By establishing a new trilinear estimate, we show a regularity criterion for the three dimensional micropolar fluid system via the velocity in homogeneous Besov spaces. This improves [B. Q. Dong, Z. L. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal. 73(2010), 2334-2341] in some sense

Zhang, Zujin

Electronic journal of qualitative theory of differential equations

Zujin Zhang

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A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces

Zhang Zujin

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2016, No. 104, 1–6; doi: 10.14232/ejqtde.2016.1.104 http://www.math.u-szeged.hu/ejqtde/A regularity criterion for the three-dimensionalmicropolar fluid system in homogeneous Besov spacesZujin ZhangBSchool of Mathematics and Computer Sciences, Gannan Normal UniversityGanzhou 341000, Jiangxi, P.R. ChinaReceived 7 September 2016, appeared 21 November 2016Communicated by Maria Alessandra RagusaAbstract. By establishing a new trilinear estimate, we show a regularity criterion forthe three dimensional micropolar fluid system via the velocity in homogeneous Besovspaces. This improves [B. Q. Dong, Z. L. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal. 73(2010), 2334–2341] in some sense.Keywords: micropolar fluid flows, regularity criteria, Besov spaces.2010 Mathematics Subject Classification: 35Q35, 35B44.1 IntroductionIn this paper, we study the following three-dimensional micropolar fluid system with unitviscosities: ∂tu+ (u · ∇)u−4u+∇P−∇×w = 0,∂tw+ (u · ∇)w−4w+ 2w−∇divw−∇× u = 0,∇ · u = 0,(u,w)|t=0 = (u0,w0),(1.1)where u = (u1, u2, u3) is the fluid velocity field, P is the pressure arising from the usualassumption of incompressibility divu = 0, w = (w1, w2, w3) is the micro-rotation vector field,u0 and w0 are the prescribed initial data satisfying ∇ · u0 = 0.The modern theory of micropolar fluid system was initialled by Eringe [9], which modelssome physical phenomena that cannot be treated by the classical Navier–Stokes equations, forexample, the motion of animal blood, liquid crystals and dilute aqueous polymer solutionsfor example. Mathematically, many authors [3, 4, 7, 10, 20, 22, 24] treated the well-posednessand large-time behaviour of solutions to system (1.1). However, the issue of global regularityof weak solutions to (1.1) remains an open problem. Therefore it is important to study theBEmail: zhangzujin361@163.com2 Z. J. Zhangregularity criterion on the some physical quantities, such as velocity, vorticity and pressure.Dong and Zhang [6] showed the following regularity conditionu ∈ L 21+r (0, T; Br∞,∞(R3)), −1 < r < 1. (1.2)Here and in what follows, Ḃsp,q(R3) (resp. Bsp,q(R3)) with s ∈ R, p, q ∈ [1, ∞] is the inhomo-geneous (homogeneous) Besov space, whose definition, fine properties and its utilization influid dynamical systems can be found in [1]. Later, Wang and Yuan [23] and He and Wang [17](with [17, Equation 1.10] replaced by [26]) then considered the following regularity criterionin a logarithmically