suitable weak solutions to (1.1)-(1.3).

[211.

THEOREM.1. There is an absolute constant E > 0 with the following property:If ( u , p ) is a suitable weak solution of the Nauier-Stokes system near ( x , t ) withf E Lq, q > $, and iffinally

then u is essentially bounded in a neighbourhood of ( x , t ) .See [2], Proposition 2, p. 776; the definition of suitable weak solution is givenin [2], p. 779ff. In particular, Theorem 1.1 implies that any suitable weak solution

of (1.1)-(1.3) on an open set Q in space-time is locally bounded off a singular setS of Hausdorff dimension dim S 6 1 (with respect to the metric S ( ( x , t ) , y , s ) )= ( x - ( + d m ) ; ee [2], Theorem B, p. 772.

~ The notions of weak solutions and suitable weak solutions essentiallyW e r in that the latter notion also requires a generalized energy inequality andan estimate for the pressure p . By using Solonnikovs estimates [17], Theorem 17,p. 115, Caffarelli, Kohn and Nirenberg succeed in proving the existence ofsuitable weak solutions for Sl = R 3 or Sl e R3 (bounded); see [2], Theorem A,p. 772ff. Soh and von Wahl have recently obtained estimates for the pressure inexterior domains, as well; see [16], p. 436.Note that Scheffer has recently constructed an example of a singular solutionu to the Navier-Stokes equations with a highly irregular force f which at allpoints in space-time is acting against the motion of the fluid, i.e., the build-up ofa singularity seems to be mediated by the pressure; see [14].In this note, we want to bring out more clearly the role the pressure plays inregularity. While we also require a version of the energy inequality, our resultsindicate that for the local partial regularity result Theorem 1.1 estimates for thepressure are not essential. To support this view, we prove local a priori estimatesfor weak solutions to the Navier-Stokes equations under a smallness condition(1.5)-but using no information about the pressure. In order to simplify theproblem while keeping the essential difficulties, instead of considering the insta-tionary problem (1.1)-(1.3) in n = 3 space dimensions, we study the time-inde-pendent problem (1.1)-(1.2) in n = 5 dimensions. (By the dimensions table in [2],


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440 M. TRUWEx E Cl there is R , > 0 such that(1.11) r - l i lVu12 dy 5 , fo r all r < R,,x -y l c rthen u is Holder-continuous near x for some exponent a > 0 , and its a-Holder normis bounded in terms of the L2-norms of u, V U , the Lq-norm o f f , and R , . (Higherregularity for smooth f then follows. In particular, u i s a classical solution to(1.6)-(1.7) of a singular set of Hausdorf dimension less than or equal to 1.)

For the proof of Theorem 1.2 we rely on standard techniques for provingregularity of solutions of elliptic systems as developed by Morrey 1111,or Giaquinta [5], and Lions-Magenes [lo]. Moreover, we use estimates ofSolonnikov [18] for the linearized Stokes' equations

divv = 0.A similar approach to the stationary Navier-Stokes equations in dimensionsn 4 was taken by Giaquinta and Modica [6] in order to obtain generalizationsof Gerhardt's [4] regularity result to systems of Navier-Stokes type. The extensionof this method to n = 5 dimensions is possible by the use of Lemma 2.4;moreover, we heavily exploit exact dependence of solutions to the linearizedStokes problem on the data of all orders. (Our Theorem 2.7 in this respect servesas an extension of Theorem 1.3 of [6], p. 183.) It would be interesting to know ifanalogous partial regularity results hold in dimensions n > 5.

We point out that potential theoretical estimates and (local) estimates for thepressure also appear in the proofs of the generalizations of Serrin's resultmentioned earlier. As a side line in Section 3 we observe that these results may beobtained by a simple extension of Serrin's original method, where the pressure iseliminated from the equations (1.1) by considering the vorticity w = curl u (indimension n = 3) instead of u.

Thus it seems that local regularity properties are not influenced by thenonlocal effects of the pressure-as long as we are interested only in bounded-ness and spatial regularity of solutions. Note, however, that given any boundedmeasurable function a ( t ) and any harmonic function \k the functionu ( x , ) = a ( t ) v * ( x )

is a local weak solution to (1.1)-(1.2). This example due to Serrin [15] makes itclear that higher regularity in time will in general require global estimates-alsofor the pressure.


