ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS KAZUO YAMAZAKI 1 2 Abstract. We generalize Leray-alpha type models studied in [3] and [8] via fractional Laplacians and employ Besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation. Keywords: Leray-alpha model, Besov space, fractional Laplacian (1) 1. Introduction We study the generalized Leray-α model defined on T 3 R +, T = [0, L] 3 : { t v + u v + p + νl 2 1 v = 0 v = (1 + α 2 L 2 2 )u; u = v = 0, v t=0 = v 0 where u is velocity of fluid, p scalar pressure and ν > 0 a dissipative coefficient. The operators L i, i = 1, 2, 3 have Fourier symbols m i (ξ) = ξ γ i g i (ξ) where g i, i = 1, 2, 3 are radially symmetric nondecreasing functions bounded from below by 1. We also study the generalized Leray-α MHD model: (2) t v + u v + 3 =1 v u + p + 1 2 B 2 B B + νl 2 1 v = 0 t B + u B B u + ηl 2 2 B = 0 v = (1 + α 2 L 2 3 )u, u = v = B = 0, (v, B) t=0 = (v 0, B 0 ), where η > 0 is a magnetic diffusivity coefficient. Hereafter we assume ν = η = α = 1. The system (1) in the case g i (ξ) 1, γ i = 1, i = 1, 2 was studied in [3] as a new model for three-dimensional turbulence. In the limit as α approaches zero, (1) becomes the well-known Navier-Stokes equation (NSE) which enoys a rescaling property of 1 2000MSC : 35B65, 35Q35, 35Q86 2 Department of Mathematics, Oklahoma State University, 01 Mathematical Sciences, Stillwater, OK 7078, USA 1

2 KAZUO YAMAZAKI u λ (x, t) = λu(λx, λ 2 t), p λ (x, t) = λ 2 p(λx, λ 2 t) and in terms of fractional Laplacian, γ 1 5 being considered as critical and subcritical regimes. The system (1) was inspired by the LANS-α model of turbulence and compares successfully with empirical data from turbulence channel and pipe flows for a wide range of Reynolds numbers. The authors stated a theorem concerning the global regularity of the unique solution to (1) in R 3 citing the work in [7]. In [] the authors studied (1), LANS-α and Bardina model in comparison with the NSE in their Reynolds numbers. The system (2) in the case g i 1, γ i = 1, i = 1, 2, 3 was introduced in [8] along with others as a sub-grid scale turbulence model of MHD equations without enhancing the dissipation for the magnetic field and was shown to be globally-wellposed. The study of another model, namely (2) without 3 =1 v u in two dimension was studied in [19] and they showed global regularity of the solution when one of the two equations is inviscid. For discussion concerning similar systems in the case g i 1, i = 1, 2, 3, we refer readers to e.g. [15] and [10]. Concerning other cases, we have recently seen intensive study: [2] on Euler equation, [5] Euler-Boussinesq system, [11] Schrödinger equation, [12] NSE, [13] wave equation, [15] MHD equation and [18] anisotropic NSE. We now present our results: Theorem 1.1. Suppose (3) 2γ 1 + γ 2 5 2, γ 1 1 2, and g 1, g 2 satisfy 1 dξ ξg 1 (ξ)g2 2 (ξ) = Then, any initial data v 0 H m, m > max{ 5 2, 1+2γ 1}, generates a unique global smooth solution to (1). Theorem 1.2. Suppose () γ 1 + γ 2 + γ 3 5 2, min{γ 1, γ 3 } > γ 2 > 0 and g i, i = 1, 2, 3 satisfy 1 dξ ξg 1 (ξ)g2 2 (ξ)g 3 (ξ) = Then, any initial data (v, B) H m H m, γ 3 + 2γ 1 + 1 > m > max{ 5 2, 1 + 2γ i }, i = 1, 2, generates a unique global smooth solution pair to (2).