improved version∫ T0‖P‖2Ḃr∞,∞1 + ln(e + ‖P‖Ḃr∞,∞) dt < ∞, −1 ≤ r < 1. (1.3)For interested readers, please refer to [5, 8, 11–16, 18, 25, 29].The purpose of this paper is to improve (1.2) from inhomogeneous Besov spaces to homo-geneous ones. Before stating the precise result, let us recall the weak formulation of (1.1).Definition 1.1. Let u0 ∈ L2(R3) with divu0 = 0, w0 ∈ L2(R3). A measurable R3-valuedpair (u,w) is called a weak solution to system (1.1) on (0, T), provided the following twoconditions hold,(1) (u,w) ∈ L∞(0, T; L2(R3)) ∩ L2(0, T; H1(R3));(2) (u,w) verifies (1.1) in the distributional sense.Now, our result reads as follows.Theorem 1.2. Let T > 0, 0 < r < 1 and (u,w) be a weak solution pair of system (1.1) with initialdata (u0,w0) ∈ H1(R3). Ifu ∈ L 21+r (0, T; Ḃr∞,∞(R3)), (1.4)then the solution (u,w) is regular on (0, T).Remark 1.3. Due to the fact that Bsp,q(R3) = Lp(R3)∩ Ḃsp,q(R3) for any s > 0 and 1 ≤ p, q ≤ ∞(see [2, Theorem 6.3.2]), we point that, in the case 0 < r < 1, (1.4) is an improvement of (1.2).Remark 1.4. The refinement u ∈ L1(0, T; Ḃ1∞,∞(R3)) of (1.2) in case r = 1 was already estab-lished in [5, Theorem 1.3]; while the gainu ∈ L2(0, T; Ḃ0∞,∞(R3)) (1.5)of (1.2) in case r = 0 can be similarly verified just as [27, Theorem 1.1].Remark 1.5. By observing the following fact from [21, Section 1.3]:s < 0, p, q ≥ 1⇒ Ḃsp,q(R3) ⊂ Bsp,q(R3),we see that the condition u ∈ L 21+r (0, T; Ḃr∞,∞(R3)) with −1 < r < 0 could also ensure thesmoothness of the solution. One is referred to [19, 28] for similar results of the Navier–Stokesequations.Regularity criterion for the micropolar fluid system 3Before showing Theorem 1.2 in Section 2, let us end this introduction by proving thefollowing trilinear estimate, which could have its own interest.Lemma 1.6. For f ∈ Ḃr∞,∞(R3), g, h ∈ H1(R3) and any ε > 0, 0 < r < 1, k ∈ {1, 2, 3}, we have∫R3∂k f · gh dx ≤ C ‖ f ‖21+rḂr∞,∞‖(g, h)‖2L2 + ε ‖∇(g, h)‖2L2 . (1.6)Proof. ∫R3∂k f · gh dx = −∫R3f · ∂k(gh)dx = −∫R3Λr f ·Λ−r∂k(gh)dx(Λ = (−4) 12)≤ C ‖Λr f ‖Ḃ0∞,∞∥∥Λ−r∂k(gh)∥∥Ḃ01,1 (by [1, Proposition 2.29])≤ C ‖ f ‖Ḃr∞,∞ ‖gh‖Ḃ1−r1,1 (by [1, Lemma 2.1])≤ C ‖ f ‖Ḃr∞,∞(‖g‖L2 ‖h‖Ḃ1−r2,1 + ‖g‖Ḃ1−r2,1 ‖h‖L2)(by analogues of [1, Corollary 2.54])≤ C ‖ f ‖Ḃr∞,∞(‖g‖L2 ‖h‖rḂ02,∞‖h‖1−rḂ12,∞ + ‖g‖rḂ02,∞‖g‖1−rḂ12,∞ ‖h‖L2)(by [1, Proposition 2.22])≤ C ‖ f ‖Ḃr∞,∞(‖g‖L2 ‖h‖rL2 ‖∇h‖1−rL2 + ‖g‖rL2 ‖∇g‖1−rL2 ‖h‖L2)(by [1, Proposition 2.39])≤ C ‖ f ‖Ḃr∞,∞ ‖(g, h)‖1+rL2 ‖∇(g, h)‖1−rL2≤ C ‖ f ‖21+rḂr∞,∞‖(g, h)‖2L2 + ε ‖∇(g, h)‖2L2 .2 Proof of Theorem 1.2In this section, we give the proof of Theorem 1.2. Taking the inner product of (1.1)1 with −4uin L2(R3), we obtain12ddt‖∇u‖2L2 + ‖4u‖2L2 =∫R3[(u · ∇)u] · 4udx−∫R3(∇×w) · 4udx. (2.1)On the other hand, testing (1.1)2 by −4w, we get12ddt‖∇w‖2L2 + ‖4w‖2L2 + 2 ‖∇w‖2L2 + ‖∇divw‖2L2=∫R3[(u · ∇)w] · 4w dx−∫R3(∇× u) · 4w dx.(2.2)Plugging (2.1) and (2.2) together, integrating by parts then yields12ddt‖∇(u,w)‖2L2 + ‖4(u,w)‖2L2 + 2 ‖∇w‖2L2 + ‖∇divw‖2L2=[−3∑i=1∫R3[(∂iu · ∇)u] · ∂iudx−3∑i=1∫R3[(∂iu · ∇)w] · ∂iw dx]−2∫R3(∇×w) · 4udx≡ I + J.