In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana

Gala, Sadek

Ragusa, Maria

Théra, Michel

English

Federation ResearchOnline

  Page 1 of 1 Federation University ResearchOnline https://researchonline.federation.edu.au Copyright Notice  This is the published version of the following article:   Gala, Sadek, Ragusa, Maria Alessandra, & Théra, Michel. (2021). A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure. Bollettino Della Unione Matematica Italiana (2008), 14(2), 331–337.   Copyright @ Bolletino dell Unione Matematica Italiana  This is the published version of the work.  It is posted here with the permission of the publisher for your personal use.  No further use or distribution is permitted.  https://doi.org/10.1007/s40574-020-00257-2     CRICOS 00103D RTO 4909   See this record in Federation ResearchOnline at: https://researchonline.federation.edu.au/vital/access/manager/Index  Bollettino dell’Unione Matematica Italiana (2021) 14:331–337https://doi.org/10.1007/s40574-020-00257-2A new regularity criterion of weak solutions to the 3Dmicropolar fluid flows in terms of the pressureSadek Gala1,2 ·Maria Alessandra Ragusa2,3 ·Michel Théra4,5Received: 16 April 2020 / Accepted: 8 September 2020 / Published online: 21 September 2020© Unione Matematica Italiana 2020AbstractIn this study,we establish a new regularity criterion ofweak solutions to the three-dimensionalmicropolar fluid flows by imposing a critical growth condition on the pressure field.Keywords Micropolar fluid flows · Weak solutions · Pressure criterion · Besov spacesMathematics Subject Classification 35Q35 · 35B65 · 76D051 Introduction and themain resultIn this paper we consider the following Cauchy problem for the incompressible micropolarfluid equations in R3:⎧⎪⎪⎨⎪⎪⎩∂t u − u + (u · ∇) u + ∇π − ∇ × ω = 0,∂tω − ω − ∇∇ · ω + 2ω + (u · ∇)ω − ∇ × u = 0,∇ · u = 0,u(x, 0) = u0(x), ω(x, 0) = ω0(x),(1.1)where u = u(x, t) ∈ R3,ω = ω(x, t) ∈ R3 andπ = π (x, t) denote the unknown velocity ofthe fluid, the micro-rotational velocity of the fluid particles and the unknown scalar pressureof the fluid at the point (x, t) ∈ R3 × (0, T ), respectively, while u0, ω0 are given initial datasatisfying ∇ · u = 0 in the sense of distributions.This model for micropolar fluid flows proposed by Eringen [6] enables to consider somephysical phenomena that cannot be treated by the classical Navier–Stokes equations for theB Maria Alessandra Ragusamaragusa@dmi.unict.it1 University of Mostaganem, P. O. Box 270, 27000 Mostaganem, Algeria2 Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125Catania, Italy3 RUDN University, 6 Miklukho-Maklay St, Moscow 117198, Russia4 XLIM UMR-CNRS 7252 Université de Limoges, Limoges, France5 Centre for Informatics and Applied Optimisation, Federation University Australia, Ballarat, Australia123332 S. Gala et al.viscous incompressible fluids, such as for example, themotion of animal blood,muddy fluids,liquid crystals and dilute aqueous polymer solutions, colloidal suspensions, etc.When