Regularity Criteria for a Turbulent Magnetohydrodynamic Model

and Applied Analysis 3 for any fixed T > 0. Then v,H can be extended beyond T > 0 if one of the following conditions is satisfied: 1 u ∈ C ( 0, T ;Ln/3 ) , 1.11 2 u ∈ L 0, T ;L with 2 p n q 3, n 3 < q ≤ n, 1.12 3 ∇u ∈ C ( 0, T ;Ln/4 ) , 1.13 4 ∇u ∈ L 0, T ;L with 2 p n q 4 with n 4 < q ≤ n 2 . 1.14 Remark 1.5. If we delete the harmless lower order terms ∂tu − Δu and ∂tB − ΔB in 1.1 and 1.2 , then we have −∂tΔu Δ2u u · ∇ u ∇π B · ∇ B, −∂tΔB Δ2B u · ∇ B − B · ∇ u 0, 1.15 then the system 1.15 has the following property: if u, B, π is a solution of 1.15 , then for all λ > 0, uλ, Bλ, πλ x, t ( λ3u, λ3B, λ6π )( λx, λ2t ) 1.16 is also a solution. In this sense, our conditions 1.8 and 1.11 – 1.14 are scaling invariant optimal . Equations 1.8 and 1.12 do not hold true for q > n. But we also can establish regularity criteria for q > n in nonscaling invariant forms. In Section 2, we will prove Theorem 1.2. In Section 3, we will prove Theorem 1.4. 2. Proof of Theorem 1.2 Since it is easy to prove that the problem 1.1 – 1.5 has a unique local smooth solution, we only need to establish the a priori estimates. The proof of the case n ≤ 4 is easier and similar and thus we omit the details here, we only deal with the case n ≥ 5. Testing 1.1 by u, using 1.3 and 1.4 , we find that 1 2 d dt ∫ u2 |∇u|dx ∫ |∇u| |Δu|dx ∫ B · ∇ B · udx. 2.1 4 Abstract and Applied Analysis Testing 1.2 by B, using 1.3 and 1.4 , we see that 1 2 d dt ∫ B2 |∇B|dx ∫ |∇B| |ΔB|dx ∫ B · ∇ u · B dx − ∫ B · ∇ B · udx. 2.2 Summing up 2.1 and 2.2 , we easily get 1.6 . I Let 1.8 hold true. In the following calculations, we will use the product estimates due to Kato and Ponce 3 : ∥Λsfg∥Lp ≤ C ∥Λsf∥Lp1 ∥g∥Lq1 ∥f∥Lp2 ∥Λsg∥Lq2 ) , 2.3 with s > 0, Λ : −Δ 1/2 and 1/p 1/p1 1/q1 1/p2 1/q2. The proof of the case q n is easier