AN INTRODUCTION TO CONVEX INTEGRATION. 1. Divergence-free vector fields

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1 AN INTRODUCTION TO CONVEX INTEGRATION WOJCIECH O A SKI In ths note we brey dscuss the method of convex ntegraton. We rst prove a theorem regardng dvergence-free vector elds, whch demonstrates the man dea of the method whle beng elementary. We then move to the context of the three-dmensonal ncompressble Euler equatons, where we follow De Lells & Székelyhd Jr. (2009) to construct nntely many compactly supported weak solutons to the Euler equatons. 1. Dvergence-free vector felds Let Ω R 3 be open and bounded. In ths note we wll prove the followng. Theorem 1. There exst nntely many u L (R 3 ; R 3 ) such that dv u = 0 n D (R 3 ), u(x) = 1 Ω (x) for almost every x R 3. (1) Here 1 Ω denotes the ndcator functon of Ω and D (R 3 ) denotes the space of dstrbutons on R 3, so that the rst lne of (1) s equvalent to beng satsed for all φ C 0 (R 3 ). We wll need the followng two lemmas. R 3 u φ = 0 Lemma 1. If x n x n a real Hlbert space, then x n x f and only f x n x. Proof. Wrtng x n x 2 = x n 2 2(x n, x) + x 2 we see that (x n, x) x 2 as n, and so x n x 2 0 as n f and only f the norms x n converge to x. Lemma 2. If (X, d) s a complete metrc space and J : X R s a pontwse lmt of contnuous functons on X (that s there exst J k C(X) such that J k (u) J(u) for each u X) then the set S of ponts of contnuty of J s dense n X. The lemma s a standard result n functonal analyss, and the man element of the proof (whch can be found below Theorem 4.6 n Székelyhd Jr. (2013)) uses Bare's theorem (that s: A countable unon of nowhere dense sets n a complete metrc space s nowhere dense as well.) We can now prove the man theorem. Proof of Theorem 1. Date: 7th Nov

2 AN INTRODUCTION TO CONVEX INTEGRATION 2 Step 1. Specfy the functonal setup. Let denote the set of subsolutons. Let X 0 := {u C 0 (Ω): dv u = 0 and u < 1} X := weak closure of X 0 n L 2 (Ω). Note that X s bounded n L 2 (Ω) (by Ω ) and so X, together wth the weak L 2 topology, s metrzable (snce B(0, Ω ) L 2 (Ω) equpped wth ths topology s metrzable (see for example Theorem 5.1 n Secton V.5 nconway (1990)) and X s ts closed subset). Let d denote the correspondng metrc. Then (X, d) s n fact a compact metrc space (by the Banach-Alaoglu theorem, see for example Theorem 3.1 n Secton V.3 n Conway (1990)). Let I : X R, I(u) := Ω ( 1 u 2 ). Clearly, I s contnuous on X wth respect to the strong L 2 topology. Note that any u X satsfyng I(u) = 0 s a soluton to (1). In the followng steps we wll show that {I(u) = 0} s dense n X. (2) Ths shows that the set of solutons to (1) s dense n X, and so n partcular proves the theorem. Step 2. Gven u X 0 and Ω Ω, add oscllatons to u on Ω. Namely, we wll construct a sequence {u k } such that (a) u k X 0 for sucently large k, (b) u k u n L (Ω), (c) lm nf k u k 2 u 2 + I Ω(u) 2 /8 Ω, where denotes the L 2 (Ω) norm and I Ω(u) ( := 1 u 2 ). Ω Note that (b) gves n partcular u k u n L 2 (Ω), and that (c) gves n partcular that u k does not converge strongly to u (see Lemma 1). Thus let ξ, η R n be such that ξ η, ξ = η = 1, φ C 0 (Ω; [0, 1]) such that φ = 1 on Ω and v k := η 2k (1 u(x) 2 )φ(x) sn(kx ξ). We wll show that u k := u + curl v k satses (a), (b), (c). As for (a) note that u k C 0 (Ω), dv u k = 0 (recall the dentty dv curl = 0) and that u k = u < 1 outsde supp φ. Moreover, let δ > 0 be such that u 1 δ on supp φ, and note that curl v k = (ξ η) (1 u(x) 2 )φ(x) cos(kx ξ) + O(1/k). (3) 2