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS3 Remark 1.1. (1) In comparison to the results in [3] and [8], Theorem 1.1 only requires in this case 2γ 1 + γ 2 5 2, also allowing g to grow logarithmically and hence in a logarithmically supercritical regime, similarly in Theorem 1.2. (2) Theorem 1.1 and 1.2 in comparison to the work in [12] and [1] reflects on the fact that the global regularity relies not only on the dissipation term but delicate balance between that and the nonlinear term (cf. [6], [16] on the β-generalized quasi-geostrophic equation (QG)). (3) The restriction in the case of LANS α model in [10] is also the restriction of 2γ 1 + γ 2 5 2 ; our result is logarithmically sharper and this method can be generalized to extend their result logarithmically as well. () If we take γ 2 = 0, g 2 1, then (1) becomes the hyper-dissipative NSE considered in [12]; Theorem 1.1 generalizes the result. (5) It would be of much interest if the well-posedness of the systems (1) and (2) can be extended to Besov space. It is well-known that global well-posedness of modified active scalars such as the modified QG in critical Besov space is an outstanding open problem (cf. [9], [17]). (6) Our proof is an appropriate modification of the approach of Terence Tao in [12], subsequently applied elegantly in the Besov setting by Wu in [1]. However, at the estimate of I 3 below, we will have to compromise to γ 1 + γ 2 5 2 from which we obviate by diverging from the proof in [1]. The structure of (2) forces the additional powers of g i s. Our notations, preliminaries and local results are standard, to be discussed in Appendix for completeness, only in case of (1). We now prove Theorem 1.2 and subsequently sketch the proof of Theorem 1.1 as they are similar. Firstly, it is standard to obtain 2. Proof of Theorem 1.2 t ( u 2 L 2 + L 3 v 2 L 2 + B 2 L 2 ) + L 2 B 2 L 2 + L 1 u 2 L 2 + L 1 L 3 u 2 L 2 = 0 and therefore the energy dissipation bound of T 0 L 1 B 2 L 2 + L 1 u 2 L 2 + L 1 L 3 u 2 L 2 dt c 2.1. a priori estimate of v. We take a curl on the first equation of (1) and obtain with w = curlv, z = curlb, (5) t w + L 2 1w = u w + w u + B z z B

KAZUO YAMAZAKI We apply Littlewood-Paley operator to obtain (6) t w+l 2 1 w = (u w)+ (w u)+ (B z) (z B) We multiply by w and integrate in space to obtain (7) 1 2 t w 2 L + L 2 1 w 2 L 2 = (u w) wdx + (w u) wdx + (B z) wdx (z B) wdx We multiply by 2 2s where s = m 1 and sum over 1 to obtain 1 2 t w 2 s + 2 2s L 1 w 2 L = 2 2s 2 (u w) wdx + 2 2s (w u) wdx + 2 2s (B z) wdx 2 2s (z B) wdx = I 1 + I 2 + I 3 + I 2.1.1. Estimate on I 1. By Bony product, (cf. [1]) (u w) = + (S k 1 u k w) k 3 ( k u S k 1 w) + ( k u k w) k 3 k 1 which we can further decompose the first term into 3 parts to obtain

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS5 I 11 = 2 I 12 = 2 I 13 = 2 I 1 = 2 I 15 = 2 2 2s 2 2s 2 2s 2 2s 2 2s w [, S k 1 u ] k wdx w k 3 k 3 (S k 1 u S u) k wdx w S u wdx = 0 ( u = 0) w ( k u S k 1 w)dx w k 3 k 1 ( k u k w)dx We take I 11 : I 11 c c c c 2 2s w L 2 [, S k 1 u ] k w L 2dx 2 2s w L 2 2 2s w L 2 k 3 k 3 k 3 2 2s w 2 L 2 S k 1 u L 2 S k 1 u L k w L 2 xφ 0 L 1 2 k S k 1 u L k w L 2 c 2 2s w 2 L 2 ( 5 2 2 )m m u L 2 m 2 1 2 2s 22γ 1 g 2 1 (2 ) w 2 L + c g1(2 2 )2 2s 2 w 2 L 2 2γ1 [ 2 ( 5 2 2 )m m u L 2] 2 m 2 1 2 2s L 1 w 2 L + c g 1(2 2 )2 2s 2 w 2 L 2 2γ1 [ 2 ( 5 2 2 )m m u L 2] 2 m 2 = 1 2 2s L 1 w 2 L + I 2 111 + I 112 by Lemma.5 and Bernstein s inequality; we shall call I 111 and I 112 lower and higher frequency respectively which we estimate:

6 KAZUO YAMAZAKI I 111 = c N c N g 2 1(2 )2 2(s+γ 2 γ 1 ) w 2 L 2 [ g 2 1(2 )2 2(s+γ 2 γ 1 ) w 2 L 2 m 2 2 (m )γ 2+( 5 2 γ 2)m m u L 2] 2 sup 2 2(γ 1+γ 3 )m m u 2 L 2 m 2 cg1(2 N )g3(2 2 N ) w 2 s sup L 1 L 3 u 2 L 2 mn by (). On I 112, we rewrite and estimate I 112 = c >N c >N c >N g 2 1(2 )2 2(s γ 1+γ 2 ) w 2 L 2 [ m 2 g 1(2 )g 2 3(2 )2 2(s γ 1+γ 2 ) w 2 L 2 [ g 1(2 )g 2 3(2 )2 2(s γ 1+γ 2 ) w 2 L 2 2 (m )γ 2 2 m( 5 2 γ 2) m u L 2] 2 m 2 2 (m )γ 2 m L 1 L 3 u L 2] 2 sup m L 1 L 3 u 2 L 2 m 2 Now as g grows logarithmically, we have that for large N, (8) g 1(2 )g 2 3(2 )2 2(s γ 1+γ 2 ) cg 1(2 N )g 2 3(2 N )2 2s by () and therefore, (9) I 112 cg 1(2 N )g 2 3(2 N ) w 2 s L 1 L 3 u 2 L 2 with which we conclude I 11 1 2 2s L 1 w 2 L + cg 1(2 N )g3(2 2 N ) w 2 s L 2 1 L 3 u 2 L 2 We next estimate I 12 term. Similar computations give I 12 1 2 2s L 1 w 2 L + c g 1(2 2 )2 2(s+ 5 2 2 γ1) w 2 L 2 u 2 L 2 = 1 2 2s L 1 w 2 L + I 2 121 + I 122 and as before

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS7 I 121 c N g 1(2 )g 2 3(2 )2 2(s+ 5 2 γ 1) w 2 L 2 2 2(γ 1+γ 3 ) (sup N cg1(2 N )g3(2 2 N ) sup L 1 L 3 u 2 L 2 2(s+ 5 2 2 2γ 1 γ 3 ) w 2 L 2 N N cg1(2 N )g3(2 2 N ) sup L 1 L 3 u 2 L w 2 2 s N The estimate on I 122 is also similar; we conclude (10) I 12 1 2 2s L 1 w 2 L + cg 1(2 N )g3(2 2 N ) L 2 1 L 3 u 2 L w 2 2 s The estimate on I 1 is also similar: I 1 1 2 2s L 1 w 2 L +c 2 g 2 1(2 )2 2s u 2 L 2 2 2γ 1 [ 2 2(γ 1+γ 3 ) g 2 1 (2 )g 2 3 (2 ) u 2 L 2 ) m 2 As before we split the second sum into low and high frequencies and I 11 = N (11) C N g 2 1(2 )2 2(γ 3+γ 1 ) u 2 L 2 [ m 2 2 ( 5 2 )m m w L 2] 2 2 (m )(γ 3+2γ 1 s)+m( 5 2 +s 2γ 1 γ 3 ) m w L 2] 2 g1(2 )g3(2 2 ) L 1 L 3 u 2 L ( sup 2 m( 5 2 2 +s 2γ 1 γ 3 ) m w L 2) 2 m 2 Cg1(2 N )g3(2 2 N ) L 1 L 3 u 2 L w 2 2 s where the first inequality made use of m 2 2 (m )(γ 3+2γ 1 s) < Next, estimating higher frequency side as before, we conclude I 1 1 2 2s L 1 w 2 L + cg 1(2 N )g3(2 2 N ) w 2 s L 2 1 L 3 u 2 L 2 With I 15 bounded by I 15 1 2 2s L 1 w 2 L + Cg 1(2 N )g3(2 2 N ) L 2 1 L 3 u 2 L w 2 2 s we conclude this subsection with I 1 2 2s L 1 w 2 L 2 + cg 1(2 N )g 2 3(2 N ) w 2 s L 1 L 3 u 2 L 2

8 KAZUO YAMAZAKI 2.1.2. Estimate on I 2. By Bony product again, we can decompose I 2 to I 21 = 2 I 22 = 2 I 23 = 2 I 2 = 2 I 25 = 2 2 2s w [, S k 1 w ] k udx 2 2s 2 2s 2 2s 2 2s w k 3 k 3 (S k 1 w S w) k udx w S w udx w ( k w S k 1 u)dx w k 3 k 1 We take I 21 via similar procedure as before: ( k w k u)dx I 21 1 2 2s L 1 w 2 L + c g 1(2 2 )2 2s 2 u 2 L 2 2γ1 [ 2 = 1 2 2s L 1 w 2 L + I 2 211 + I 212 We estimate: 2 ( 5 2 )m m w L 2] 2 m 2 (12) I 211 cg 1(2 N )g 2 3(2 N ) L 1 L 3 u 2 L 2 w 2 s On I 212, we have (13) I 212 c >N g 1(2 )g 2 3(2 ) L 1 L 3 u 2 L 2 sup 2 2m( 5 2 +s 2γ 1 γ 3 ) m w 2 L 2 m 2 Taking advantage of logarithmic growth of g s again, we conclude (1) I 21 1 2 2s L 1 w 2 L + cg 1(2 N )g3(2 2 N ) w 2 s L 2 1 L 3 u 2 L 2 We next estimate I 22 term. The computation here is similar: (15) I 22 1 2 2s L 1 w 2 L + I 2 221 + I 222 where I 221 cg1(2 N )g3(2 2 N ) sup L 1 L 3 u 2 L w 2 2 s N