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NAVIER-STOKES EQUATIONS 4412. Proof of Theorem 1.2

2.1. Notations. We are interested in the local behavior of weak solutions uto the stationary Navier-Stokes equations (1.6)-(1.7) in a smooth open domain0 c W5. t will suffice to restrict u to balls

Moreover, by invariance of (1.6)-(1.7) under translation and scalingu( ' ) R u ( - ? k )= u R ( x ) ,

we may always shift the center of attention to the point x = 0 and we mayassume that u solves (1.8) on the unit ball

Also let

with boundaryS, = aB,.

For a domain P c R", 15 p 4 m, k E N let LP(P; W k ) enote the Lebesguespace of measurable functions cp: 0 + R k such that

For m E N let H'",J'(P;R ) denote the Sobolev space of functions cp withdistributional derivatives Iv"rpIE LP(P) for all la1 6 m and

The Sobolev space Hr*P(P;R k ) s defined as the 11 I)Hm.p-closuref CF(P; R k ) :the space of smooth functions with compact support in 9.For any bounded domain P, Poincar6's inequality holds,I l cp l lHm'P(Q) 6 4% , )llv'"cpIILP(n)

for all cp E H,"%P(Q; ~ ) .


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442 M.STRUWEMoreover, we recall the trace and embedding theorems (for smooth 5 2 ) :

In case p = 2 the spaces L 2 ( Q , R k ) , tc. are Hilbert spaces. The dual ofH$y2(52,R k ) s denoted by H -" (Q , W") with the dual norm

The definition of Hmp2(i2;2 k, can be extended to arbitrary real m by interpola-tion; see Lions-Magenes [lo], Chapter I. The trace theorem continues to hold truefor m > f and m - f 6C N; the embedding theorem remains true for m 2 .For ease of notation we shall often abbreviate LP = LP(52;R"), etc. ifdom ain and/o r range are clear from the context.2.2. The key lemmata. Let u E H'*2(52;R') be a solution of (1.8)-(1.9)satisfying (1.10) in a domain 52 c R 5 with f E Lq(0), q > H. We may assume

that B c 0.LEMMA.1.ing property ; Let There exist absolute constants 0 < A , < 1,e, > 0 with the follow-

then


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NAVIER-STOKES EQUATIONS 443A trivial consequence of Lemma 2.1 is the following:OBSERVATION.2. Suppose u E H1?(B;R s )satisfies (1.8)-(1.10) in B withf E L9(B ) , q > 4, and suppose that, for some R , > 0, the condition

holds. Then for any E > 0 there is a number 0 < R < R, depending only on E,the L2-norm of u and the L9-norm of f, and the number R , such that theestimate

is valid.Proof: In Lemma 2.1 chooseS < e, r, = min{Rl( l l f llLv, S), R , } , r k +l = Airk,

k E No,and iterate inequality (2.2). This proves the statement.Theorem 1.2 will now be a consequence ofLEMMA.3. For any q > $, any a -= 2 - 5 / q there exists a constant e, > 0depending only on a with the following property: If u E H1 i2 (B;Ws) satisJies(1.8)-(1.10) in B with f E L 9 (B) a n d i f , for some radius 0 < R 5 R , =

RAllf I lLqB) ,4,

then there exists a neighborhood U f 0 in B and a constant C depending only on R ,11f l l L v ( B ) , q, and a such tha t fo r all x E U and all 0 < r s R the Dirichlet growthcondition

Proc- of Theorem 2: Take e, = el. By Observation 2 (with R,=inf( R,, R3(ll I I L o ( B ) , a)}) and assumption (L l l ) , condition (2.4) is satisfied forsome radius R = R(ll f l l L v ( B ) , a,R,, and therefore Lemma 2.3 yieldsthe estimate (2.5), uniformly in a neighborhood U of x, = 0, for r 5 R . ByMorreys Dirichlet growth theorem [ l l ] , Theorem 3.5.2, p. 79, condition (2.5)implies that u is Halder continuous with exponent a in U,with Holder norm


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444 M. STRUWEbounded by C. Tracing back the dependence of C through R on the variousquantities listed in Lemma 2.3, respectively O bservation 2.2, we find that

as claimed.alizes the w ell-known Courant-Lebesgue lemma to arbitrary d imensions.2.3. A lemma. The following easy applica tion of Fubini's theorem gener-

LEMMA .4. Suppose u E H', , (B) . Then there is a radius r E [i,1 such thatuIs, E H'l2(S,) and