3 AN INTRODUCTION TO CONVEX INTEGRATION 3 Thus on supp φ u k u (1 u )(1 + u ) + C k u + (1 u )(1 δ/2) + C k = 1 + δ 2 ( u 1) + C k 1 δ2 2 + C k, and so u k < 1 for sucently large k, whch gves (a). Clam (b) follows mmedately, snce the rst part of the rght hand sde of (3) s an oscllatory term wth the frequency ncreasng wth k (and so tends to zero weakly- n L by the Remann-Lebesgue lemma, see for example Lemma 1.4 n Duoandkoetxea (2001)) and the second part tends strongly to 0 n L as k. Clam (c) follows by a drect calculaton. Indeed, smlarly as n (b) we obtan (curl v k, u) 0 as k and u k 2 = u 2 + 2(curl v k, u) + curl v k 2 = u (1 u(x) 2 ) 2 φ(x) 2 cos(kx ξ) 2 dx + o(1) 4 Ω u (1 u(x) 2 ) 2 dx + o(1), 8 Ω u ( 2 (1 u(x) 2 ) dx) + o(1), 8 Ω Ω where we used the dentty cos α 2 = (1 + cos 2α)/2, the Cauchy-Schwarz nequalty and we denoted any term tendng to 0 as k by o(1). Takng lm nf k gves (c). Step 3. Observe that the set S := {u X : any sequence weakly convergent to u n L 2 (Ω) converges strongly} s dense n (X, d). Indeed, lettng J(u) := u we see (usng Fubn's theorem) that J s a pontwse lmt of J k (u) := ρ 1/k u, where we extended u by zero outsde Ω, the symbol denotes the convoluton, ρ ε (x) := ε 3 ρ(x/ε) and ρ s a standard mollfyng kernel. Moreover, each J k s a contnuous functon on (X, d) (that s f w n w then J k (w n ) J k (w) as n for each k). Thus the clam follows from Lemma 2. Step 4. Show that S {I(u) = 0}. (Note that ths ncluson together wth Step 3 prove (2), as requred.) Suppose otherwse that there exsts u S such that I(u) > 0. Then there exsts Ω Ω such that I Ω(u) > 0. Snce u s a weak lmt of a sequence {u k } X 0 (by denton of X), we can apply Step 2 for each u k to obtan u k X 0 such that d(u k, u k ) 1/k and u k 2 u k 2 + I Ω(u k ) 2 /8 Ω 1/k.

4 AN INTRODUCTION TO CONVEX INTEGRATION 4 Notng that n fact u k converges to u strongly n L 2 (by denton of S) we see that u k converges strongly to u as well and so takng the lmt k n the last nequalty gves u 2 u 2 + I Ω(u) 2 /8 Ω > u 2, a contradcton. 2. Euler equatons We wll wrte prevously or as before to relate back to the problem (1). In ths secton we paraphrase the man result from De Lells & Székelyhd Jr. (2009), usng also some tools from De Lells & Székelyhd Jr. (2010). Theorem 2. Let Ω R n R be open and bounded. Then there exsts nntely many pars v, p L (R n R) such that t v + (v )v + p = 0 n D (R 3 R) dv v = 0 n D (R 3 R), (4) v = 1 Ω a.e. n R n R, p = 0 outsde Ω. Proof. Step 1. Reformulate the problem. Namely, problem (4) s equvalent to ndng v, u, q L supported n Ω such that t v + dv u + q = 0, dv v = 0, (5) (v, u) K := {(v, u) S n 1 S0 n : u = v v v 2 I n /n} a.e. n Ω, (6) where v v = v v T denotes the matrx, whose, j-th entry s v v j, S n 1 = B(0, 1) denotes the n 1 dmensonal sphere, S0 n denotes the space of n n symmetrc matrces wth zero trace and I n denotes the n-dmensonal dentty matrx. The equvalence (4) (5,6) s clear by substtutng q = p + v 2 /n, snce then dv u + q = dv (v v) dv ( v 2 I n /n) + q = (v )v ( v 2 /n) + q = (v )v + p. Step 2. Observe that K co = {(v, u) B(0, 1) S0 n : e(v, u) 1/2}, (7) where K co denotes the convex hull of K and e: R n S0 n R s dened by e(v, u) := n 2 λ max (v v u) and λ max (A) denotes the largest egenvalue of a symmetrc matrx A (recall the fact that symmetrc matrces have the full set of egenvalues, so that e(v, u) s well-dened).

