On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

Transcription

1 On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals Fanghua Lin Changyou Wang Dedicated to Professor Roger Temam on the occasion of his 7th birthday Abstract For any n-dimensional compact Riemannian manifold M, g without boundary and another compact Riemannian manifold N, h, we establish the uniqueness of the heat flow of harmonic maps from M to N in the class C[, T, W 1,n. For the hydrodynamic flow u, d of nematic liquid crystals in dimensions n = or 3, we show the uniqueness holds for the class of weak solutions provided either i for n =, u L t L x L t H 1 x, P L 4 3 t L 4 3 x, and d L t L x L t H x; or ii for n = 3, u L t L x L t H 1 x C[, T, L n, P L n t L n x, and d L t L x C[, T, L n. This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary. 1 Introduction and statement of results For geometric nonlinear evolution equations or systems with critical nonlinearities, it is well-known that the short time smooth solutions may develop finite time singularities. The natural classes of solutions to such systems usually involve weak solutions in various larger function spaces. Although the existence of such weak solutions may be established, the uniqueness and regularity often remain to be very challenging. Courant Institute of Mathematical Sciences, New York University, New York, NY 11 Department of Mathematics, University of Kentucky, Lexington, KY 456 Both authors are partially supported by NSF 1

2 Here we mention two examples. The first one is the celebrated work made by Leray [14 in 1934 on the existence of so-called Leray-Hopf type weak solutions to the Naiver-Stokes equation. Both uniqueness and regularity for the Leray-Hopf type weak solutions to NSE in dimension three still remain largely open. The second example is the heat flow of harmonic maps. It is well-known that in dimensions two or higher, the heat flow of harmonic maps can indeed develop singularities in finite time, see for example the works by Chang-Ding-Ye [3 for dimension two and Chen- Ding [ in dimensions at least three. On the other hand, weak solutions that allow possible singularities to the heat flow of harmonic maps have been established by Struwe [1 and Chang [1 in dimension two and by Chen-Struwe [6 and Chen-Lin [5 in dimensions higher. While Freire [9 proved that Struwe s solution is unique in the class of weak solutions whose energies are monotonically decreasing in dimension two, whether Chen-Struwe s solution is unique in certain classes in higher dimensions is unknown. These two examples motivate us to investigate the uniqueness issue of weak solutions to both the heat flow of harmonic maps and the equation of liquid crystal flows in certain critical L p spaces. The later equation is a simplified version of the Ericksen-Leslie system modeling the hydrodynamics of liquid crystal materials developed by Ericksen [8 and Leslie [13 in 196 s. It is a macroscopic continuum description of the time evolution of the material under the influence of both the flow filed and the macroscopic description of the microscopic orientation configurations of rod-like liquid crystals. Mathematically, it is a strongly coupled system between the Navier-Stokes equation and the transported heat flow of harmonic maps into sphere. Now let s describe the problems and our results. First, we describe the heat flow of harmonic maps. Let M, g be a n-dimensional compact or complete Riemannian manifold without boundary, N, h R k be a compact Riemannian manifold without boundary, isometrically embedded into the Euclidean space R k. Consider the

3 heat flow of harmonic maps u : M R + N: u t u = Au u, u 1.1 u t= = u 1. where A, is the second fundamental form of N, and u : M N is a given map. For 1 p < +, recall the Sobolev space W 1,p M, N is defined by W 1,p M, N = For < T, H 1 M [, T, N is defined by H 1 M [, T, N = { } v W 1,p M, R k : vx N a.e. x M. { } v H 1 M [, T, R k : vx, t N a.e. x, t M [, T. For u W 1, M, N and < T +, recall that a map u H 1 M [, T, N is a weak solution of 1.1 and 1. if u satisfies 1.1 in the sense of distributions and 1. in the sense of trace. Our first result is the following uniqueness theorem. Theorem 1.1 For n, < T, and u W 1,n M, N, suppose that u, v H 1 M [, T, N C[, T, W 1,n M, N are two weak solutions to 1.1 on M, T such that u t= = v t= = u on M. Then u v on M [, T. Remark 1. We would like to point out that when considering the heat flow of harmonic maps on manifolds M with boundaries, Theorem 1.1 remains to be true under the initial condition and the boundary condition: u = u on M, T, provided that u C M, N. The interested readers can check that slight modifications of the proof presented in will achieve this. Next we start to describe the liquid crystal flows in dimensions two and three. For n = or 3, let Ω R n be either a bounded smooth domain or R n. First, let s briefly recall that the equation of hydrodynamic flow of nematic liquid crystals on Ω. The interested readers can refer to [8, [13, [15, and [17 for the detailed background. For < T +, let u : Ω [, T R n be the fluid velocity field, and d : Ω [, T S be the director field of the nematic liquid crystals. Then the 3