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS9 The estimate on I 222 is also similar and we conclude (16) I 22 1 2 2s L 1 w 2 L + cg 1(2 N )g3(2 2 N ) L 2 1 L 3 u 2 L w 2 2 s Unlike I 13, we do not get the estimate on I 23 free by divergence-free property, but this estimate is not hard: I 23 1 2 2s L 1 w 2 L + Cg 1(2 N )g3(2 2 N ) L 2 1 L 3 u 2 L w 2 2 s The estimate on I 2 and I 25 are both similar. We conclude I 2 2 2s L 1 w 2 L 2 + cg 1(2 N )g 2 3(2 N ) w 2 s L 1 L 3 u 2 L 2 2.1.3. Estimate on I 3. We will slightly diverge here from the proof of [1] to not compromise our hypothesis (). We estimate as follows: I 31 c c 2 2s w L 2 k 3 2 s 2 γ 1 g 1 (2 ) w L 22 (s γ 1+ 7 2 ) g 1 (2 ) S k 1 B L k z L 2 xφ 0 L 1 m 2 m B L 2 B L 2 where we used B = -curl curlb + grad div B = -curl curl B ; hence z L 2 c curl curlb L 2 = c B L 2 due to the following the classical estimate from [1] Lemma 2.1. Suppose the vector function f is divergence-free and g = f. Then there exists constants C independent of f such that (17) f L p C g L p 1 < p < Now we continue the estimate by 1 2 2s L 1 w 2 L + c 2 1 2 2s L 1 w 2 L + c 2 1 2 2s L 1 w 2 L + c 2 1 2 2s L 1 w 2 L + c 2 2 2(s γ 1+ 7 2 ) g 2 1(2 ) B 2 L 2 [ 2 2(s γ 1+ 7 2 +ɛ 0) g 2 1(2 ) B 2 L 2 [ m 2 m 2 m B L 2] 2 2 2(s γ 1+ 7 2 +ɛ0) g1(2 2 ) B 2 L 2 sup m B 2 L 2 m 2 2 2(s γ 1+ 7 2 +ɛ 0) g 2 1(2 ) B 2 L 2 2 (m )ɛ 0 m B L 2] 2

10 KAZUO YAMAZAKI with ɛ 0 > 0 to be specified subsequently. Here the last inequality used and hence we bound I 31 by m B(, t) 2 L 2 m B 0 ( ) 2 L 2 c < (18) 1 2 2s L 1 w 2 L + c 2 2 2(s γ 1+ 7 2 +ɛ 0) g 2 1(2 ) B 2 L 2 I 32, I 33, I 3, I 35 can all be bounded similarly; thus, I 3 2 2s L 1 w 2 L + c 2 2(s+ 7 2 2 γ 1+ɛ 0 ) g1(2 2 ) B 2 L 2 1 1 2.1.. Estimate on I. This is similar to I 3 ; the bound is identical. 2.2. a priori estimate of B. We now take (19) B t + u B B u + L 2 2B = 0 We apply, multiply by B, integrate with respect to x, multiply by 2 2τ g 2 1 (2 ) where τ = s + 7 2 γ 1 + ɛ 0 where ɛ 0 > 0 satisfies (20) min{2γ 1 + γ 3 5 2, γ 1 + 2γ 3 5 2 } > ɛ 0 > 0 and sum over to study 1 2 t (g 1 (2 )2 τ B L 2) 2 + (2 τ g 1 (2 ) L 2 B L 2) 2 = 2 2τ g1(2 2 ) (u B) Bdx + 2 2τ g1(2 2 ) (B u) Bdx = J 1 + J 2 2.2.1. Estimate on J 1. We estimate