5 AN INTRODUCTION TO CONVEX INTEGRATION 5 An easy calculaton shows that e s convex and for v R n, u S n 0 where. denotes the operator norm of a matrx, and u 2n 1 e(v, u), (8) n v 2 v 2 e(v, u) wth = f and only f u = v v 2 n I n. (9) These facts are, essentally, a consequence of the fact that u s trace-free, see Lemma 3 n De Lells & Székelyhd Jr. (2010) for an enlghtenng proof. From (9) we see that K = {(v, u) S n 1 S n 0 : e(v, u) = v 2 /2} S 1, where S 1 denotes the rght-hand sde of (7). Snce S 1 s convex (as e(v, u) s a convex functon) we obtan K co S 1. It remans to show the opposte ncluson. To ths end one can use the fact that u has zero trace to show that the set E of extremal ponts of S 1 s contaned n K, see Lemma 3 (v) n De Lells & Székelyhd Jr. (2010) for a proof. Thus, snce (8), (9) gve n partcular that S 1 s compact, we see that S 1 s the closure of the convex hull of E (by the Kren-Mlman theorem (see Secton XII.1 n Yosda (1965) for example)) and so S 1 E co K co = K co, where the last equalty follows from the fact that K s compact (recall that the convex hull of a compact set n R n s compact). Step 3. Show that 0 U := Int K co. Observe that Step 2 mmedately gves 0 K co. In order to see that 0 belongs to the nteror of ths set one needs to work a bt harder. Namely, one can prove t usng Jensen's nequalty, the facts that S n 1 z dz = 0, S n 1 (z z I n /n)dz = 0, and the Open Mappng Theorem, see Lemma 4.2 n De Lells & Székelyhd Jr. (2009) for the detals. Here we omt the proof. Step 4. Specfy the functonal setup. Let X 0 := {(v, u, q) C 0 (R n R): supp (v, u, q) Ω, (v, u, q) solves (5) and (v, u) U, q < 1 a. e.} be the set of subsolutons and let X := weak closure of X 0 n L 2. Smlarly as before, X, equpped wth the L 2 weak topology, s metrzable (wth some metrc d) and the resultng space (X, d) s a compact metrc space. Observe that any trple (v, u, q) X s supported wthn Ω, solves (5) and (v, u) K co, q 1 almost everywhere.

6 AN INTRODUCTION TO CONVEX INTEGRATION 6 Let I : (X, d) R + be dened by I(v, u, q) := (1 v 2 ). Note that I s contnuous wth respect to the strong L 2 topology. Moreover I(v, u, q) = 0 f and only f (v, u, q) s the requred soluton, that s solve (5), (6). Indeed, the mplcaton s trval. As for the note that I(v, u, q) = 0 gves n partcular that v = 1 a. e. n Ω, and so Ω 1 2 = v 2 2 e(v, u) 1 2, where the two nequaltes follow from (9) and Step 2, respectvely. Thus (9) gves that u = v v v 2 I n /n. Thus, except for the equaton (5) the trple (v, u, q) also satses the constrant (6), as requred. In the followng steps we wll verfy that whch (smlarly as before) proves the theorem. Step 5. Dene the wave cone Λ. Let {(v, u, q): I(v, u, q) = 0} s dense n (X, d), (10) { ( )} Λ := (v, u, q) R n S0 n u + qi n v R: det. v T 0 What s the wave cone? That Λ s a cone (that s αa Λ for all α > 0, a Λ) s clear. As for the wave part note that (v, u, q) Λ f and only f there exsts ξ R n+1 such that ( ) u + qi n v ξ = 0. v T 0 An elementary calculaton shows that w(x, t) := (v, u, q) h(ξ (x, t)) (11) satses (5) for any choce of h: R R. Thus t s helpful to thnk of the wave cone Λ as of the set of all drectons (v, u, q) n whch there exst a planewave. The wave cone s an object of crucal mportance, snce t s by addng nntely many oscllatons, each n a drecton from the wave cone, that solutons to (5), (6) are obtaned (that s the densty of {I = 0} n (X, d) wll be shown). In fact, the noton of the wave cone s used n a wde famly of equatons (the famly of derental nclusons, of whch the Euler equatons and (1) are partcular cases), see Secton 5.3 n Székelyhd Jr. (2013). Note however that the plane waves of the form (11) are not compactly supported. Rather, the support of w s a slce n R 3 R that s orthogonal to ξ and whose wdth equals the length of supp h. Ths s an unfortunate property (snce we are nterested only n the oscllatons supported wthn Ω), and ths problem s resolved n the next step. Step 6. Construct localsed oscllatons n a drecton from the wave cone.

