This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $ (0, T] $ provided that either the norm $ \left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))} $ with $ \frac{2}{\alpha }+ \frac{3}{\beta } = 2 $ and $ \frac{3}{2} < \beta < \infty $ or $ \left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))} $ with $ \frac{2}{\alpha }+\frac{3}{\beta } = 3 $ and $ 1 < \beta < \infty $ is sufficiently small.
Citation: Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa. Regularity results for solutions of micropolar fluid equations in terms of the pressure[J]. AIMS Mathematics, 2023, 8(9): 21208-21220. doi: 10.3934/math.20231081
This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $ (0, T] $ provided that either the norm $ \left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))} $ with $ \frac{2}{\alpha }+ \frac{3}{\beta } = 2 $ and $ \frac{3}{2} < \beta < \infty $ or $ \left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))} $ with $ \frac{2}{\alpha }+\frac{3}{\beta } = 3 $ and $ 1 < \beta < \infty $ is sufficiently small.
| [1] |
H. Beirão da Veiga, A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 99–106. http://doi.org/10.1007/PL00000949 doi: 10.1007/PL00000949
|
| [2] | H. Beirão da Veiga, Concerning the regularity of the solutions to the Navier-Stokes equations via the truncation method Ⅱ, In: Équations aux dérivées partielles et applications, Paris: Gauthier-Villars, Éd. Sci. Méd. Elsevier, 1998,127–138. |
| [3] | J. Bergh, J. Löfström, Interpolation spaces: An introduction, Berlin, Heidelberg: Springer, 1976. http://doi.org/10.1007/978-3-642-66451-9 |
| [4] |
L. C. Berselli, G. P. Galdi, Regularity criteria involving the pressure for the weak solutions of the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585–3595. http://doi.org/10.1090/S0002-9939-02-06697-2 doi: 10.1090/S0002-9939-02-06697-2
|
| [5] |
S. Bosia, V. Pata, J. C. Robinson, A weak-$L^{p}$ Prodi-Serrin type regularity criterion for the Navier-Stokes equations, J. Math. Fluid Mech., 16 (2014), 721–725. http://doi.org/10.1007/s00021-014-0182-5 doi: 10.1007/s00021-014-0182-5
|
| [6] |
J. Chen, Z. M. Chen, B. Q. Dong, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619–1635. http://doi.org/10.1088/0951-7715/20/7/005 doi: 10.1088/0951-7715/20/7/005
|
| [7] |
Q. Chen, C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equ., 252 (2012), 2698–2724. https://doi.org/10.1016/j.jde.2011.09.035 doi: 10.1016/j.jde.2011.09.035
|
| [8] |
Z. M. Chen, W. G. Price, Decay estimates of linearized micropolar fluid flows in $\mathbb{R}^{3}$ space with applications to $L^{3}$ -strong solutions, Int. J. Eng. Sci., 44 (2006), 859–873. https://doi.org/10.1016/j.ijengsci.2006.06.003 doi: 10.1016/j.ijengsci.2006.06.003
|
| [9] |
B. Q. Dong, Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525. http://doi.org/10.1063/1.3245862 doi: 10.1063/1.3245862
|
| [10] |
B. Q. Dong, W. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal., 73 (2010), 2334–2341. https://doi.org/10.1016/j.na.2010.06.029 doi: 10.1016/j.na.2010.06.029
|
| [11] |
B. Q. Dong, Y. Jia, Z. M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Method. Appl. Sci., 34 (2011), 595–606. http://doi.org/10.1002/mma.1383 doi: 10.1002/mma.1383
|
| [12] | A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. |
| [13] |
S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space, Nonlinear Anal. Real, 12 (2011), 2142–2150. https://doi.org/10.1016/j.nonrwa.2010.12.028 doi: 10.1016/j.nonrwa.2010.12.028
|
| [14] |
S. Gala, M. A. Ragusa, A regularity criterion for 3D micropolar fluid flows in terms of one partial derivative of the velocity, Ann. Pol. Math., 116 (2016), 217–228. http://doi.org/10.4064/ap3829-11-2015 doi: 10.4064/ap3829-11-2015
|
| [15] |
S. Gala, J. Yan, Two regularity criteria via the logarithmic of the weak solutions to the micropolar fluid equations, J. Part. Diff. Eq., 25 (2012), 32–40. http://doi.org/10.4208/jpde.v25.n1.3 doi: 10.4208/jpde.v25.n1.3
|
| [16] |
S. Gala, A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure, Math. Method. Appl. Sci., 34 (2011), 1945–1953. http://doi.org/10.1002/mma.1488 doi: 10.1002/mma.1488
|
| [17] |
G. P. Galdi, S. Rionero, A note on the existence and uniqueness of solutions of micropolar fluid equations, Int. J. Eng. Sci., 15 (1977), 105–108. https://doi.org/10.1016/0020-7225(77)90025-8 doi: 10.1016/0020-7225(77)90025-8