4 initial value problem for the equation of hydrodynamic flow of liquid crystals is given by u t + u u u + P = d d 1.3 u = 1.4 d t + u d d = d d. 1.5 u, d t= = u, d 1.6 where P : Ω [, T R is the pressure function, d d = d x i, d x j 1 i,j n is the stress tensor induced by the director field d, denotes the divergence operator, u L Ω, R n, with u =, is the initial velocity field, and d : Ω S, with d L Ω, R 3n, is the initial director field. When Ω is a bounded smooth domain of R n, we will consider the system 1.3, 1.4, 1.5, and 1.6 along with the boundary condition: u, d =, d on Ω, T. 1.7 For n =, we will establish the uniqueness for the class of Leray-Hopf type weak solutions to the equation of hydrodynamic flow of nematic liquid crystals. precisely, we have More Theorem 1.3 For < T +, u L Ω, R with u =, and d : Ω S with d L Ω, R 6, suppose that for i = 1,, u i L t L x L t HxΩ 1 [, T, R, P i L 4 3 t L 4 3 x, and d i L t Ḣx 1 L t Ḣ xω [, T, S are a pair of weak solutions 1 to 1.3, 1.4, 1.5 under either i when Ω = R, the same initial condition: u i, d i t= = u, d, i = 1,, 1.8 or ii when Ω R is a bounded domain, the same initial and boundary conditions: u i, d i = u, d on Ω {}, u i, d i =, d on Ω, T, i = 1,, 1.9 with d C,β Ω, S for some β, 1. Then u 1, d 1 u, d in Ω [, T. 1 The reader can refer to [19 Definition 1.1 for the exact definition of weak solutions. 4

5 We would like to point out that theorem 1.3 answers affirmatively that the global weak solutions obtained by Lin-Lin-Wang [19 Remark 1.5 i is unique in the same class of weak solutions. For simplicity, when n = 3, we only consider the uniqueness of the Cauchy problem of the hydrodynamic flow of nematic liquid crystals in the entire space, i.e. Ω = R n. Theorem 1.4 For n = 3 and < T +, u L n R n, R n with u =, and d : R n S with d L n R n, R 3n, suppose u i L t L x L t L xr n [, T C[, T, L n R n, P i L n t L n x R n [, T, and d i L t Ḣ1 x C[, T, Ẇ 1,n R n, S, i = 1,, are a pair of weak solutions to 1.3, 1.4, 1.5 under the same initial condition: u i, d i t= = u, d, i = 1,. 1.1 Then u 1, d i u, d on R n [, T. Remark 1.5 i When we consider the equation of hydrodynamic flow of liquid crystals 1.3, 1.4, 1.5 and 1.6 on smooth bounded domains Ω R 3, the uniqueness theorem 1.4 remains to be true under the boundary condition 1.7, provided that d C,β Ω, S for some β.1. The interested readers can check that this follows from an ɛ -boundary regularity estimate similar to lemma 3., which can be proved by suitable modifications of the interior ɛ -regular lemma 3.. ii It is also true that both Theorem 1.4 and i remain to hold for n 4. The rest of this paper is organized as follows. In, we first establish a small energy regularity for 1.1 and then prove Theorem 1.1. In 3, we first establish a uniqueness result under the extra assumption on the blow up rate of ut L + dt L, and then verify that this assumption holds for the class of weak solutions dealt in both Theorem 1.3 and 1.4. Here Ḣ1 and Ẇ 1,n denotes the homogeneous Sobolev space on R n. 5