Proof: By Fubini's theorem, u, v u E L*(S , )for a.e. r E $, 11. Let A1,, bethe set of all values r E [f , 1 such that (2.6), respectively (2.7), hold, and letXl , , = [ f , 1\ . 2 .The estimate

and a similar estimate for the measure lA21show that

i.e., Al n A, # 0 , and we can find r as claimed.We apply Lemma 2.4 to our solution u. Scaling with (2.1) we may assumethat the conclusion of Lemma 2.4 holds true with r = 1, i.e., u E H 1 l 2 ( B ) ,u Is E H','(S) . We now decompose u = u + w, where u satisfies(2.8) - A v + v q = O in B,(2-9) divu = 0 in B,(2.10) u = u on S ,while w solves(2.11) - Aw + ( U * V ) U + v ( p - q ) = f in B,(2.12) divw = 0 in B ,(2.13) w = 0 on S.


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NAVIER-STOKES EQUATIONS 445Note that by (1.2) the necessary compatibility condition for the existence of o issatisfied.

Of course, we may give weak interpretations to (2.8)-(2.10), respectively(2.11)-(2.13), analogous to (1.8)-(1.9). For example, u E H 1 , 2 ( B )weakly solves(2.8)-(2.10) if the following relations hold: u - u E Ho>(B)and(2.14) b u v u * d u = 0 for all v * E C Z ( B ) :divu* = 0,

(2.15) /,. v q * dx = 0 for all q* E C?(B).2.4. The Stokes problem. In this section we recall Solonnikovs results forthe linearized stationary problem (2.8)-(2.10) (Stokes problem) from which wederive dual and intermediate existence and regularity results by the methodof transposition and interpolation developed by Lions and Magenes [lo]. Inparticular, we obtain the followingPROPOSITION.5. Suppose u E H1y2(B;W 5 ) with uIs E Hp2(S;R 5 ) satisfiesdiv u = 0. Then there exists a unique solution u E H 3 / 2 , 2n H*/(B; W 5 ) toStokes problem (2.8)-(2.10) and

Moreover, u is analytic in B and for any 0 < r < f the estimates(2.17)

(2.18)

hold. The constants c are independent of the da ta u.First we recall the following result by Solonnikov [18]-see also Lady-zenskaya 171, Theorem 111, p. 78-for the inhomogeneous Stokes problem:

(2.19) - A v * + v q * = f * in B ,(2.20) divv* = 0 in B, I(2.21) u * = u * on S.Since (2.19)-(2.21) is elliptic in the sense of Agmon, Douglis and Nirenberg (seealso Solonnikov [18], p. 197ff.) by the estimates in [l],Chapter 10, p. 75ff.,Theorem 2.6 below is valid for any dimension n E M.


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446 M. TRUWETHEOREM .6. Suppose f * E H"I~(B; "), u* E H m + 2 - 1 / 2 , 2S;") forsome integer m g 0 and suppose that

(2.22) b* d a = 0 ,where n is the exterior unit normal to S; hen there exists a unique solutionu* E H"+2*2(B; " ) to problem (2.19)-(2.21) and(2.23) IlU*llHm+2.2(B) 4llf llH".2(B) + Ilu*llHm+2-1/2.2(s)),with a constant c independent o f f * and u * .

Note that if u* extends to a function u* E H1*'(B) with div u* = 0, thenthe compatibility condition (2.22) is automatically satisfied.Following the method of Lions and Magenes we now extend Theorem 2.6 tom E R . By density of Cm(B) n H".'(B), respectively Hhm(B) or all m 2 , itsuffices to establish the a priori estimate (2.23) for real m . Moreover, for ourpurposes we need only consider the homogeneous case f = 0.For ease of notation we introduce the spaces

{ u E Hmi2(B)ldivu 0} if m 2 ,{ u E H"(B)Jdivu = 0} if m < 0,(2.24) Z " = {

where div u is understood in the sense of distributions. Similarly, we may definetrace spaces

with induced norms 11 I l r m , where(u, ) ( s )= (u, )p-w(S)XH-m+1/z29

denotes the dual pairing of a distribution u with the (smooth) normal vectorfield n.Let u E Z" be a solution to (2.8)-(2.10) with boundary data u E r",m E R . Decompose u 7 u1 + VS,where(2.25) A s = O in B,(2.26) a- s = u * nan on S.