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS11 J 11 = 2 J 12 = 2 J 13 = 2 J 1 = 2 J 15 = 2 2 2τ g1(2 2 ) 2 2τ g1(2 2 ) 2 2τ g1(2 2 ) 2 2τ g1(2 2 ) 2 2τ g1(2 2 ) B [, S k 1 u ] k Bdx B k 3 k 3 (S k 1 u S u) ( k B)dx B S u ( B)dx = 0 ( u = 0) B ( k u ( S k 1 B))dx B k 3 k 1 ( k u ( k B))dx We start with J 11 as before: J 11 1 2 2τ g 2 1(2 ) L 2 B 2 L 2 + c 2 2(τ γ 2) g 2 1(2 )g 2 2(2 ) B 2 L 2 [ The lower frequency side is estimated as follows: c N 2 2τ g 2 1(2 )g 2 2(2 ) B 2 L 2 [ m 2 m 2 m u L ] 2 2 (m )γ 2 2 m( 5 2 γ 2) m u L 2] 2 cg2(2 2 N ) 2 2τ g1(2 2 ) B 2 L sup 2 2m( 5 2 2 γ2) m u 2 L 2 m 2 N cg 2 1(2 N )g 2 2(2 N )g 2 3(2 N ) 2 2τ g 2 1(2 ) B 2 L 2 L 1 L 3 u 2 L 2 The higher frequency side as done as before. Take 0 < ɛ < γ 2 this time: c 2 2(τ γ2) g1(2 2 )g2(2 2 ) B 2 L 2 2ɛ [ 2 (m )ɛ 2 m( 5 2 2 ɛ) m u L 2] 2 >N c >N 2 2τ 2 2N(γ 2 ɛ) g 2 1(2 )g 2 2(2 ) B 2 L 2 [ m 2 m 2 cg1(2 2 N )g3(2 2 N )g2(2 2 N )2 2N(γ 2 ɛ) g1(2 2 )2 2τ B 2 L w 2 2 s 2 (m )ɛ 2 m( 5 2 ɛ) m u L 2] 2 Next,

12 KAZUO YAMAZAKI J 12 1 2 2τ g 1(2 2 ) L 2 B 2 L + c 2 2τ g1(2 2 )g2(2 2 ) 2 B 2 L 2 2( 5 2 2 γ2) u 2 L 2 Splitting the second sum into 2 parts as before, we estimate (21)J 121 c L 1 L 3 u 2 L 2 g 2 1(2 N )g 2 2(2 N )g 2 3(2 N ) N 2 2τ g 2 1(2 ) B 2 L 2 and the higher frequency: (22)J 122 = c >N 2 2τ g 2 1(2 )g 2 2(2 )2 2(γ 2 ɛ) B 2 L 2 2 2( 5 2 ɛ) u 2 L 2 cg 2 1(2 N )g 2 2(2 N )g 3(2 N )2 2N(γ 2 ɛ) >N 2 2τ g 2 1(2 ) B 2 L 2 w 2 s J 1, J 15 can be done similarly; we conclude for 0 < ɛ < γ 2 (23) J 1 1 2 2τ g 1(2 2 ) L 2 B 2 L 2 + cg1(2 2 N )g2(2 2 N )g3(2 2 N ) L 1 L 3 u 2 L 2 2τ g1(2 2 ) 2 B 2 L 2 + cg 1(2 N )g 2 2(2 N )g 3(2 N )2 2N(γ 2 ɛ) w 2 s 2 2τ g 2 1(2 ) B 2 L 2 2.3. Estimate on J 2. The estimate on J 2 is very similar to J 1 ; (2) J 2 c 2 2τ g 2 1(2 ) L 2 B 2 L 2 + c L 1 L 3 u 2 L 2 g 2 1(2 N )g 2 2(2 N )g 2 3(2 N ) 2 2τ g 2 1(2 ) B 2 L 2 + cg2(2 2 N )g3(2 N )2 2N(γ2 ɛ) w 2 s 2 2τ g1(2 2 ) B 2 L 2 where γ 2 > ɛ > 0. We now let (25) 2 N = E s = w 2 s + 2 2τ g 2 1(2 ) B 2 L 2 so that combining all estimates

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS13 (26) 1 2 te s cg 1(E s )g 2 2(E s )g 3(E s ) L 1 L 3 u 2 L 2 E s Gronwall s inequality implies the desired result. Notice because g 1, (27) B 2 H τ = 2 2τ B(, t) 2 L 2 2 2τ g 2 1(2 ) B(, t) 2 L 2 C < Importantly for our proof 3. Proof of Theorem 1.1 (28) T 0 A(t)dt = T 0 L 1 v(, s) 2 L 2 ds 1 2 v 0 2 L 2 where v 0 = (1 + L 2 2 )u 0(x). Via similar procedure we obtain (29) t v 2 s + 2 2s L 1 v 2 L 2 = 2 2s 2 (u v) vdx for s large. By Bony s product as usual we split into I 1 = 2 I 2 = 2 I 3 = 2 I = 2 I 5 = 2 2 2s 2 2s 2 2s 2 2s 2 2s v [, S k 1 u ] k vdx v k 3 k 3 (S k 1 u S u) ( k v)dx v S u vdx = 0 v ( k u (S k 1 v))dx v k 3 k 1 ( k u ( k v))dx The following can be obtained:

1 KAZUO YAMAZAKI I 1 1 2 2s L 1 v 2 L 2 + cg1(2 N )g2(2 2 N ) sup L 1 (1 + L 2 )u 2 L v 2 2 s + cg1(2 2 N )g2(2 2 N )2 2N(γ1 δ) v s N I 2 1 2 2s L 1 v 2 L 2 + cg1(2 N )g2(2 2 N ) sup L 1 (1 + L 2 )u 2 L v 2 2 s + cg1(2 2 N )g2(2 2 N )2 2N(s γ 1 γ 2 ) v s N I 1 2 2s L 1 v 2 L 2 + cg1(2 N )g2(2 2 N ) v 2 s sup L 1 v 2 L + cg1(2 2 N )g2(2 2 N )2 2N(γ 1+γ 2 δ) v 2 s N I 5 1 2 2s L 1 v 2 L 2 + cg1(2 2 N )g2(2 2 N )2 2N(γ1+δ) v s + cg1(2 N )g2(2 2 N ) sup L 1 v 2 L v 2 2 s N for 0 < δ < γ 1. Combining estimates, with 2 N = E s (t) = v 2 s, we have (30) t E s cg 1(E s )g 2 2(E s )A(t)E s by (3). Gronwall s inequality completes the proof.. Appendix.1. Notations and preliminaries. Let F be the space of Fourier series (31) {u = k T û k φ k : û k C 3 } where T = {mα 1 0 : m Z 3 \ {0}}, α 0 = L/(2π) and φ k = e ik x. Let V be the space of divergence-free trigonometric polynomials consisting of all u F such that k û k = 0 k T and û k = 0 for all but finitely many values of k T. We define H s = {u F : u s <, û k = û k and û 0 = 0} V s = {u H s : k û k = 0} with u 2 s = L 3 k T k 2s û k 2 When s 0, the spaces H s correspond to spaces of periodic functions on T 3 with zero mean. Furthermore, V s are those elements of H s that are divergence-free; note also that V s is the closure of V in H s with respect to s norm. The inner product is defined by (u, v) = L 3 k T (û k, ˆv k );

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS15 H is the Hilbert space of periodic L 2 -integrable functions on T 3 that are divergence free with mean zero. Importantly throughout the proof, due to Poincare s inequality, if for example v = (1 + α 2 L 2 2 )u and g 2 1, then (32) α 2 u 2γ2 v L 2 {α 2 0 + α 2 } u 2γ2 We now discuss useful inequalities and Besov spaces. Firstly, Lemma.1. For all m Z + {0}, there exists c m > 0 such that (33) (3) 0 β m fg H m C m ( f L D m g L 2 + D m f L 2 g L ) D β (fg) fd β g L 2 c m ( f L D m 1 g L 2 + D m f L 2 g L ) We denote by S(R d ) the Schwartz class functions and S (R d ), its dual, the space of tempered distributions. We define S 0 to be the subspace of S (35) S 0 = {φ S, φ(x)x γ dx = 0, γ = 0, 1, 2,...} R d Its dual S 0 is given by S 0 = S/S 0 = S /P where P is the space of polynomials. For Z we define A = {ξ R d : 2 1 < ξ < 2 +1 } It is a well-known that there exists a sequence {Φ } S(R d ) such that (36) (37) suppˆφ A, ˆΦ (ξ) = ˆΦ 0 (2 ξ) or Φ (x) = 2 d Φ 0 (2 x) { 1 if ξ R ˆΦ d \ {0} k (ξ) = 0 if ξ = 0 k= Consequently, for any f S 0, k= Φ k f = f To define the homogeneous Besov space, we set f = Φ f, = 0, ±1, ±2,... With such we can define for s R, 1 p, q, the homogeneous Besov space Ḃs p,q = {f S 0 : f Ḃ < } where p,q s { ( f Ḃs = (2s f L p) q ) 1/q for q < p,q sup 2 s f L p for q = To define inhomogeneous Besov space, let Ψ C 0 (Rd ) be even such that