6 Proof of Theorem 1.1 For the simplicity of presentation, we assume that M, g = R n, dx is the n- dimensional euclidean space equipped with the standard metric. For x R n, t >, and R >, let B R x be the ball in R n with center x and radius R and denote B R = B R ; and let P R x, t = B R x [t R, t be the parabolic ball in R n+1 with center x, t and radius R and denote P R = P R,. The proof of theorem 1.1 relies on the following two lemmas. The first is an ɛ -regularity estimate. Lemma.1 There is ɛ > such that if u H 1 P 1, N L [ 1,, W 1,n B 1, N is a weak solution to 1.1 satisfying u L [ 1,,L n B 1 ɛ,.11 then u C P 1, N and u C m P 1 Cm, ɛ u L P 1, m..1 Proof. The reader can refer to Wang [ for the proof in the critical dimension n =. Here we present a proof, which is valid for n 3. For any x, t P 1 and < r < 1, it follows from.11 that u L [t r,t,l n B rx ɛ. Let v : P r x, t R k solve v t v = in P r x, t v = u on p P r x, t.13 where p P r x, t = B r x [t r, t B r x {t r } denotes the parabolic boundary of P r x, t. 6

7 Multiplying both 1.1 and.13 by u v, subtracting the resulting equations, and integrating over P r x, t, we obtain u v P rx,t P rx,t t u u v t r u L B rx u L n B rx u v L B rx u L [t r,t,l n B rx t = n n u L B rx u v L B rx t r Cɛ u L P rx,t u v L P rx,t, where we have used the Sobolev embedding inequality. Hence we have u v Cɛ u..14 P rx,t P rx,t On the other hand, by the standard theory on the heat equation, we have that for any θ, 1, θr n P θr x,t v θ r n P rx,t Combining.14 with.15 yields θr n u C θ + ɛ θ n r n P θr x,t u..15 P rx,t u.16 for any x, t P 1, < r 1, and θ, 1. For any α, 1, first choose θ, 1 such that Cθ θα and then choose ɛ such that Cɛ θn+α, we obtain θ r n u θ α r n P θ rx,t P rx,t u, x, t P 1, < r By iterating.17, we conclude that for any α, 1, it holds 3 r n u Cr P α u, x, t P 1, < r P rx,t 3 We would like to point out that.19 would imply the Hölder continuity of u, provided that u satisfies the following local energy inequality: Z Z r n u t r n P rx,t P r x,t u..18 However,.18 doesn t seem to hold automatically for the class of weak solutions of 1.1, u L [ 1,, W 1,n B 1 for n = 3. 7

8 To conclude.1 from.19 without the local energy inequality.18, we employ the estimate of parabolic Riesz potentials in the parabolic Morrey spaces that was established by Huang-Wang [11 recently. For the convenience of readers, we outline the main steps. First recall the parabolic Morrey spaces on R n+1. For 1 p < +, λ n +, and an open set U R n+1, the Morrey space M p,λ U is defined by { } M p,λ U = f L p loc U : f p M p,λ U sup r λ n+ f p < +. z U,r> P rz U It is clear that.19 implies that for any α, 1, u M, α P 1 and u M, α P 1 C u L P 1.. Now we have Claim. u L q P 1 for any 1 < q < + and u L q P 1 [1 Cq + u L P 1..1 To show this claim, let η C P 1 be such that η 1, η 1 on P 1, and η t + η + η 64. Set vz = ηzuz. Then v t v = F, F [ηau u, u uη t η u η. Then we have vz = Gz wf w dw R n+1 where G is the fundamental solution of the heat equation on R n. By [11 lemma 3., we have vz = Gz wf w R n+1 F w C δz, w n+1 dw = CI 1 F z, R n+1 8