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NAVIER-STOKES EQUATIONS 447Note that since u E r", the compatibility condition for the existence of s issatisfied. From Lions-Magenes [lo], Theorems 11.5.4, p. 1.165, 11.6.7, p. 1.179,11.7.4,p. 1.188, we infer moreover the estimate

for all real m, nd thus that vs as a trace on S such that(2.28)(Note that, if rn - E 2, estimate (2.28) cannot be derived directly from (2.27)via the standard trace theorems; see [lo], Theorem 1.4.3, p.I.26. However, ssolves an elliptic equation, and (2.28) follows using the above results of Lions-Magenes for the Dirichlet problem for equation (2.25).)The function u, solves a Stokes problem (2.8)-(2.9) with pressure q1 = q andboundary data

u1 = u - v s on S.Observe that by (2.26) we have u1 n = 0 on S , and (2.22) is satisfied.We now estimate u, in H - " ( B ) for integer m 2 0. By duality,

For any such f* let u * E Xm + 2 be the solution to (2.19)-(2.21) with u * = 0.Upon integrating by parts we see that

Since d i v u * = divu, = 0 in B, u* = 0, u1 n = 0 on S , the last two termsvanish. By (2.23), (2.28) and the trace theorem we therefore conclude that


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448 M. TRUWEThat is, it follows by (2.27), (2.29) that

By interpolation we finally obtainTHEOREM.7. The Stokes problem (2.8)-(2.10) establishes on algebraic andtopological isomorphism between the spaces 2" and I?" for all m E R and alln >= 2.Notice that our definition of 8", r rn y a result of Triebel [19],Chapter I.17.1. Theorem 1, p. 118, is consistent with interpolation.Proof of Proposition 2.5: The first assertion follows from Theorem 2.7 (withm = $) and the Sobolev embedding theorem. By linearity, vu is not changed ifwe subtract the mean value

from the data. Thus we obtain

as claimed.To verify the remaining assertions note that q is represented by an H1/292-function on B. By (2.9), q is harmonic, in particular, analytic in B, and so is u.Normalizing q by requiring the mean value of q to be 0, we have moreover from

(2.8)

The right inequalities simply follow upon testing (2.8) with u - u ; we use ourchoice of S.Let cp E C?(B) be a smooth localizing function, m E N . Multiplying theequation Aq = 0 by P - ' q cp and integrating by parts we obtain the bound


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NAVIER-STOKES EQUATXONS 449Letting m = 1,2,3,. we eventually arrive at the estimate

for all m E N and any compact subdomain B' @ B. For m > )n + k, Vk+'qwill be locally bounded by the Sobolev embedding theorem andl lv411H*.-(B. ) $ c(dist(S, B ' ) , k ) lVUlfL~(S)

for any B' Q B, k E No.Choose a radius r E $, 11 such that (2.6)-(2.7) hold for u and decomposeu = 6 + i,where A6 = 0 in B, while t? E H,'*'( ,). By the mean value propertyof harmonic functions, for any p < +I ,

and the right-hand sides may be further estimated in terms of I l ~ ] l & ~ , , respec-tively IIvu11&), via Theorem 2.7 (with m = 0). Finally, from Schauder's esti-mates for the solution 6 of Dirichlet's problemA ~ = A u V q in B,,

we also deduce that, for p < r , 0 < a! < 1,

and similarly that

IlVV=ll'(6.J CPSllV41 2 ( S ) .


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450 M. TRUWE2.5 Proofs of Lemmata 2.1, 2.3: Multiply equation (2.11) by w and integrateby parts to obtain the identity

p w 1 2 d r + L ( u v u ) w d x + /,. v w w d x = if.dx .Since u = u + w , where v u E L5I2,and on account of (l.lO), this can be justifiedby an approximation procedure. Note that by (2.12)-(2.13) any terms involvingthe pressure disappear. By (1.10) also the third term on the left is non-negative,and by Holder's inequality for f E Lq, 4 > f , we arrive at the estimate

with s = 4/(4- 1) c $. By the Poincar6-Sobolev inequality,

Young's inequality therefore gives

where we have also used Proposition 2.5. Hence if in addition we take (2.7) intoaccount we see that

Combining this estimate with our Campanato-type estimates (2.17)-(2.18) for thefunction u we infer that, for p < $,

Choosing a larger constant c1 if necessary, we may assume that (2.31) holds forall p c = 2. Moreover, using (2.1), we may scale back to the unit ball r = 1.Proof of Lemma 2.1: Choose 0 < A, 5 1 and then el > 0 such that

(2.32) c1 * ( A; + E l ) 4 +A;.