|
| [18] | L. Grafakos, Classical Fourier analysis, 2 Eds., New York: Springer, 2008. http://doi.org/10.1007/978-0-387-09432-8 |
| [19] |
X. Ji, Y. Wang, W. Wei, New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier-Stokes equations, J. Math. Fluid Mech., 22 (2020), 13. http://doi.org/10.1007/s00021-019-0476-8 doi: 10.1007/s00021-019-0476-8
|
| [20] |
Y. Jia, W. Zhang, B. Q. Dong, Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure, Appl. Math. Lett., 24 (2011), 199–203. https://doi.org/10.1016/j.aml.2010.09.003 doi: 10.1016/j.aml.2010.09.003
|
| [21] |
Y. Jia, J. Zhang, B. Q. Dong, Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the pressure, Abstr. Appl. Anal., 2012 (2012), 395420. http://doi.org/10.1155/2012/395420 doi: 10.1155/2012/395420
|
| [22] |
H. Kozono, M. Yamazaki, Exterior problem from the stationary Navier-Stokes equations in the Lorentz space, Math. Ann., 310 (1998), 279–305. http://doi.org/10.1007/s002080050149 doi: 10.1007/s002080050149
|
| [23] | G. Łukaszewicz, Micropolar fluids: Theory and applications, Boston, MA: Birkhäuser, 1999. http://doi.org/10.1007/978-1-4612-0641-5 |
| [24] |
M. Loayza, M. A. Rojas-Medar, A weak-$L^{p}$ Prodi-Serrin type regularity criterion for the micropolar fluid equations, J. Math. Phys., 57 (2016), 021512. http://doi.org/10.1063/1.4942047 doi: 10.1063/1.4942047
|
| [25] | J. Malý, Advanced theory of differentiation-Lorentz spaces, Lect. Notes, 2003, 8. |
| [26] |
B. Pineau, X. Yu, A new Prodi-Serrin type regularity criterion in velocity directions, J. Math. Fluid Mech., 20 (2018), 1737–1744. http://doi.org/10.1007/s00021-018-0388-z doi: 10.1007/s00021-018-0388-z
|
| [27] |
B. Pineau, X. Yu, On Prodi-Serrin type conditions for the 3D Navier-Stokes equations, Nonlinear Anal., 190 (2020), 111612. https://doi.org/10.1016/j.na.2019.111612 doi: 10.1016/j.na.2019.111612
|
| [28] |
M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301–319. http://doi.org/10.1002/mana.19971880116 doi: 10.1002/mana.19971880116
|
| [29] |
T. Suzuki, Regularity criteria of weak solutions in terms of the pressure in Lorentz spaces to the Navier-Stokes equations, J. Math. Fluid Mech., 14 (2012), 653–660. http://doi.org/10.1007/s00021-012-0098-x doi: 10.1007/s00021-012-0098-x
|
| [30] |
T. Suzuki, A remark on the regularity of weak solutions to the Navier-Stokes equations in terms of the pressure in Lorentz spaces, Nonlinear Anal., 75 (2012), 3849–3853. https://doi.org/10.1016/j.na.2012.02.006 doi: 10.1016/j.na.2012.02.006
|
| [31] | H. Triebel, Theory of function spaces, Basel: Birkhäuser, 1983. http://doi.org/10.1007/978-3-0346-0416-1 |
| [32] |
Y. Wang, H. Zhao, Logarithmically improved blow up criterion for smooths solution to the 3D micropolar fluid equations, J. Appl. Math., 2012 (2012), 541203. http://doi.org/10.1155/2012/541203 doi: 10.1155/2012/541203
|
| [33] |
N. Yamaguchi, Existence of global strong solution to the micropolar fluid equations, Math. Method. Appl. Sci., 28 (2005), 1507–1526. http://doi.org/10.1002/mma.617 doi: 10.1002/mma.617
|
| [34] |
B. Yuan, On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025–2036. http://doi.org/10.1090/S0002-9939-10-10232-9 doi: 10.1090/S0002-9939-10-10232-9
|
| [35] |
Y. Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann., 328 (2004), 173–192. http://doi.org/10.1007/s00208-003-0478-x doi: 10.1007/s00208-003-0478-x
|
| [36] |
Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb{R}^{3}$, Proc. Amer. Math. Soc., 134 (2006), 149–156. http://doi.org/10.1090/S0002-9939-05-08312-7 doi: 10.1090/S0002-9939-05-08312-7
|
| [37] |
Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb{R}^{3}$, Z. angew. Math. Phys., 57 (2006), 384–392. http://doi.org/10.1007/s00033-005-0021-x doi: 10.1007/s00033-005-0021-x
|
| [38] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Nonlin. Mech., 41 (2006), 1174–1180. https://doi.org/10.1016/j.ijnonlinmec.2006.12.001 doi: 10.1016/j.ijnonlinmec.2006.12.001
|
Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa. Regularity results for solutions of micropolar fluid equations in terms of the pressure[J]. AIMS Mathematics, 2023, 8(9): 21208-21220. doi: 10.3934/math.20231081
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