and similar, we omit the details here. Now we assume n/3 < q < n. Applying Λ to 1.1 , testing by Λv, using 1.4 , we deduce that 1 2 d dt ∫ |Λsv|dx ∫ ∣∣Λs 1v ∣∣ 2 dx ∫ Λ B · ∇ B −Λs u · ∇ u Λv dx ∫ Λ div B ⊗ B − u ⊗ u Λv dx. 2.4 Similarly, applying Λ to 1.2 , testing by ΛH, using 1.4 , we infer that 1 2 d dt ∫ |ΛsH|dx ∫ ∣∣Λs 1H ∣∣ 2 dx ∫ Λ curl u × B ·ΛsH dx. 2.5 Summing up 2.4 and 2.5 , using 2.3 , we get 1 2 d dt ∫ |Λsv| |ΛsH|dx ∫ ∣∣Λs 1v ∣∣ 2 ∣∣Λs 1H ∣∣ 2 dx ∫ Λ div B ⊗ B − u ⊗ u Λv dx ∫ Λ curl u × B ·ΛsH dx ≤ C ( ‖B‖Lq ∥∥Λs 1B ∥∥ Lt1 ‖u‖Lq ∥∥Λs 1u ∥∥ Lt1 ) ‖Λv‖Lt2 C ( ‖u‖Lq ∥∥Λs 1B ∥∥ Lt1 ‖B‖Lq ∥∥Λs 1u ∥∥ Lt1 ) ‖ΛH‖Lt2 Abstract and Applied Analysis 5 ≤ C ‖u‖Lq ‖B‖Lq ∥∥Λs 1u ∥∥ Lt1 ∥∥Λs 1B ∥∥ Lt1 ) ‖Λv‖Lt2 ‖ΛH‖Lt2 C‖ u, B ‖Lq ∥∥Λs 1 u, B ∥∥ Lt1 ‖Λ v,H ‖Lt2 ( 1 q 1 t1 1 t2 1, 2 ≤ t2 ≤ 2n n − 2 < t1 ≤ 2n n − 4 )and Applied Analysis 5 ≤ C ‖u‖Lq ‖B‖Lq ∥∥Λs 1u ∥∥ Lt1 ∥∥Λs 1B ∥∥ Lt1 ) ‖Λv‖Lt2 ‖ΛH‖Lt2 C‖ u, B ‖Lq ∥∥Λs 1 u, B ∥∥ Lt1 ‖Λ v,H ‖Lt2 ( 1 q 1 t1 1 t2 1, 2 ≤ t2 ≤ 2n n − 2 < t1 ≤ 2n n − 4 ) ≤ C‖ u, B ‖Lq ∥∥Λs−1 v,H ∥∥ Lt1 ‖Λ v,H ‖Lt2 ≤ C‖ u, B ‖Lq‖Λ v,H ‖1−θ1 L2 ∥∥Λs 1 v,H ∥∥ θ1 L2 ‖Λ v,H ‖1−θ2 L2 ∥∥Λs 1 v,H ∥∥ θ2 L2 C‖ u, B ‖Lq‖Λ v,H ‖2−θ1−θ2 L2 ∥∥Λs 1 v,H ∥∥ θ1 θ2 L2 ≤ 1 2 ∥∥Λs 1 v,H ∥∥ 2 L2 C‖ u, B ‖ 2−θ1−θ2 Lq ‖Λ v,H ‖L2 , 2.6 which implies ‖ v,H ‖L∞ 0,T ;Hs ‖ v,H ‖L2 0,T ;Hs 1 ≤ C. 2.7 Here we have used the following Gagliardo-Nirenberg inequalities: ∥∥Λs−1 v,H ∥∥ Lt1 ≤ C‖Λ v,H ‖1−θ1 L2 ∥∥Λs 1 v,H ∥∥ θ1 L2 , ( − n t1 1 − θ1 ( 1 − n 2 ) θ1 ( 2 − n 2 )) ‖Λ v,H ‖Lt2 ≤ C‖Λ v,H ‖1−θ2 L2 ∥∥Λs 1 v,H ∥∥ θ2 L2 , ( − n t2 1 − θ2 ( − 2 ) θ2 ( 1 − n 2 )) . 2.8 II Let 1.9 hold true. In the following calculations, we will use the following commutator estimates due to Kato and Ponce 3 : ∥∥Λs(fg) − fΛsg∥Lp ≤ C ∥∇f∥Lp1 ∥∥Λs−1g ∥∥ Lq1 ∥Λsf∥Lp2 ∥g∥Lq2 ) , 2.9 with s > 0 and 1/p 1/p1 1/q1 1/p2 1/q2. The proof of the case q n/2 is easier and similar, we omit the details here. Now we assume n/4 < q < n/2. 