the micro-rotation effects are neglected or ω = 0, (1.1) reduces to the incompress-ibleNavier–Stokes equations, and it iswell known that the regularity criteria forweak solutionfor the fluid dynamical models attracts more and more attention. There are many importantresults on the velocity or vorticity or pressure blow-up criteria for Navier–Stokes equations,micropolar fluid equations and MHD equations and so on (see e.g., [1–4,7–10,14,16,17] andthe references therein).As for the pressure criterion, let us first recall some interesting results on pressure regularityof Navier–Stokes equations. In Ref. [13], He and Gala proved regularity of weak solutionsunder the condition ∫ T0‖π(·, t)‖2Ḃ−1∞,∞dt < ∞. (1.2)Here and thereafter, Ḃ−1∞,∞ stands for the homogeneous Besov space, (for the definition seee.g. [12,13]). Later on, Guo and Gala [12] refined the condition (1.2) to∫ T0‖π(·, t)‖2Ḃ−1∞,∞1 + log(e + ‖π(·, t)‖Ḃ−1∞,∞) dt < ∞. (1.3)Motivated by the paper ofGuo andGala [12], the aimof this paper is to give a new regularitycriterion for weak solutions to the 3Dmicropolar fluid flows in terms of the pressure in criticalBesov spaces. More precisely, our main result reads as follows.Theorem 1.1 Let T > 0 and (u0, ω0) ∈ L2(R3) ∩ L4(R3) with ∇ · u0 = 0 in the sense ofdistributions. Assume that (u, ω) is a weak solution of the 3D micropolar fluid flows (1.1) on(0, T ). If the pressure π satisfies the following condition :∫ T0‖π(·, t)‖2Ḃ−1∞,∞(e + log(e + ‖π(·, t)‖Ḃ−1∞,∞))log(e + log(e + ‖π(·, t)‖Ḃ−1∞,∞)) dt < ∞, (1.4)then (u, ω) is regular on (0, T ], i.e., (u, ω) ∈ C∞(R3 × (0, T ]).Remark 1.1 This result provides a new information concerning the question of the regularityof weak solutions of the micropolar fluid equations and extends those of [13] and [12]. Inparticular, the double-logarithm estimate (1.4) is sharper than any other results [12,13].Before stating our result, let us recall what we mean by a weak solution.Definition 1.2 ([14]) Let (u0, ω0) ∈ L2(R3)and ∇ · u0 = 0. A measurable function(u(x, t), ω(x, t)) is called a weak solution to the 3D micropolar flows equations (1.1) on(0, T ) if (u, ω) satisfies the following properties :(1) (u, ω) ∈ L∞ ((0, T ) ; L2 (R3)) ∩ L2 ((0, T ) ; H1 (R3)) for all T > 0;(2) (u(x, t), ω(x, t)) verifies (1.1) in the sense of distribution;(3) The following energy inequality holds :‖u(·, t)‖2L2 + ‖ω(·, t)‖2L2 + 2∫ t0(‖∇u(·, τ )‖2L2 + ‖∇ω(·, τ )‖2L2 + ‖∇ · ω(·, τ )‖2L2)dτ≤ ‖u0‖2L2 + ‖ω0‖2L2 ,for all 0 ≤ t ≤ T .123A new regularity criterion of weak... 333By a strong solution we mean a weak solution (u, ω) such that(u, ω) ∈ L∞ ((0, T ) ; H1 (R3)) ∩ L2 ((0, T ) ; H2 (R3)) .It is well known that strong solutions are regular (say, classical) and unique in the class ofweak solutions.In order to prove Theorem 1.1, we first establish some estimates between pressure andvelocity. Taking div and ∇div to both sides of the micropolar fluid flows for smooth solution(u, π), separately, we get the well-known pressure-velocity relation in R3, given byπ = (−)−13∑i, j=1∂2∂xi∂x j(uiu j ) and ∇π = (−)−13∑i, j=1∂2∂xi∂x j(∇(uiu j )).Then, the Calderón-Zygmund inequality implies that for any 1 < α < +∞ :‖π‖Lα ≤ C‖u‖2L2α and ‖∇π‖Lα ≤ C‖ |u| ∇u‖Lα . (1.5)2 Proof of Theorem 1.1Now we are in the position to prove Theorem 1.1. Firstly, by means of the local existenceresult, which is similar to the one of the Navier–Stokes equations (refer to Giga [11], seealso Dong et al. [5]), and the standard local solution extension technique, Eq. (1.1) with(u0, ω0) ∈ L2(R3) ∩ L4(R3) admits a unique L4-strong solution (u, ω) on a maximal timeinterval. For the notation simplicity, we may suppose that the maximal time interval is [0, T ).Thus, in order to prove Theorems 1.1, it remains to show thatlimt→T (‖u(t)‖4 + ‖w(t)‖4) < ∞.This will lead to a contradiction to the estimates to be derived below.Proof Before going into the proof, we recall the following inequality established in Ref. [15](see also [12]) :‖ f ‖2L4 ≤ C ‖ f ‖ ·B−1∞,∞‖∇ f ‖L2 . (2.1)Testing (1.1)1 by u |u|2 and using ( 1.1)3, we get14ddt‖u‖4L4 + ‖|u| ∇u‖2L2 +12∥∥∇ |u|2∥∥2L2=∫R3(∇ × ω) · u |u|2 dx −∫R3(u · ∇π)|u|2dx=∫R3ω[∇ × (u |u|2)]dx +∫R3πu · ∇|u|2dx (2.2)Testing (1.1)2 by ω |ω|2 and using (1.1)3, we infer that14ddt‖ω‖4L4 + ‖|ω| ∇ · ω‖2L2 +12∥∥∇ |ω|2∥∥2L2=∫R3(∇ × u) · ω |ω|2 dx − 2 ‖ω‖4L4=∫R3u[∇ × (ω |ω|2)dx − 2 ‖ω‖4L4 , (2.3)123334 S. Gala et al.wherewe have used the following identities due to the divergence free property of the velocityfield u : ∫R3(u · ∇)u · u |u|2 dx = 0 =∫R3(u · ∇)ω · ω |ω|2 dx .Summing up (2.2) and (2.3), it follows that14ddt(‖u‖4L4 + ‖ω‖4L4) + ‖|u| ∇u‖2L2 +12∥∥∇ |u|2∥∥2L2 + ‖|ω| ∇ · ω‖2L2 +12∥∥∇ |ω|2∥∥2L2=∫R3ω[∇ × (u |u|2)]dx +∫R3u[∇ × (ω |ω|2)dx − 2 ‖ω‖4L4 +∫R3πu · ∇|u|2dx .(2.4)Using the Hölder inequality and the Young inequality and integration by parts, we derive theestimate of the first three terms on the right-hand side of (2.4) as follows:∫R3ω[∇ × (u |u|2)]dx +∫R3u[∇ × (ω |ω|2)dx − 2 ‖ω‖4L4≤ ‖u‖L4 ‖ω‖L4 (‖|u||∇u‖L2 + ‖|ω||∇ω‖L2) − 2 ‖ω‖4L4≤ ‖|u||∇u‖2L2 + ‖|ω||∇ω‖2L2 + C ‖u‖4L4 . (2.5)To estimate the last term of the right-hand side of (2.4), we have after integrating by partsand employing the Hölder inequality and the Young inequality,∫R3πu · ∇|u|2dx ≤∫R3|π | |u| ∣∣∇|u|2∣∣ dx≤ C ‖π‖2L4 ‖u‖2L4 +14∥∥∇ |u|2∥∥2L2≤ C ‖π‖Ḃ−1∞,∞ ‖∇π‖L2 ‖u‖2L4 +14∥∥∇ |u|2∥∥2L2≤ C ‖π‖Ḃ−1∞,∞ ‖u‖2L4 ‖|u| ∇u‖L2 +14∥∥∇ |u|2∥∥2L2≤ C ‖π‖2Ḃ−1∞,∞‖u‖4L4 +12‖|u||∇u‖2L2 +14∥∥∇ |u|2∥∥2L2 . (2.6)and henceddt(‖u(t)‖4L4 + ‖ω(t)‖4L4) + ‖|u| ∇u‖2L2 +∥∥∇ |u|2∥∥2L2 + ‖|ω| ∇ · ω‖2L2 + ‖|ω||∇ω‖2L2≤ C(1 + ‖π‖2Ḃ−1∞,∞) ‖u‖4L4Taking the Gronwall inequality into consideration, one shows that‖u(·, t)‖4L4 + ‖ω(·, t)‖4L4≤ (‖u0‖4L4 + ‖ω0‖4L4 ) exp(C∫ t0(1 + ‖π(·, τ )‖2Ḃ−1∞,∞)dτ). (2.7)Taking the inner product of (1.1)1 with −u, (1.1)2 with −ω in L2(R3), and addingthe resulting equations together, we obtain12ddt(‖∇u(·, t)‖2L2 + ‖∇ω(·, t)‖2L2 ) + ‖u‖2L2 + ‖ω‖2L2 + ‖∇∇ · ω‖2L2 + 2 ‖∇ω‖2L2=∫R3(u · ∇)u · udx −∫R3(∇ × ω) · udx +∫R3(u · ∇)ω · ωdx −∫R3(∇ × u) · ωdx123A new regularity criterion of weak... 