16 KAZUO YAMAZAKI 1 = ˆΨ(ξ) + ˆΦ k (ξ), Ψ f + Φ k f = f k=0 for any f S. We further set 0 if 2 f = Ψ f, if = 1 Φ f, if = 0, 1, 2,... and define for any s R, 1 p, q, inhomogeneous Besov space Bp,q s = {f S : f B s p,q < } where { ( = 1 f B s p,q = (2s f L p) q ) 1/q, if q <, sup 1< 2 s f L p, if q = We use notations S k< k, k=0 The following lemma is classic in harmonic analysis: Lemma.2. (cf. [1]) Bernstein s Inequality: Let f L p (R d ) with 1 p q and 0 < r < R. Then for all k Z + {0}, and λ > 0, there exists a constant C k > 0 such that (a) sup β f L q Cλ k+d( 1 p 1 q ) f L p β =k if suppff {ξ : ξ λr}, (b)c 1 k λk f L p sup β =k β f L p C k λ k f L p if suppff {ξ : λr ξ λr}. and if we replace derivative β by the fractional derivative, the inequalities remain valid only with trivial modifications. Lemma.3. Besov Embedding (cf. [1]) Assume s R and p, q [1, ]. (1) If 1 q 1 q 2, then Ḃs p,q 1 (R d ) Ḃs p,q 2 (R d ). (2) If 1 p 1 p 2 and s 1 = s 2 + d( 1 p 1 1 p 2 ), then Ḃs 1 p 1,q(R d ) Ḃ s 2 p 2,q(R d ). Finally, we recall mollifiers: given an arbitrary radial function ρ( x ) C 0 (R3 ), ρ 0, R ρ dx = 1, we define the mollifer operator T 3 ɛ : L p (R 3 ) C (R 3 ), 1 p, ɛ > 0, by (T ɛ f)(x) = ɛ 3 ρ( x y )f(y)dy f L p (R 3 ). R 3 ɛ Lemma.. For m Z + {0}, s R, k R+. Then in particular (a) For all f H s (R 3 ), lim ɛ 0 T ɛ f f H s = 0, T ɛ f f H s 1 cɛ f H s. (b) For all f H m (R 3 ), T ɛ f H m+k cɛ k f H m, T ɛ D k f L cɛ (1+k) f L 2.

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS17 In the subsequent subsection, by (a), (b) we mean those of Lemma.. The following lemma due to [1] was used often: Lemma.5. For any 1, p [1, ], (38) [, f ]g L p C f L q g L r xφ L σ C2 ( 1+d(1 1 σ )) f L q g L r xφ 0 L σ where [, f ]g denotes (f g) f g and q, r and σ satisfy q, r, σ [1, ], 1 + 1 p = 1 q + 1 r + 1 σ, 1 r + 1 σ + 1 d > 1. As special cases, (39) [, f ]g L p C2 f L g L p xφ 0 L 1 for any p [1, ] and (0) [, f ]g L p C2 ( 1+ d r ) f L p g L r xφ 0 L r for any p [1, ], r p and 1 r + 1 r = 1.2. Local result. We regularize (1) and study (1) { vt ɛ + T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ )) + p ɛ + T ɛ (T ɛ L 2 1 vɛ ) = 0 v ɛ t=0 = v 0 (x); u ɛ = (1 + α2 2L2 2 ) 1 v ɛ also written after taking the proection P as (2) v ɛ t = T 2 ɛ L 2 1v ɛ P [T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ ))] = F 1 ɛ (v ɛ ) F 2 ɛ (v ɛ ) = F ɛ (v ɛ ) Now F (H m ) H m and F ɛ is locally Lipschitz on H m using (b): (3) F ɛ (v 1 ) F ɛ (v 2 ) H m = (F 1 ɛ F 2 ɛ )(v 1 ) (F 1 ɛ F 2 ɛ )(v 2 ) H m c v 1 v 2 H m, for c = c( v i L 2, ɛ), i = 1, 2; i.e. F is locally Lipshitz on any open set O M = {v H m : v H m < M}. By Picard Theorem, we obtain a unique solution v ɛ C 1 ([0, T ɛ ], H m O M ) fo some T ɛ > 0. Now clearly we have () if v ɛ C 1 ([0, T ], L 2 ), then sup v ɛ L 2 v 0 L 2 0tT We want to show that the local solution v ɛ can be continued for all t > 0. We invoke the continuation property of ODEs on Banach spaces that solutions can be continued in provided F is time independent. To show an a priori bound on v ɛ (, t) H m, we take (3) with v 1 = v ɛ, v 2 = 0 and get