9 where δz, w = max{ x y, t s } is the parabolic distance between z = x, t and w = y, s, and I 1 is the parabolic Riesz potential of order one 4. Since F outside P 1, it is not hard to see from. that F M 1, α R n+1 and F M 1, α R n+1 [1 C + u L P 1. Hence, by the estimate of Riesz potential in Morrey spaces see [11 Theorem 3.1, we have that v L α 1 α, R n+1 5 and Since lim α 1.1 holds. v α C F L 1 α, R n+1 M 1, α R n+1 [1 C + u L P 1. α 1 α = +, we conclude that u Lq P 1 for any 1 < q < + and It is readily seen that claim implies u C P 1 and.1 holds. This completes the proof. By suitable translations and dilation of lemma.1, we can obtain the blow-up rate of ut L R n as t tends to zero. More precisely, we have Lemma. For T > and u W 1,n R n, N, suppose u H 1 R n [, T, N C[, T, W 1,n R n, N is a weak solution to 1.1 and 1., then there exists < t T depending on u, n such that u C R n, t, N and and sup t ut L R n < +,. <t t lim t ut L t + R n =..3 Proof. Since u C[, T, W 1,n R n and u = u W 1,n R n, there exist r = r u > and < t = t u min{r, T } such that sup x R n, t t ut L n B r x ɛ,.4 4 The parabolic Riesz potential of order β < n + is defined by Z I β fz = R n+1 fw dw. δz, w n+ β 5 Here L p, R n+1 denotes the weak L p -space for p 1. 9

10 where ɛ > is given by lemma.1. In particular, we have that for any x R n and < τ t, u L [,τ,l n B τ x ɛ. Define vy, s = ux + τy, τ + τ s for y, s P 1. Then v solves 1.1 on P 1 and v L [ 1,,L n B 1 ɛ. Applying lemma.1, we conclude that v C P 1 and v L P 1 C v L P 1. Back to the original scales, this implies u C P τ x, τ and τ u L P τ x,τ Cτ n u 1 C u L [,τ,l n B τ x..5 P τ x,τ Taking supremum over all x R n and < τ t yields.. To see.3, observe that for any < ɛ ɛ, there exist τ ɛ > and r ɛ > such that and 1 sup ut u n n t τ ɛ R n sup x R n,<r r ɛ Hence there exists t ɛ > such that 1 u n n B rx ɛ, ɛ. Hence.5 yields sup u L [,τ,l n B τ x ɛ. x R n, τ t ɛ sup <τ t ɛ τ uτ L R n Cɛ. This clearly implies.3. The proof is now complete. Proof of Theorem 1.1. First, by interpolation inequalities,. and.3 imply that for any n < p +, n 1 sup t p ut L p R n < +, lim <t t t 1 t n p ut L p R n =..6 1

11 Set w = u v. Then w C[, t, W 1,n R n solves w t w = Au u, u Av v, v in R n, t w t= =. Direct calculations imply Au u, u Av v, v C [ u + v w + v w..7 By the Duhamel s formula, we have t wt = e t s Au u, u Av v, vs ds t e t s [ u + v w + v w s ds,.8 and wt = t t e t s Au u, u Av v, vs ds e t s [ u + v w + v w s ds..9 To proceed with the proof, we need three claims. Claim 1. For any < δ < 1, there exists C = Cδ > such that for < t t, t δ wt L n C δ R sup us n L n R n + vs L n R n..3 s t To see it, applying the standard estimate of the heat kernel 6 to.8 yields wt L n δ R n t t s δ t t s δ = Ct δ us L n R n + vs L n R n ds ds sup us L n R n + vs L n R n s t sup us L n R n + vs L n R n. s t 6 For 1 p q +, e t f t p 1 q 1 n e f L q R n L p R n, t f t 1+ p 1 q 1 n f L q R n L p R n. 11

