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Global well-posedness and large-time decay for the 2D micropolar equations. (English) Zbl 1361.35143
The authors are concerned with the $$2D$$ micropolar equations with only angular velocity dissipation,
$$\partial_tu + \kappa u-2\kappa \nabla \times w + \nabla \pi + u\cdot \nabla u =0,$$
$$\nabla \cdot u =0,$$
$$\partial_t w-\gamma \Delta w+4\kappa w-2\kappa \nabla \times u+u\cdot \nabla w=0,$$
where $$u=(u_1,u_2)$$ and $$w$$ are the usual unknown functions. The first result (Theorem 1.1) establishes the global existence and uniqueness of the solutions to the above PDE system. The next result (Theorem 1.2) establishes explicit time decay rates for the solutions of the same PDE system, without the term $$4\kappa w$$, under the condition $$\gamma > 4\kappa$$. Theorem 1.3 is an additional result on the decay rate of $$\| w(\cdot , t)\|_{L^2}$$ as $$t\rightarrow \infty$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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##### References:
 [1] Amann, H., Maximal regularity for nonautonoumous evolution equations, Adv. Nonlinear Stud., 4, 417-430, (2004) · Zbl 1072.35103 [2] Chen, Q.; Miao, C., Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252, 2698-2724, (2012) · Zbl 1234.35193 [3] Chen, Z.; Price, W. G., Decay estimates of linearized micropolar fluid flows in $$\mathbb{R}^3$$ space with applications to $$L^3$$-strong solutions, Internat. J. Engrg. Sci., 44, 859-873, (2006) · Zbl 1213.76012 [4] Cowin, S. C., Polar fluids, Phys. Fluids, 11, 1919-1927, (1968) · Zbl 0179.56002 [5] Dong, B.; Chen, Z., Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50, 103525, (2009) · Zbl 1283.76016 [6] Dong, B.; Chen, Z., Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst., 23, 765-784, (2009) · Zbl 1170.35336 [7] Dong, B.; Zhang, Z., Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249, 200-213, (2010) · Zbl 1402.35220 [8] Erdogan, M. E., Polar effects in the apparent viscosity of suspension, Rheol. Acta, 9, 434-438, (1970) [9] Eringen, A. C., Theory of micropolar fluids, J. Math. Mech., 16, 1-18, (1966) · Zbl 0145.21302 [10] Eringen, A. C., Micropolar fluids with stretch, Internat. J. Engrg. Sci., 7, 115-127, (1969) · Zbl 0164.27507 [11] Guo, Y.; Wang, Y., Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37, 2165-2208, (2012) · Zbl 1258.35157 [12] Galdi, G.; Rionero, S., A note on the existence and uniqueness of solutions of micropolar fluid equations, Internat. J. Engrg. Sci., 14, 105-108, (1977) · Zbl 0351.76006 [13] Jiu, Q.; Miao, C.; Wu, J.; Zhang, Z., The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46, 3426-3454, (2014) · Zbl 1319.35193 [14] Kato, T., Strong $$L^p$$ solutions of the Navier-Stokes equations in $$\mathbb{R}^m$$, with application to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [15] Lemarie-Rieusset, P. G., Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics Series, (2002), CRC Press · Zbl 1034.35093 [16] Lukaszewicz, G., Micropolar fluids. theory and applications, Modeling and Simulation in Science, Engineering and Technology, (1999), Birkhäuser Boston · Zbl 0923.76003 [17] Popel, S.; Regirer, A.; Usick, P., A continuum model of blood flow, Biorheology, 11, 427-437, (1974) [18] Schonbek, M. E., $$L^2$$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031 [19] Stokes, V. K., Theories of fluids with microstructure, (1984), Springer New York [20] Xue, L., Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34, 1760-1777, (2011) · Zbl 1222.76027 [21] Yamazaki, K., Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35, 2193-2207, (2015) · Zbl 1308.35232 [22] Yuan, B., Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci., 30B, 1469-1480, (2010) · Zbl 1240.35421
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