335≤ ‖u‖L4 ‖∇u‖L4 ‖u‖L2 + ‖∇ω‖L2 ‖u‖L2 + ‖u‖L4 ‖∇ω‖L4 ‖ω‖L2 + ‖∇u‖L2 ‖ω‖L2≤ C ‖u‖65L4‖u‖95L2+ ‖ω‖12L2‖ω‖12L2‖u‖L2 + ‖u‖L4 ‖ω‖15L4‖ω‖95L2+ ‖u‖12L2‖u‖12L2‖ω‖L2≤ C ‖u‖12L4 + C ‖ω‖2L2 + C(‖u‖12L4 + ‖ω‖12L4 ) + C ‖u‖2L2 +12(‖u‖2L2 + ‖ω‖2L2 )≤ C(1 + ‖u‖12L4 + ‖ω‖12L4 ) +12(‖u‖2L2 + ‖ω‖2L2 ),where we have used the Gagliardo–Nirenberg inequalities:‖∇u‖L4 ≤ C ‖u‖15L4‖u‖45L2and ‖∇u‖L2 ≤ C ‖u‖12L2‖u‖12L2.This yieldsddt(‖∇u(·, t)‖2L2 + ‖∇ω(·, t)‖2L2) + ‖u‖2L2 + ‖ω‖2L2 + ‖∇∇ · ω‖2L2 + 2 ‖∇ω‖2L2≤ C(1 + ‖u‖12L4 + ‖ω‖12L4).Integrating the above inequality over (0, t), we have‖∇u(t)‖2L2 + ‖∇ω(t)‖2L2 +∫ t0(‖u(τ )‖2L2 + ‖ω(τ)‖2L2 + ‖∇∇ · ω(τ)‖2L2 + 2 ‖∇ω(τ)‖2L2 )dτ≤ ‖∇u0‖2L2 + ‖∇ω0‖2L2 + C∫ t0(1 + ‖u(τ )‖12L4 + ‖ω(τ)‖12L4 )dτ. (2.8)On the other hand, by a Sobolev embedding theorem Ḣ1(R3) ↪→ L6(R3), (2.8) and (1.5),we obtain thate + ‖π(·, t)‖L3 ≤ e + C ‖u(·, t)‖2L6 ≤ e + C ‖∇u(·, t)‖2L2≤ e + C(‖∇u0‖2L2 + ‖∇ω0‖2L2) + C∫ t0(1 + ‖u(·, τ )‖12L4)dτ≤ e + C(‖∇u0‖2L2 + ‖∇ω0‖2L2) + C(e + t) sup0≤τ≤t(1 + ‖u(·, τ )‖12L4)≤ C (e + ‖∇u0‖2L2 + ‖∇ω0‖2L2)(e + t) sup0≤τ≤t(1 + ‖u(·, τ )‖12L4)≤ C0(e + t) exp(C∫ t0(1 + ‖π(·, τ )‖2Ḃ−1∞,∞)dτ), (2.9)where the constant C0 = C(e, ‖∇u0‖L2 , ‖∇ω0‖L2 , ‖u0‖L4 , ‖ω0‖L4). Using the fact thatL3(R3) ⊂ Ḃ−1∞,∞(R3), it follows thate + ‖π(·, t)‖Ḃ−1∞,∞ ≤ C(e + t) exp(C∫ t0(1 + ‖π(·, τ )‖2Ḃ−1∞,∞)dτ)(2.10)Now, taking the logarithm on both sides of (2.10), we can conclude thatlog(e + ‖π(·, t)‖Ḃ−1∞,∞) ≤ log(C(e + t)) + C∫ t0(1 + ‖π(·, τ )‖2Ḃ−1∞,∞)dτ. (2.11)For simplicity, letZ(t) = log(e + ‖π(·, t)‖Ḃ−1∞,∞),E(t) = log(C(e + t)) + C∫ t0(1 + ‖π(·, τ )‖2Ḃ−1∞,∞)dτ, (2.12)123336 S. Gala et al.with E(0) = log(Ce). Then, the above inequality (2.11) implies that0 < Z(t) ≤ E(t)and we easily get(e + Z(t)) log(e + Z(t)) ≤ (e + E(t)) log(e + E(t)).On the other hand, we haveddtlog(e + E(t)) = 1e + E(t)(1e + t + C(1 + ‖π(·, t)‖2Ḃ−1∞,∞))≤ 1e2+ C1 + ‖π(·, t)‖2Ḃ−1∞,∞e + E(t)= 1e2+ C1 + ‖π(·, t)‖2Ḃ−1∞,∞(e + E(t)) ln(e + E(t)) log(e + E(t))≤ 1e2+ C1 + ‖π(·, t)‖2Ḃ−1∞,∞(e + Z(t)) ln(e + Z(t)) log(e + E(t))Applying the Gronwall inequality to log(e + E(t)), we findlog(e + E(t))≤ log(e + E(0)) exp⎛⎝Te2+ C∫ t01 + ‖π(·, τ )‖2Ḃ−1∞,∞(e + Z(τ )) log(e + Z(τ ))dτ⎞⎠ ,which yieldse + E(t) ≤ (e + E(0))exp⎛⎝ Te2+C ∫ t01+‖π(·,τ )‖2Ḃ−1∞,∞(e+Z(τ )) log(e+Z(τ )) dτ⎞⎠and from (2.12), we deduce thatt∫0‖π(·, τ )‖2Ḃ−1∞,∞dτ ≤ (e + E(0))exp⎛⎝ Te2+ 1C∫ t01+‖π(·,τ )‖2Ḃ−1∞,∞(e+Z(τ )) log(e+Z(τ )) dτ⎞⎠< ∞. (2.13)Hence by virtue of (2.7), (2.8) and (2.13), we conclude that(u, ω) ∈ L∞ ((0, T ) ; H1(R3)) ∩ L2 ((0, T ) ; H2(R3)) ,which completes the proof of Theorem 1.1. Acknowledgements The publication was prepared with the support of the “RUDN University Program 5-100”. Part of the work was carried out while S. Gala was a long-term visitor at the University of Catania.The hospitality of Catania University is graciously acknowledged. Research of M. Théra is supported bythe Australian Research Council (ARC) grant DP160100854 and benefited from the support of the FMJHProgram PGMO and from the support of EDF. The funding was provided by Gruppo Nazionale per l‘AnalisiMatematica, la Probabilità e le loro Applicazioni (2019).123A new regularity criterion of weak... 337References1. Chen, J., Chen, Z.