12 For < t t, set Then we have At = sup <s t B δ t = sup <s t s us L R n + vs L R n, s 1 δ us n + vs L δ R n L n. δ R n Claim. There exists C = Cδ > such that for any < t t, t δ wt L n δ R n C[A t sup <s t s δ ws L n δ R n +B δ t sup ws L n R n..31 s t This is a refinement of claim 1. By.8 and.7, we have wt L n δ R n + t t = I + II. I can be estimated by t I t s 1 s 1 δ since t s 1 us L n δ R n + vs L n δ R n ws L n R n ds vs L R n ws L n δ R n ds ds sup ws L n R n s t sup <s t t δ Bδ t sup ws L n R n, s t t II can be estimated by t [ II s 1+ δ ds t s 1 s 1 δ ds = t δ sup <s t s 1 δ 1 t δ A t sup s δ ws L n. δ R <s t n us n + vs L δ R n L n δ R n 1 s 1 s 1 δ ds = Ct δ. [ s vs L R n sup s δ ws L n δ R <s t n Putting these two estimates together yields.31. Finally, we need Claim 3. There exists C = Cδ > such that for any < t t, wt L n R n C[At sup vs L n R n sup s δ ws L n δ R s t <s t n +B δ t sup s t ws L n R n..3 1

13 To show.3, observe that.9 and the standard estimate on the heat kernel imply + wt L n R n t t t s 1+δ t s 1+δ = III + IV. us n + vs L δ R n L n ws δ R n L n R n ds vs L n R n ws L n δ R n ds III can be estimated by since III t t B δ t IV can be estimated by since IV t s 1+δ s 1 δ ds sup ws L n R n s t t s 1+δ s 1 δ ds = t 1 B δ t sup, s t ws L n R n 1 s 1+δ s 1 δ ds < +. t t s 1+δ δ 1 s ds At sup vs L n R n sup s δ ws L n δ R s t <s t n At sup vs L n R n sup s δ ws L n δ R, s t <s t n t s 1+δ δ 1 s ds = 1 Putting these two estimates together yields.3. Now define the function Φ :, t R + by 1 s 1+δ δ 1 s ds < +. [ Φt = sup ws L n R n + sup s δ ws L n δ R, < t t s t <s t n. Without loss of generality, we may assume that Φt = [ wt L n R n + t δ wt L n δ R. n 13

14 Then.31 and.3 imply Φt C [ 1 + sup s t vs L n R n At + B δ t Φt. It follows from.6 that there exists sufficiently small < t 1 t such that Hence C [ 1 + sup vs L n R n At 1 + B δ t 1 1 s t 1. Φt 1 1 Φt 1. This implies Φt 1 =. Thus u v on R n [, t 1. Repeating the above argument at t = t 1, we can conclude u v on R n [, T. This completes the proof. 3 Proof of Theorem 1.3 and Theorem 1.4 In this section, we will present the proof of the uniqueness theorem for the hydrodynamic flow of liquid crystals. There are two steps to prove Theorem 1.3 and 1.4: i we establish the uniqueness under the extra assumption that [ t ut L Ω + dt L Ω, as t, and ii we verify that this assumption holds for the class of weak solutions we consider in Theorem 1.3 and 1.4. Lemma 3.1 For n = or 3 and < T < +, suppose that for i = 1,, u i, d i : Ω [, T R n S are a pair of weak solutions of 1.3, 1.4, 1.5, 1.6 and 1.7 when Ω R n is a bounded domain with u i L t L x L t HxΩ 1 [, T and d i L Ω [, T. There exists ɛ > such that if for some < t T max i=1, [ sup t ui t L Ω + d i L Ω ɛ, 3.33 <t t then u 1 u and d 1 d on Ω [, t 7. [19. 7 It is known that the weak solutions u i, d i, i = 1,, are smooth in Ω, t, see for example 14