7 AN INTRODUCTION TO CONVEX INTEGRATION 7 Namely, gven (v, u, q) Λ, k > 0 and Ω 1 Ω 2 R n R, there exsts (v k, u k, q k ) C 0 (Ω 2 ) that solves (5) and dst ((v k, u k, q k )(x, t), J (v,u,q) ) < 1/k for(x, t) Ω 2, Ω 2 v k α v Ω 1, (v k, u k, q k ) 0 n L, where α > 0 s a dmensonal constant and J z = [ z, z] denotes the closed lne segment jonng z and z. The proof of ths step conssts of a number of elementary calculatons (and, as the reader mght expect, a choce of a cuto functon φ C 0 (Ω 2 ; [0, 1]) such that φ = 1 on Ω 1 ), and thus s omtted n ths note. We refer the reader to Proposton 2.2 n De Lells & Székelyhd Jr. (2009) for a proof n the case Ω 1 = B(0, 1/2), Ω 2 = B(0, 1). Step 7. Show that there s sucently many drectons n Λ. Namely show that for every (v 0, u 0 ) U there exsts (v, u) such that (v, u, 0) Λ, v C(1 v 0 2 ) and (12) (v 0, u 0 ) + J (v,u) U wth dst ((v 0, u 0 ) + J (v,u), U) 1 2 dst ((v 0, u 0 ), U). (13) In ths note we only sketch the man deas of the proof of ths step (see Lemma 4.3 n De Lells & Székelyhd Jr. (2009) for the detals). Frst use Carathéodory's Theorem to choose (v, u) such that (12) holds and moreover Then observe that for every d > 0 (v 0, u 0 ) + J 2(v,u) K co. (14) (v 0, u 0 ) + J (v,u) + B(0, d/2) ( B((v 0, u 0 ), d) J 2(v,u) ) co, (15) whch s a smple geometrc fact descrbed n Fgure 1. Takng d := dst ((v 0, u 0 ), U) we see from (14) that whch s equvalent to (13). (v 0, u 0 ) + J (v,u) + B(0, d/2) K co, Fnally, one can perform a smple matrx calculaton to see that (v, u, 0) Λ. Step 8. Gven (v, u, q) X 0 and Ω Ω add oscllatons to (v, u, q) on Ω. Namely, we wll construct a sequence (v k, u k, q k ) such that (a) (v k, u k, q k ) X 0 for sucently large k, (b) (v k, u k, q k ) (v, u, q) n L, (c) lm nf k v k 2 v 2 + βi Ω(v)/ Ω, where denotes the L 2 ( Ω) norm, β := 4 n 1 α 2 C 2 and I Ω(v) := (1 v 2 ). Ω

8 AN INTRODUCTION TO CONVEX INTEGRATION 8 d J + B(0, d/2) 2J O J d/2 Fgure 1. Consder a ball B(O, d) and a lne segment whose mddle pont concdes wth O. Then J + B(0, d/2) (B(O, d) 2J) co, whch follows from the fact that the ball centred at the rght endpont of J and radus d/2 s tangent to the lne tangent to B(O, d) and passng through 2J (snce the former s a homothetc transform of B(O, d) wth respect to he endpont of 2J and rato 1/2). Thus (15) follows by takng O := (v 0, u 0 ), J := (v 0, u 0 ) + J (v,u). To ths end observe that snce (v, u, q) X 0 we have (v, u)(x 0, t 0 ) U, and, by contnuty, there exsts δ > 0 such that dst ((v, u)(x 0, t 0 ), U) δ for (x 0, t 0 ) Ω. Moreover, step 7 gves that for every (x 0, t 0 ) Ω there exsts of (v, u)(x 0, t 0 ) such that (v, u, 0)(x 0, t 0 ) Λ, and v(x 0, t 0 ) C(1 v(x 0, t 0 ) 2 ) (16) dst ((v, u)(x 0, t 0 ) + J (v,u)(x0,t 0), U) δ/2. Usng the last nequalty and the unform contnuty of (v, u) we see that there exsts ε > 0 such that dst ((v, u)(x, t) + J (v,u)(x0,t 0), U) δ/4 (17) whenever (x, t), (x 0, t 0 ) Ω are such that (x, t) (x 0, t 0 ) ε. Now let {B }, B = B (x, t ), be a (nte) famly of parwse dsjont ε-balls that are contaned n Ω and (1 v 2 ) 2 (1 v(x, t ) 2 ) B ; (18) Ω for ths take ε smaller, f requred. For each k 1 and apply step 6 (wth Ω 1 := B /2, Ω 2 := B ) to obtan (v,k, u,k, q,k ) C 0 (B ) such that () dst ((v,k, u,k, q,k )(x, t), J (v,u,0)(x,t ) 1/k for (x, t) B, () B v,k α v(x, t ) B /2, and () (v,k, u,k, q,k ) 0 n L (B ) as k, for each.