(2.3)4 Z. J. ZhangBy Lemma 1.6,I ≤ C ‖u‖21+rḂr∞,∞‖∇(u,w)‖2L2 +14‖4(u,w)‖2L2 . (2.4)Meanwhile,J ≤ 2 ‖∇×w‖L2 ‖4u‖L2 ≤ 2 ‖∇w‖2L2 +12‖4u‖2L2 . (2.5)Putting (2.4) and (2.5) into (2.3), we deduce12ddt‖∇(u,w)‖2L2 +14‖4(u,w)‖2L2 + ‖∇divw‖2L2 ≤ C ‖u‖21+rḂr∞,∞‖∇(u,w)‖2L2 .Applying Gronwall’s inequality with the fact (1.4), we find‖∇(u,w)(t)‖2L2 ≤ ‖∇(u0,w0)‖2L2 · exp[∫ T0‖u(τ)‖21+rḂr∞,∞dτ]< ∞, ∀ 0 ≤ t < T.By Sobolev’s inequality,u ∈ L∞(0, T; L6(R3)) ⊂ L4(0, T; L6(R3)).By [5, Theorem 1.1], we complete the proof of Theorem 1.2.AcknowledgementsThis work is supported by the Natural Science Foundation of Jiangxi (grant no.20151BAB201010), the National Natural Science Foundation of China (grant nos. 11501125,11361004) and the Supporting the Development for Local Colleges and Universities Founda-tion of China – Applied Mathematics Innovative Team Building.References[1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differentialequations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Springer, Heidel-berg, 2011. MR2768550; url[2] J. Bergh, J. Löfström, Interpolation spaces: an introduction, Grundlehren der Mathema-tischen Wissenschaften, Vol. 223, Springer-Verlag, Berlin, New York, Heidelberg, 1976.MR0482275; url[3] J. W. Chen, Z.M. Chen, B.Q. Dong, Uniform attractors of non-homogeneous micropolarfluid flows in non-smooth domains, Nonlinearity 20(2007), 1619–1635. MR2335076; url[4] B. Q. Dong, Z. M. Chen, Asymptotic profiles of solutions to the 2D viscous incompress-ible micropolar fluid flows, Discrete Contin. Dyn. Syst. 23(2009), 765–784. MR2461826; url[5] B. Q. Dong, Z .M. Chen, Regularity criteria of weak solutions to the three-dimensionalmicropolar flows, J. Math. Phys. 50(2009), 103525, 13 pp. MR2572698; url[6] B. Q. Dong, Z. L. Zhang, On the regularity criterion for three-dimensional micropolarfluid flows in Besov spaces, Nonlinear Anal. 73(2010), 2334–2341. MR2674209; urlRegularity criterion for the micropolar fluid system 5[7] B. Q. Dong, Z. F. Zhang, Global regularity of the 2D micropolar fluid flows with zeroangular viscosity, J. Differential Equations 249(2010), 200–213. MR2644133; url[8] L. L. Du, D. Q. Zhou, Global well-posedness of two-dimensional magnetohydrodynamicflows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal. 47(2015), 1562–1589. MR3338000; url[9] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16(1966), 1–18. MR0204005[10] G. P. Galdi, S. Rionero, A note on the existence and uniqueness of solutions of themicropolar fluid equations, Internat. J. Engrg. Sci. 15(1977), 105–108. MR0467030; url[11] S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in thecritical Morrey–Campanato space, Nonlinear Anal. Real World Appl. 12(2011), 2142–2150.MR2801007; url[12] S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in theMorrey–Campanato space, Nonlinear Differ. Equ. Appl. 17(2010), 181–194. MR2639150; url[13] S. Gala, M. A. 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A regularity criterion for the three-dimensional