15 Proof. For 1 < p < +, let E p be the closure in L p Ω, R n of all divergence-free vector fields with compact support in Ω. Let P : L Ω, R n E be the Leray projection operator. It is well-known that P can be extended to a bounded operator from L p Ω, R n to E p for all 1 < p < +. Let A = P be the Stokes operator 8 Let w = u 1 u and d = d 1 d. Applying P 9 to both sides of 1.3 for u 1 and u and subtracting the resulting equations, it is not hard to see that w, d satisfies: and w t Aw = P w u 1 + u w + d d 1 + d d 3.34 w = 3.35 d t d = [ d 1 + d dd 1 + d d [w d 1 + u d 3.36 w, d t= =,, 3.37 when Ω R n is a bounded domain. For < t t, set w, d =, on Ω, T 3.38 A i t = t [ u i t L Ω + d i L Ω, i = 1,, Ct = [ u 1 t L n Ω + u t L n Ω + d 1 t L n Ω + d t L n Ω, and for fixed < δ < 1, D δ t = t 1 δ u 1 t L nδ Ω + u t L nδ Ω + d 1 t L nδ Ω + d t L nδ Ω. Then we have, by interpolation inequalities, that By the Duhamel formula, we have wt = t D δ t Ct δ A 1 t + A t 1 δ, < t t e t sa P w u 1 + u w + d d 1 + d d s, Note that if Ω = R n, then A = on E p W,p R n, R n. 9 This is possible, since the assumption 3.33 can imply that for i = 1,, d i W,1 q R n [t 1, t for any 1 < q < + and hence we can choose the pressure P i such that P i L R n [t 1, t for any t 1 >. 15

16 and dt = t e t s [ d 1 + d dd 1 + d d w d 1 u d s Similar to the proof of Theorem 1.1, we can estimate d as follows. estimate To proceed, we first claim t δ dt L n δ Ω δ sup t dt L n δ Ω. <t t We need to sup Ct, < t t. 3.4 <t t In fact, since d d 1 + d =, 3.41 and the standard estimate on the heat kernel imply that for < t t, This yields 3.4. dt L n δ Ω t t s δ [ d i s L n Ω + u i s L Ω n ds i=1 t t s δ ds sup Cs s t t δ sup Cs. s t Next we want to refine the above estimate as follows and the standard estimate on the heat kernel imply that for < t t, + + dt L n δ Ω t t t s 1 [ d i s n L δ Ω + u i s n L δ Ω ds L n Ω + ws L n Ω i=1 t s 1 d s L Ω d s L n n Ω ds L δ Ω t t s 1 δ 1 s ds sup D δ s sup ds L n Ω + ws L n Ω t t s 1 δ 1 s ds [ t δ sup <s t [ sup <s t <s t sup A s <s t D δ s + sup <s t A s s t sup s t sup Cs <s t Cs sup <s t s δ ds L n δ Ω ds L n Ω + ws L n Ω + s δ ds L n δ Ω,

17 where we have used the inequality t t s 1 δ 1 s ds t δ. Applying of both sides of 3.41 and employing the standard L p -estimate of e t, we have that for < t t, dt L n Ω t t t t s 1+δ d1 s n L δ Ω + d s n L δ Ω ds L Ω ds t s 1+δ d1 s n L δ Ω ws L n Ω + u s n L δ Ω ds L n Ω ds t s 1+δ d s L Ω d s L n n Ω ds L δ Ω ds t t s 1+δ δ 1 s ds sup D δ s sup ws L n Ω + d L n Ω t C sup t s 1+δ δ 1 s ds <s t D δ s sup s t <s t sup A s <s t s t sup Cs <s t ws L n Ω + d L n Ω sup s δ ds L n δ Ω <s t + C sup A s sup Cs sup s δ ds L n δ Ω, 3.44 <s t <s t <s t where we have used t t s 1+δ δ 1 s ds = 1 1 s 1+δ δ 1 s ds < +. Now we want to estimate wt L n Ω. Before doing it, we need to recall the following L p L q estimate of e ta P : e ta P f 1+ n p n q L q Ω t f L p Ω 1 < p q < The reader can find the proof of 3.45 by Kato[1 when Ω = R n, and by Giga [1 when Ω R n is a bounded domain. 17