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NAVIER-STOKES EQUATIONS 45Scaling inequality (2.31) o arbitrary radii r 5 1,p = A,r, we see that, if

with y = 6 - o /q > 2, .e.,

Hence Lemma 2.1 follows if we let

In order to prove Lemma 2.3 we note that (2.30) and Proposition 2.5 alsoimply that, for p < r = 1,

We may assume that c3 2 l .(Y < min(l,2 - 5 / 4 } is given. Choose 0 < A,The remainder of the proof is similar to that of Lemma 2.1: Assume that1 and E~ > 0 such that(2.36) C 3 (A \ + E 2 ) 6 x32+2a=< $A',.


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452 M. TRUWEScaling to arbitrary r 1, p = A2r, if

we obtain, analogous to the estimates (2.33)-(2.34), that

Moreover, (2.36) implies (2.32) and hence (2.34). Therefore, ifr s ~ l ( l l f I l L 9 ( B ) , 2 ) ,

condition (2.37) reproduces itself.Choose p: 2a < p < y - 2, and define

and if (2.37) holds for r , we infer the estimate@ ( h 2 r )5 AZ,"@(r)+ (Arr' - + C4A;'rr-2)11f11i9(B) 5 Z,"@(r),

while also (2.37) is reproduced for p = A,r.By iteration we obtain that, for any k E N, ny r R , ,@( Ak,r) s r ) ,

provided only (2.37) holds for r. Now, for any p 5; r there exists a uniquenumber k E N such that p / r ]A:, A:-] Hence, for such p ,


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NAVIER-STOKES EQUATIONS 453such that

for all p s r , provided (2.37) holds for r. Finally, by continuity, if (2.37) holdswith center xo = 0 for some r > 0, this condition will hold true in an openneighborhood U of x o .3. Serrin's Regularity Result

Let Q be an open domain in R", n 2 , Q = Q x 10, T[. Denote byL P * q (Q ;Rk) he space of measurable functions cp: Q -- j W such that

If p = 4 we simply write LP(Q),etc.(1.1)-(1.2) in the sense that div u = 0 as distribution and thatWe consider weak solutions u E L2ym(Q;" ) with l vu l E L2(Q) toJd{ - u ar(P+ VUVcp - ( u ' V ) q ) u } d x d t = J d f cp d x d tfor all cp E C r ( Q )with divcp = 0.The following result is due essentially to Serrin [15]. (The limit cases werestudied by Fabes, Jones, Riviere, respectively Sohr, and von Wahl; see theIntroduction.)

THEOREM 3.1. Suppose u E L2'"(Q;W") with l vu l E L2(Q) s a weak solu-tion to the instationary Nauier-Stokes equations in Q = i2 X 10, T [ for an opendomain $2C W", n 2 , and satisJies either th e conditionor (i) u E LP? (Q ) , where 1 < p , 4 < bo, n / p + 2/4 s 1,

(ii) u E L"?"(Q)and there is an R > 0 such that

uniformly for all t 10, T [ for some absolute constant E > 0. Then u is locallyessentially bounded (and hence regular) in Q.Proof: Following Serrin [15] we let w = curl u (= *du, if n > 3) denote thevorticity of u; w weakly satisfies the equation (if n = 3)

a(34 a x j,d - A@' + -(uiw' - @'a') = curl f ,


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454 M. STRUWE(see Semn [15], (l l ) , p. 190). For n > 3 a similar equation holds (see [15], (23), p.194). First assume that u is uniformly bounded while w E L2.Testing (3.1) with

(and a similar testing function in dimensions greater than 3), wheres 2 1,cp E C $( Q) , cp 2 , we obtain

This estimate holds true for all dimensions (provided the integrals exist). ByHolder's inequality and Sobolev's embedding theorem applied to the function1wI"cp E L2,"(Q)with o(~o~"cp)L 2 ( Q ) ,we have 1 ~ 1 " c pE L"lp(Q) for all T,psuch that

n 2 1T p 2 '+ - = - nand

(3.3)

where


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NAVIER-STOKES EQUATIONS 455Also applying Holder's inequality to the right side of (3.1) yields

where

i.e.,n 2 1-* + --p T n + 1 - (5 i),and the latter is at least i n if n / p + 2/q 5 1. In this case, (3.2)-(3.3) yield theestimate

Hence if l1412Lp.'7(,ppcp) < e is sufficiently small-which by absolute continuity ofthe Lebesgue integral for finite p, q < co can always be achieved by choosingsupp cp sufficiently small-we obtain

By (3.2)-(3.4) this also implies that

for all 8, p such that n/n + 2/p 2 i n ; in particular we can boundlwlscp E L zB (Q ) ,where /3 =n+ 2 > l .