18 KAZUO YAMAZAKI F ɛ (v ɛ ) H m c v ɛ H m. for some constant c = c( v ɛ L 2, ɛ) which implies by (2), d (5) dt vɛ H m c v ɛ H m Thus, Gronwall s inequality gives us the a priori bound v ɛ (T, ) H m e CT We ust proved the global existence and uniqueness of solutions to the regularized equation; i.e. for all ɛ > 0, there exists v ɛ C 1 ([0, ); H m ), m Z + {0} to (2). We now show that if m > max{5/2, 1 + 2γ 1 }, there exists a subsequence convergent to a limit function v that solves (1) up to some T > 0. We already have that solutions to the regularized equation satisfy (). However, note that (5) depends on ɛ. Thus, we first prove a priori higher derivative estimates that are also independent of ɛ: (6) 1 d 2 dt vɛ 2 H m + T ɛl 1 v ɛ 2 H m c m,γ 2 T ɛ v ɛ L v ɛ 2 H m This can be shown by taking proection P on (1), applying D β, β m, multiplying by D β v ɛ and integrating in space, summing over β m, using Hölder s inequality and (3). To show that (v ɛ ) the family of solution is uniformly bounded in H m, we divide (6) by v ɛ H m first and then use Sobolev imbedding to obtain d dt vɛ H m c m,γ2 v ɛ 2 H m for m > 5 2. Thus, for all ɛ > 0, we have (7) sup 0tT v ɛ v 0 H m H m 1 c m,γ2 T v 0 H m which implies that for T < 1 C m,γ2 v 0 m, (v ɛ ) is uniformly bounded in C([0, T], H m ). Next, we show that the solutions v ɛ form a contraction in the low norm C([0,T], L 2 (T 3 )). Take v ɛ t v ɛ t = (T 2 ɛ L 2 1v ɛ T 2 ɛ L 2 1v ɛ ) P [T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ )) P T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ ))] and multiply by v ɛ v ɛ and integrate to get (vt ɛ v ɛ t, v ɛ v ɛ ) = (Tɛ 2 L 2 1v ɛ T ɛ 2 L 2 1v ɛ, v ɛ v ɛ ) (P [T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ )) P T ɛ ((T ɛ u ɛ ) (T ɛ v ɛ ))], v ɛ v ɛ ) = I II By Hölder s inequality and (a)

ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS19 I cmax{ɛ, ɛ}max{ L 2 1v ɛ H 1, L 2 1v ɛ H 1} v ɛ v ɛ L 2 max{ɛ, ɛ} max{ v ɛ H m, v ɛ H m} v ɛ v ɛ L 2 if m > 1 + 2γ 1. Next, we decompose For example, II = II 1 + II 2 + II 3 + II + II 5 = ((T ɛ T ɛ ){(T ɛ u ɛ ) (T ɛ v ɛ )}, v ɛ v ɛ ) + (T ɛ {(T ɛ T ɛ )u ɛ (T ɛ v ɛ )}, v ɛ v ɛ ) + (T ɛ {T ɛ (u ɛ u ɛ ) (T ɛ v ɛ )}, v ɛ v ɛ ) + (T ɛ {(T ɛ u ɛ ) (T ɛ T ɛ )v ɛ }, v ɛ v ɛ ) + (T ɛ {(T ɛ u ɛ ) T ɛ (v ɛ v ɛ )}, v ɛ v ɛ ) II 1 cmax{ɛ, ɛ}( u ɛ L v ɛ H 1 + u ɛ H 1 v ɛ L ) v ɛ v ɛ L 2 cmax{ɛ, ɛ} v ɛ 2 H m vɛ v ɛ L 2 by Hölder s inequality, (a), (33) and Sobolev embedding. A similar estimate holds for II 2, II 3 and II while II 5 = 0. From these we obtain (8) d dt vɛ v ɛ L 2 C(M)(max{ɛ, ɛ} + v ɛ v ɛ L 2) for M an upper bound from (7). Gronwall s inequality gives (9) sup tt v ɛ v ɛ L 2 C(M, T )max{ɛ, ɛ} We deduce that {v ɛ } is Cauchy in C([0, T], L 2 (T 3 )) and converges to v C([0, T ], L 2 (T 3 )). By interpolation inequality on v ɛ v, (7) and (9), sup v ɛ v H s C( v 0 H m, T, s)ɛ 1 s m tt which gives v C([0, T ], H s (T 3 )), 0 s < m. Thus, we see that vt ɛ converges to -L 2 1 v u v in C([0, T ], C(T3 )). As v ɛ v, the distribution limit of vt ɛ must be v t ; i.e. v is a classical solution (1). The uniqueness is proven by using the difference of two solutions and Gronwall s inequality. 5. Acknowledgment The author expresses gratitude to Professor Jiahong Wu and Professor David Ullrich for their teaching.

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