18 Applying 3.45 with p = n 1+δ and q = n to 3.4, we have that for < t t, wt L n Ω + t t t s 1+δ u1 s n L δ Ω + u s n L δ Ω ws L n Ω ds t s 1+δ d1 s n L δ Ω + d s n L δ Ω ds L n Ω ds t t s 1+δ δ 1 s ds sup D δ s <s t sup ws L n Ω + d L n Ω s t [ C sup <s t D δ s sup s t ws L n Ω + d L n Ω Finally, set the function Φ :, t R + by Φt = sup ds L n Ω + ws L n Ω + s δ ds L nδ Ω. <s t Combining the inequalities together, we obtain that for < t t, [ Φt C sup D δ s + sup A 1 s + A s sup Cs Φt <s t <s t <s t 1 Φt 3.47 provided that ɛ > is sufficiently small such that [ C sup D δ s + sup A 1 s + A s sup <s t <s t Cs <s t Cδ[ɛ 1 δ + ɛ 1. This implies Φt for < t t. Hence u 1, d 1 u, d on Ω [, t. Proof of Theorem 1.3: First it follows [19 Theorem 1. that i u i, d i C Ω, T for i = 1,, and ii for Ω R a bounded domain, since d C,β Ω, S for some β, 1, u i, d i C,1 β Ω, T 1. Moreover, by a simple scaling argument, we have that for any < t T, A i t sup <s t s ui s L Ω + d i s L Ω < +, i = 1, Here C,1 β Ω, T denotes the spaces of C1 functions f such that xf, f t C β Ω, T. 18

19 It remains to show that lim A i t =, i = 1, t To see 3.49, recall that the weak solution u i, d i, i = 1,, in Theorem 1.3 satisfies the following energy inequality see [19: t u i t + d i t + u i + d i + d i d i Ω Ω u + d. 3.5 Ω In particular, for i = 1,, E i t = Ω u it + d i t is monotonically nonincreasing with respect to t. Hence lim E i t E u + d. t Ω On the other hand, for i = 1,, since u i t, d i t converges weakly to u, d in L Ω as t, the lower semicontinuity implies Thus lim E i t E. t lim E i t = E, i = 1, t and hence E i t C[, T for i = 1,. Now we can use the argument similar to that of Theorem 1.1 to show that lim sup t x Ω B tx [,t ui + d i =. i=1 Applying [19 Theorem 1. again, this implies It is clear that 3.49 and Lemma 3.1 imply that there exists < t < T such that u 1, d 1 = u, d on Ω [, t. For i = 1,, since u i, d i C Ω [t, T C,1 β Ω [t, T solves 1.3, 1.4, 1.5, under either the same initial condition for Ω = R or the same initial and boundary conditions for Ω R being a bounded domain, the uniqueness for classical solutions implies u 1, d 1 = u, d on Ω [t, T. This completes the proof. 19

20 In order to prove Theorem 1.4 for n = 3, we need to establish an ɛ -regularity estimate similar to lemma.1 for the heat flow of harmonic maps. More precisely, we have Lemma 3. For n = 3, there exists ɛ > such that if u L [ 1,, L n B 1, R n, P L n P 1, and d L [ 1,, W 1,n B 1, S is a weak solution of 1.3, 1.4, and 1.5 that satisfies [ u L [ 1,,L n B 1 + d L [ 1,,L n B 1 ɛ Then u, d C P 1, R n S and 4 [ u L P 14 + d L P 14 Cɛ. 3.5 Proof. It is divided into several steps. First, we have Claim 1. d L q P 1 for any 1 < q < + and d L q P 1 [1 Cq + d L P The proof of claim 1 is similar to that of lemma.1, which is sketched here. For any z = x, t P 1 and < r < 1, 3.51 implies [ u L [t r,t,l n B rx + d L [t r,t,l n B rx ɛ Let v : P r x, t R k solve v t v = in P r x, t v = d on p P r x, t 3.55 Multiplying 1.5 and 3.55 by d v and integrating the resulting equations and then subtracting each other, we obtain d v P rx,t P rx,t t [ u d + d d v [ u L n B rx + d L n B rx d L B rx d v n [ t r L u L [t r,t,l n B rx + d L [t r,t,l n B rx t d L B rx d v L B rx t r Cɛ d L P rx,t d v L P rx,t. n B rx

21 Hence we have P rx,t d v Cɛ P rx,t For v, we have that for any θ, 1, θr n v Cθ r n P θr x,t P rx,t Combining 3.56 with 3.57 yields θr n d C θ + ɛ θ n r n P θr x,t d d P rx,t d 3.58 for any x, t P 1, < r 1, and θ, 1. Similar to lemma.1, choosing sufficiently small θ = θ first and sufficiently small ɛ second and finally iterating the resulting inequality, 3.58 yields that for any α, 1, there exists C = Cɛ, α > such that r n d Cr P α d, x, t P 1, < r 1 1, 3.59 or equivalently, P rx,t d M, α P 1 C d L P Now we perform the Riesz potential estimate in Morrey spaces by the same way as in lemma.1. More precisely, let η C P 1 be a cut-off function of P 1 w = ηd. Then w satisfies and set w t w = H, H η d d u d + η t ηd η d Since H outside P 1, and u d M 1, α P 1 satisfies u d M 1, α P 1 C u L [ 1,,L n B 1 d L P 1, it is easy to see from 3.6 that H M 1, α R n+1 and Similar to lemma.1, we have wz C H M 1, α R n+1 C [ 1 + d L P 1. R n+1 Hw δz, w n+1 dw = CI 1 H z. 1

22 Hence w L α 1 α, R n+1 and w α C H L 1 α, R n+1 M 1, α R n+1 C [ 1 + d L P 1. Since lim α α 1 1 α = +, we can see that d Lq P 1 for any 1 < q < +, and 3.53 holds. Next we want to modify the standard argument on the small energy regularity on nonhomogeneous Naiver-Stokes equations, see for example Caffarelli-Kohn- Nirenberg [4, Lin [16, Seregin [, Escauriaza-Seregin-Sverak [7, and Lin-Liu [18, prove that Claim. u L P First observe that since 3.51 and 3.53 imply that for any 1 < q < 3, d t d = d d u d L q P Hence by the L q -estimate on the heat equation we have that d Wq,1 P 3 for any 1 < q < 3, and Since d L q P 3 d d u d [ L q P 1 Cqɛ 1 + d L P u t + u u u + P = f, in P 3, where f d d L q P 3 for any 1 < q < 3 and 8 [ f L q P 38 Cqɛ 1 + d L P Since u L t L x L t HxP 1 3 and P L n P 3, it is not hard to verify that u is a 8 8 suitable weak solution to 3.63, i.e. u satisfies ut φt + u φ B 38 B 38 [,t [ u φ t + φ + u + P u φ + f uφ 3.65 B 38 [,t for a.e. t [ 3 8, and for any nonnegative function φ C P 3 8.

23 Since P satisfies that for a.e t [ 3 8, P = f u u in B 3, 8 the same argument as [ page 1, Lemma 3.1, with the help of 3.51 and 3.64, implies that there exists θ, 1 such that 1 θ P θ z P n n Cɛ, z P Since u is a suitable weak solution of 3.63 that satisfies the smallness conditions 3.51, 3.64 for all 1 < q < 3, and 3.66, it is well-known see for example [4 [16 [ that u C α P 5 for some α, 1, and 16 u L P 5 16 Cɛ Now substituting 3.67 into 3.6, we conclude that d t d L q P 5 for all 16 1 < q < +. Hence d Wq,1 P 1 for any 1 < q < +. This and the Sobolev embedding theorem imply that d L P 1 and 4 4 d L P 14 Cɛ It is clear that 3.5 follows from 3.67 and The proof is now complete. Proof of Theorem 1.4: With the help of lemma 3., it can be done similar to that of Theorem 1.3. First, since u i, d i C[, T, L n R n for i = 1,, it follows that lim sup t + x R n P tx,t i=1 u i n + d i n = By translation and scaling, 3.69 and lemma 3. then imply lim ui L t + R n + d i L R n =, i = 1,. Hence lemma 3.1 implies that there exists < t < T such that u 1 u and d 1 d on R n [, t. Repeating the same argument at t = t can eventually lead to u 1, d 1 u, d on R n [, T. This completes the proof. 3

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