micropolar fluid system in homogeneous Besov spaces

Repository of the Academy's Library

Electronic Journal of Qualitative Theory of Differential Equations2016, No. 104, 1–6; doi: 10.14232/ejqtde.2016.1.104 http://www.math.u-szeged.hu/ejqtde/A regularity criterion for the three-dimensionalmicropolar fluid system in homogeneous Besov spacesZujin ZhangBSchool of Mathematics and Computer Sciences, Gannan Normal UniversityGanzhou 341000, Jiangxi, P.R. ChinaReceived 7 September 2016, appeared 21 November 2016Communicated by Maria Alessandra RagusaAbstract. By establishing a new trilinear estimate, we show a regularity criterion forthe three dimensional micropolar fluid system via the velocity in homogeneous Besovspaces. This improves [B. Q. Dong, Z. L. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal. 73(2010), 2334–2341] in some sense.Keywords: micropolar fluid flows, regularity criteria, Besov spaces.2010 Mathematics Subject Classification: 35Q35, 35B44.1 IntroductionIn this paper, we study the following three-dimensional micropolar fluid system with unitviscosities: ∂tu+ (u · ∇)u−4u+∇P−∇×w = 0,∂tw+ (u · ∇)w−4w+ 2w−∇divw−∇× u = 0,∇ · u = 0,(u,w)|t=0 = (u0,w0),(1.1)where u = (u1, u2, u3) is the fluid velocity field, P is the pressure arising from the usualassumption of incompressibility divu = 0, w = (w1, w2, w3) is the micro-rotation vector field,u0 and w0 are the prescribed initial data satisfying ∇ · u0 = 0.The modern theory of micropolar fluid system was initialled by Eringe [9], which modelssome physical phenomena that cannot be treated by the classical Navier–Stokes equations, forexample, the motion of animal blood, liquid crystals and dilute aqueous polymer solutionsfor example. Mathematically, many authors [3, 4, 7, 10, 20, 22, 24] treated the well-posednessand large-time behaviour of solutions to system (1.1). However, the issue of global regularityof weak solutions to (1.1) remains an open problem. Therefore it is important to study theBEmail: zhangzujin361@163.com2 Z. J. Zhangregularity criterion on the some physical quantities, such as velocity, vorticity and pressure.Dong and Zhang [6] showed the following regularity conditionu ∈ L 21+r (0, T; Br∞,∞(R3)), −1 < r < 1. (1.2)Here and in what follows, B˙sp,q(R3) (resp. Bsp,q(R3)) with s ∈ R, p, q ∈ [1,∞] is the inhomo-geneous (homogeneous) Besov space, whose definition, fine properties and its utilization influid dynamical systems can be found in [1]. Later, Wang and Yuan [23] and He and Wang [17](with [17, Equation 1.10] replaced by [26]) then considered the following regularity criterionin a logarithmically improved version∫ T0‖P‖2B˙r∞,∞1+ ln(e + ‖P‖B˙r∞,∞) dt < ∞, −1 ≤ r < 1. (1.3)For interested readers, please refer to [5, 8, 11–16, 18, 25, 29].The purpose of this paper is to improve (1.2) from inhomogeneous Besov spaces to homo-geneous ones. Before stating the precise result, let us recall the weak formulation of (1.1).Definition 1.1. Let u0 ∈ L2(R3) with divu0 = 0, w0 ∈ L2(R3). A measurable R3-valuedpair (u,w) is called a weak solution to system (1.1) on (0, T), provided the following twoconditions hold,(1) (u,w) ∈ L∞(0, T; L2(R3)) ∩ L2(0, T; H1(R3));(2) (u,w) verifies (1.1) in the distributional sense.Now, our result reads as follows.Theorem 1.2. Let T > 0, 0 < r < 1 and (u,w) be a weak solution pair of system (1.1) with initialdata (u0,w0) ∈ H1(R3). Ifu ∈ L 21+r (0, T; B˙r∞,∞(R3)), (1.4)then the solution (u,w) is regular