6 Abstract and Applied Analysis Applying Λ to 1.1 , testing by Λu, and using 1.3 and 1.4 , we deduce that 1 2 d dt ∫ |Λsu| ∣∣Λs 1u ∣∣ 2 dx ∫ ∣∣Λs 1u ∣∣ 2 ∣∣Λs 2u ∣∣ 2 dx − ∫ Λ u · ∇u − u · ∇Λsu Λu dx ∫ Λ B · ∇B − B · ∇ΛsB Λu dx ∫ B · ∇ ΛB ·Λsu dx. 2.10 Applying Λ to 1.2 , testing by ΛB, using 1.3 and 1.4 , we infer that 1 2 d dt ∫ |ΛsB| ∣∣Λs 1B ∣∣ 2 dx ∫ ∣∣Λs 1B ∣∣ 2 ∣∣Λs 2B ∣∣ 2 dx − ∫ Λ u · ∇B − u∇ΛsB ΛB dx ∫ Λ B · ∇u − B · ∇Λsu ΛB dx ∫ B · ∇ Λu ·ΛsB dx. 2.11 Summing up 2.10 and 2.11 , noting that the last terms of 2.10 and 2.11 disappeared, and using 2.9 , we obtain 1 2 d dt ∫ |Λsu| ∣∣Λs 1u ∣∣ 2 |ΛsB| ∣∣Λs 1B ∣∣ 2 dx ∫ ∣∣Λs 1u ∣∣ 2 ∣∣Λs 2u ∣∣ 2 ∣∣Λs 1B ∣∣ 2 ∣∣Λs 2B ∣∣ 2 dx − ∫ Λ u · ∇u − u · ∇Λsu Λu dx ∫ Λ B · ∇B − B · ∇ΛsB Λu dx − ∫ Λ u · ∇B − u · ∇ΛsB ΛB dx ∫ Λ B · ∇u − B · ∇Λsu ΛB dx ≤ C‖∇u‖Lq‖Λu‖L2q/ q−1 C‖∇B‖Lq‖ΛB‖L2q/ q−1 ‖Λu‖L2q/ q−1 C‖∇B‖Lq‖ΛB‖L2q/ q−1 C‖∇u‖Lq‖ΛB‖L2q/ q−1 ≤ C‖ ∇u,∇B ‖Lq‖Λ u, B ‖L2q/ q−1 ≤ C‖ ∇u,∇B ‖Lq ∥∥Λs 1 u, B ∥∥ 2 1−θ L2 ∥∥Λs 2 u, B ∥∥ 2θ L2 ≤ 1 2 ∥∥Λs 2 u, B ∥∥ 2 L2 C‖ ∇u,∇B ‖ 1−θ Lq ∥∥Λs 1 u, B ∥∥ 2 L2 , 2.12 Abstract and Applied Analysis 7 which yields ‖ u, B ‖L∞ 0,T ;Hs 1 ‖ u, B ‖L2 0,T ;Hs 2 ≤ C. 2.13and Applied Analysis 7 which yields ‖ u, B ‖L∞ 0,T ;Hs 1 ‖ u, B ‖L2 0,T ;Hs 2 ≤ C. 2.13 Here we have used the following Gagliardo-Nirenberg inequality: ‖Λ u, B ‖L2q/ q−1 ≤ C ∥∥Λs 1 u, B ∥∥ 1−θ L2 ∥∥Λs 2 u, B ∥∥ θ L2 , 2.14 with − − 1 2q n 1 − θ ( 1 − n 2 ) θ ( 2 − n 2 ) , 2n n − 2 ≤ 2q q − 1 ≤ 2n n − 4 . 2.15 This completes the proof. 3. Proof of Theorem 1.4 We only need to prove the a priori estimates. Testing 1.1 and 1.2 by v,H , using 1.3 and 1.4 , and summing up the results, we have 1 2 d dt ∫ v2 H2dx ∫ |∇v| |∇H|dx ∫ u · ∇ u ·Δudx − ∫ B · ∇ B ·Δudx ∫ u · ∇ B ·ΔB dx − ∫ B · ∇ u ·ΔB dx I1 I2 I3 I4. 3.1 Using 1.4 , we see that