9 AN INTRODUCTION TO CONVEX INTEGRATION 9 We wll show that (v k, u k, q k ) := (v, u, q) + (v,k, u,k, q,k ) satses the clams (a), (b), (c). Indeed, () gves q k < 1 and (17) and () gve (v k, u k ) U. Thus (a) follows. Clam (b) follows from () (recall that there are only ntely many 's). As for (c) we wrte v k 2 = v ( ) 2 (v,k, v) + v,k ( 2 = v 2 + v,k ) + o(1) ( 2 v v,k L1 ( Ω)) + o(1) Ω ( 2 v 2 + α2 B /2 v(x, t ) ) + o(1) Ω ( 2 v 2 + α2 C 2 B (1 v(x, t ) )) 2 + o(1) 4 n Ω v 2 + α2 C 2 ( 2 (1 v )) 2 + o(1), 4 n+1 Ω Ω where we denoted by o(1) any term tendng to 0 as k and we used (), the Cauchy-Schwarz nequalty, (), (16) and (18). Takng lm nf k gves (c). Step 9. Observe that S := {(v, u, q) X : any sequence weakly convergent to (v, u, q) n L 2 (Ω) converges strongly} s dense n (X, d). Ths step follows n the same way as step 3 n the prevous problem. Step 10. Show that S {(v, u, q): I(v, u, q) = 0}. (Note that ths and step 9 gve (10), as requred.) Suppose otherwse that there exsts (v, u, q) S such that I(v, u, q) > 0. Then there exsts a subdoman Ω Ω such that I Ω(v) > 0. By denton of X, (v, u, q) s a L 2 weak lmt of a sequence {(v k, u k, q k )} X 0. For each k we apply step 8 to obtan (v k, u k, q k ) X 0 such that d((v k, u k, q k ), (v k, u k, q k)) 1/k (19) and v k 2 v k 2 + β I Ω(v k ) 2 / Ω 1/k (20)

10 AN INTRODUCTION TO CONVEX INTEGRATION 10 Snce v k v n L 2 (Ω) the same s true for the sequence v k (by (19)), and so v k v n L2 (Ω) (by the denton of S). Thus, takng the lmt k n (20) gves v 2 v 2 + β I Ω(v) 2 / Ω, a contradcton. References Conway, J. B. (1990), A course n functonal analyss, Vol. 96 of Graduate Texts n Mathematcs, second edn, Sprnger-Verlag, New York. De Lells, C. & Székelyhd Jr., L. (2009), `The Euler equatons as a derental ncluson', Ann. of Math. (2) 170(3), De Lells, C. & Székelyhd Jr., L. (2010), `On admssblty crtera for weak solutons of the Euler equatons', Arch. Raton. Mech. Anal. 195(1), Duoandkoetxea, J. (2001), Fourer analyss, Vol. 29 of Graduate Studes n Mathematcs, Amercan Mathematcal Socety, Provdence, RI. Translated and revsed from the 1995 Spansh orgnal by Davd Cruz-Urbe. Székelyhd Jr., L. (2013), From sometrc embeddngs to turbulence, n `HCDTE lecture notes. Part II. Nonlnear hyperbolc PDEs, dspersve and transport equatons', Vol. 7 of AIMS Ser. Appl. Math., Am. Inst. Math. Sc. (AIMS), Sprngeld, MO, p. 63. Yosda, K. (1965), Functonal analyss, De Grundlehren der Mathematschen Wssenschaften, Band 123, Academc Press, Inc., New York; Sprnger-Verlag, Berln. Wojcech O»a«sk, Mathematcs Insttute, Unversty of Warwck, Coventry CV4 7AL, UK. E-mal address: W.S.Ozansk@warwck.ac.uk