on (0, T).Remark 1.3. Due to the fact that Bsp,q(R3) = Lp(R3)∩ B˙sp,q(R3) for any s > 0 and 1 ≤ p, q ≤ ∞(see [2, Theorem 6.3.2]), we point that, in the case 0 < r < 1, (1.4) is an improvement of (1.2).Remark 1.4. The refinement u ∈ L1(0, T; B˙1∞,∞(R3)) of (1.2) in case r = 1 was already estab-lished in [5, Theorem 1.3]; while the gainu ∈ L2(0, T; B˙0∞,∞(R3)) (1.5)of (1.2) in case r = 0 can be similarly verified just as [27, Theorem 1.1].Remark 1.5. By observing the following fact from [21, Section 1.3]:s < 0, p, q ≥ 1⇒ B˙sp,q(R3) ⊂ Bsp,q(R3),we see that the condition u ∈ L 21+r (0, T; B˙r∞,∞(R3)) with −1 < r < 0 could also ensure thesmoothness of the solution. One is referred to [19, 28] for similar results of the Navier–Stokesequations.Regularity criterion for the micropolar fluid system 3Before showing Theorem 1.2 in Section 2, let us end this introduction by proving thefollowing trilinear estimate, which could have its own interest.Lemma 1.6. For f ∈ B˙r∞,∞(R3), g, h ∈ H1(R3) and any ε > 0, 0 < r < 1, k ∈ {1, 2, 3}, we have∫R3∂k f · gh dx ≤ C ‖ f ‖21+rB˙r∞,∞‖(g, h)‖2L2 + ε ‖∇(g, h)‖2L2 . (1.6)Proof. ∫R3∂k f · gh dx = −∫R3f · ∂k(gh)dx = −∫R3Λr f ·Λ−r∂k(gh)dx(Λ = (−4) 12)≤ C ‖Λr f ‖B˙0∞,∞∥∥Λ−r∂k(gh)∥∥B˙01,1 (by [1, Proposition 2.29])≤ C ‖ f ‖B˙r∞,∞ ‖gh‖B˙1−r1,1 (by [1, Lemma 2.1])≤ C ‖ f ‖B˙r∞,∞(‖g‖L2 ‖h‖B˙1−r2,1 + ‖g‖B˙1−r2,1 ‖h‖L2)(by analogues of [1, Corollary 2.54])≤ C ‖ f ‖B˙r∞,∞(‖g‖L2 ‖h‖rB˙02,∞ ‖h‖1−rB˙12,∞+ ‖g‖rB˙02,∞ ‖g‖1−rB˙12,∞‖h‖L2)(by [1, Proposition 2.22])≤ C ‖ f ‖B˙r∞,∞(‖g‖L2 ‖h‖rL2 ‖∇h‖1−rL2 + ‖g‖rL2 ‖∇g‖1−rL2 ‖h‖L2)(by [1, Proposition 2.39])≤ C ‖ f ‖B˙r∞,∞ ‖(g, h)‖1+rL2 ‖∇(g, h)‖1−rL2≤ C ‖ f ‖21+rB˙r∞,∞‖(g, h)‖2L2 + ε ‖∇(g, h)‖2L2 .2 Proof of Theorem 1.2In this section, we give the proof of Theorem 1.2. Taking the inner product of (1.1)1 with −4uin L2(R3), we obtain12ddt‖∇u‖2L2 + ‖4u‖2L2 =∫R3[(u · ∇)u] · 4udx−∫R3(∇×w) · 4udx. (2.1)On the other hand, testing (1.1)2 by −4w, we get12ddt‖∇w‖2L2 + ‖4w‖2L2 + 2 ‖∇w‖2L2 + ‖∇divw‖2L2=∫R3[(u · ∇)w] · 4w dx−∫R3(∇× u) · 4w dx.(2.2)Plugging (2.1) and (2.2) together, integrating by parts then yields12ddt‖∇(u,w)‖2L2 + ‖4(u,w)‖2L2 + 2 ‖∇w‖2L2 + ‖∇divw‖2L2=[−3∑i=1∫R3[(∂iu · ∇)u] · ∂iudx−3∑i=1∫R3[(∂iu · ∇)w] · ∂iw dx]−2∫R3(∇×w) · 4udx≡ I + J.(2.3)4 Z. J. ZhangBy Lemma 1.6,I ≤ C ‖u‖21+rB˙r∞,∞‖∇(u,w)‖2L2 +14‖4(u,w)‖2L2 . (2.4)Meanwhile,J ≤ 2 ‖∇×w‖L2 ‖4u‖L2 ≤ 2 ‖∇w‖2L2 +12‖4u‖2L2 . (2.5)Putting (2.4) and (2.5) into (2.3), we deduce12ddt‖∇(u,w)‖2L2 +14‖4(u,w)‖2L2 + ‖∇divw‖2L2 ≤ C ‖u‖21+rB˙r∞,∞‖∇(u,w)‖2L2 .Applying Gronwall’s inequality with the fact (1.4), we find‖∇(u,w)(t)‖2L2 ≤ ‖∇(u0,w0)‖2L2 · exp[∫ T0‖u(τ)‖21+rB˙r∞,∞dτ]< ∞, ∀ 0 ≤ t < T.By Sobolev’s inequality,u ∈ L∞(0, T; L6(R3)) ⊂ L4(0, T; L6(R3)).By [5, Theorem 1.1], we complete the proof of Theorem 1.2.AcknowledgementsThis work is supported by the Natural Science Foundation of Jiangxi (grant no.20151BAB201010), the National Natural Science Foundation of China (grant nos. 11501125,11361004) and the Supporting the Development for Local Colleges and Universities Founda-tion of China – Applied Mathematics Innovative Team Building.References[1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differentialequations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Springer, Heidel-berg, 2011. MR2768550; url[2] J. Bergh, J. 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A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces

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