Letting so = 1, s k + l= @sk for k E No and iterating (3.5) on domains Q , =Q, Q k + l= ((x, t ) J rp k ( x , ) 2 l} with suitable functions (Pk+l E CF(Qk),weeventually see that w E L;Leirn(Q)nd that

for any compact subdomain Q' e Q and any r > 0, E > 0.


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456 M. TRUWEIn order to obtain (3.6) in the general case we approximate u by smoothfunctions uk uch that

u k + u in L 2 ( Q ) as k - , 00 ,while

Il'kllLP*q(Q) CIIUIILP-q(Q)for all k.Let wk be the solution to (3.1) for uk with initial and boundary data

w k = w on a Q x [O,T ] = Z , respectivelyQ x {O},in the sense of distributions. (Since w E L 2 ( Q )weakly solves (3.1), w has a trace

w l z ~ - 1 / 2 . - 1 / 4 (Z), wlnx(0) E H - Y Q x {O}),see Lions-Magenes [lo], Theorem IV.12.1, p. 11.60. By varying the initial andlateral boundary of Q slightly, we can even achieve that, analogously to Lemma2.4, wlz E L2(Z), * X ( O ) E L 2 ( Qx {O}).)If

IlukllLP.q(Q) r C ~ ~ u I ~ t P * ' ? ( Q )C Eis small enough-for finite p , 4 < 00 this can always be achieved by choosing asmaller domain Q' @ Q if necessay--wk is unique and a priori bounded inL2(Q) or any k E N Indeed, by duality,

lIWkllL2(Q) = ( @ k , f * ) / l l f * l l L 2 (Q ) *f*GL*(Q)\{o)

For any such f * let w * solve the backward heat equation(3.7) - a , w * - h w * = f * in Qwith initial and boundary data

W * = O on ~ Q X [ O , T ] U Q XT}.Note that by [lo], Theorem IV.6.1, p. 11.33, there exists a unique solution w* of(3.7) with a p * , v w * E L2(Q) nd(3.8) Ilatw*llLz(Q) + llvzw*11L2(Q) 4 c l l f *llL2(Q)'By the trace theorem moreover, respectively by [lo], Theorem 1.3.2, p. 1.21, wecan bound


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NAVIER-STOKES EQUATIONS 457

(In particular, also &*/ an E L 2 ( d Qx [0, I) , w* E L2(Qx (0)). Hence thereader may use our previous remark concerning w to avoid working in fractionalpower spaces, if desired. Estimates (3.8)-(3.9) can also be directly derived from(3.7) by taking the squares of both sides of (3.7) and integrating by parts.)By (3.3) and (3.8)-(3.9) we can also bound

vo* E L"%P(Q)for all numbers n, p such that n/?r + 2 / p 2 n . Thus, ntegrating by parts wehave

= (curlf- + w l - wLul),.*)

4 I l f l lLz(Q)( lVW*(IL1(Q) IlukllLP.q(Q)llwkllLz(Q)llvw*llLw,p(Q)+c ( w )Ilf * 1L2(Q)

where l/n = 1 - l/p - $ , l / p = 1 - 1/11 - $, i.e., n / l r + 2/p 2 n. It fol-lows that

ll'kllLz(Q) I; cellwkllL2(Q) + c l l f l lL2(Q)+ c ( w ) ;whence, for e > 0 ufficiently small, w k is uniformly a pnon bounded in L2(Q).Similarly, we can show that #k is unique, if E > 0 is sufficiently small. Hence@ k + w weakly in L2(Q)as k + w. By the uniform local a pnon bounds (3.6)on ok t then follows that w E Ly2e*"(Q)-and hence u E L$(Q) as in Semn'spaper 1151, Lemma 2, p. 190, and 11, p. 193, as claimed.

Acknowledgment. I wish to thank H. Amann, J. Frehse, S. Klainerman, J.Moser and W. vanWahl for their interest and helpful comments at various stagesof this work. Moreover, I am indepted to the referee for valuable suggestions.


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458 M. TRUWEBibliography

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Received May, 1987.Revised November, 1987.

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Struwe Partial Regularity

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