-M., Dong, B.-Q.: Uniform attractors of non-homogeneous micropolar fluid flows innon-smooth domains. Nonlinearity 20, 1619–1635 (2007)2. Chen, Z.-M., Price,W.: Decay estimates of linearizedmicropolar fluid flows inR3 space with applicationsto L3 -strong solutions. Int. J. Eng. Sci. 44, 859–873 (2006)3. Dong, B.-Q., Chen, Z.-M.: Regularity criteria of weak solutions to the three-dimensional micropolarflows. J. Math. Phys. 50, 103525 (2009)4. Dong, B.-Q., Zhang, W.: On the regularity criterion for the 3D micropolar fluid flows in Besov spaces.Nonlinear Anal. 73, 2334–2341 (2010)5. Dong, B.-Q., Jia, Y., Chen, Z.-M.: Pressure regularity criteria of the three-dimensional micropolar fluidflows. Math. Method Appl. Sci. 34, 595–606 (2011)6. Eringen, A.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)7. Gala, S.: On regularity criteria for the three-dimensional micropolar fluid equations in the criticalMorrey–Campanato space. Nonlinear Anal. 32, 2142–2150 (2011)8. Gala, S.: A remark on the logarithmically improved regularity criterion for the micropolar fluid equationsin terms of the pressure. Math. Method Appl. Sci. 34, 1945–1953 (2011)9. Gala, S., Yan, J.: Two regularity criteria via the logarithm of the weak solutions to the micropolar fluidequations. J. Partial Differ. Eq. 25, 32–40 (2012)10. Galdi, G., Rionero, S.: A note on the existence and uniqueness of solutions of micropolar fluid equations.Int. J. Eng. Sci. 14, 105–108 (1977)11. Giga, Y.: Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of theNavier–Stokes equations. J. Differ. Eq. 62, 186–212 (1986)12. Guo, Z., Gala, S.: Remarks on logarithmical regularity criteria for the Navier–Stokes equations. J. Math.Phys. 52, 063503 (2011)13. He, X., Gala, S.: Regularity criterion for weak solutions to the Navier–Stokes equations in terms of thepressure in the class L2(0, T ; B−1∞,∞(R3)). Nonlinear Anal. 12, 3602–3607 (2011)14. Łukaszewicz, G.: Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science,Engineering and Technology. Birkhä user, Boston (1999)15. Meyer,Y., Gerard,P., Oru,F., Inégalités de Sobolev pr écisées : Séminaire Équations aux dérivées partielles(Polytechnique) (1996–1997), Exp. No. 416. Yuan, B.: On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space.Proc. Am. Math. Soc. 138, 2025–2036 (2010)17. Zhang, W., Jia, Y., Dong, B.: Remarks on the regularity criterion of the 3Dmicropolar fluid flows in termsof the pressure. Appl. Math. Lett. 24, 199–203 (2011)Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps andinstitutional affiliations.123

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

https://researchonline.federation.edu.au/vital/access/services/Download/vital:15106/SOURCE1

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

Abstract

Similar works

Full text

Available Versions

Federation ResearchOnline