Lu, Ming; Wang, Qianying Regularity to Boussinesq equations with partial viscosity and large data in three-dimensional periodic thin domain. (English) Zbl 1373.35247 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 164, 27-37 (2017). Summary: This paper is devoted to study the partial viscosity Boussinesq equations on a 3D thin periodic domain. The global well-posedness of strong solution with the initial data \((u_0,\theta_0)\in H_{per}^2(\Omega_\epsilon)\times H_{\mathrm{per}}^1(\Omega_\epsilon)\) is established, where \(\Omega_\epsilon=[0,L_1]\times [0,L_2]\times [0,\epsilon]\) is with periodic boundary conditions. Cited in 1 Document MSC: 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics Keywords:global well-posedness; partial viscosity; three-dimensional thin periodic domain PDFBibTeX XMLCite \textit{M. Lu} and \textit{Q. Wang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 164, 27--37 (2017; Zbl 1373.35247) Full Text: DOI References: [1] Adhikari, D.; Cao, C.; Wu, J., Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations, 251, 6, 1637-1655 (2011) · Zbl 1232.35111 [2] Avrin, J. D., Large-eigenvalue global existence and regularity results for the Navier-Stokes equations, J. Differential Equations, 127, 365-390 (1996) · Zbl 0863.35075 [3] Chae, D.; Wu, J., The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230, 4-6, 1618-1645 (2011) · Zbl 1248.35156 [4] Choi, K.; Kiselev, A.; Yao, Y., Finite time blow up for a 1D model of 2D Boussinesq system, Comm. Math. Phys., 334, 1667-1679 (2015) · Zbl 1309.35072 [5] Constantin, P.; Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 5, 1289-1321 (2011) · Zbl 1256.35078 [6] Danchin, R.; Paicu, M., Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Physica D, 237, 1444-1460 (2008) · Zbl 1143.76432 [7] Danchin, R.; Paicu, M., Le théorème de Leary et le théorème de Fujita-Kato pour le systéme de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 261-309 (2008) · Zbl 1162.35063 [8] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. Math. Phys., 290, 1, 1-14 (2009) · Zbl 1186.35157 [9] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem. I, Arch. Ration. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 [10] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12, 4, 461-480 (2007) · Zbl 1154.35073 [11] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1, 1-12 (2005) · Zbl 1274.76185 [12] Iftimie, D., The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France, 127, 473-517 (1999) · Zbl 0946.35059 [13] Iftimie, D.; Raugel, G., Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations, 169, 281-331 (2001) · Zbl 0972.35085 [14] Iftimie, D.; Raugel, G., The Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J., 56, 3, 1083-1156 (2007) · Zbl 1129.35056 [15] Jiu, Q.; Wu, J.; Yang, W., Eventual regularity of the Two-Dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Sci., 25, 1, 37-58 (2015) · Zbl 1311.35221 [16] Kukavica, I.; Ziane, M., Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete Contin. Dyn. Syst., 16, 67-86 (2006) · Zbl 1115.35098 [17] Kukavica, I.; Ziane, M., On the regularity of the Navier-Stokes equation in a thin periodic domain, J. Differential Equations, 234, 485-506 (2007) · Zbl 1118.35029 [18] Larios, A.; Lunasin, E.; Titi, E. S., Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255, 2636-2654 (2013) · Zbl 1284.35343 [19] Leary, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05 [20] Lin, F., Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65, 1, 893-919 (2012) · Zbl 1426.76041 [21] Lin, F.; Zhang, P., Global small solutions to an MHD-type system: the three-dimensional case, Comm. Pure Appl. Math., 67, 1, 531-580 (2014) · Zbl 1298.35153 [22] Lin, F.; Zhang, T., Global small solutions to a complex fluid model in three dimensional, Arch. Ration. Mech. Anal., 67, 216, 905-920 (2015) · Zbl 1325.35167 [24] Majda, A. J.; Bertozzi, A. L., (Vorticity and Incompressible Flow. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27 (2002), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0983.76001 [25] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, Nonlinear Differential Equations Appl., 18, 6, 707-735 (2009) · Zbl 1235.76020 [26] Raugel, G., Dynamics of partial differential equations on thin domains, (CIME Course, Montecatini Terme. CIME Course, Montecatini Terme, Lecture Notes in Mathematics, vol. 1609 (1995), Springer: Springer Berlin), 208-315 · Zbl 0851.58038 [27] Raugel, G.; Sell, G. R., Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6, 503-568 (1993) · Zbl 0787.34039 [28] Temam, R.; Ziane, M., Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations, 499-546 (1996) · Zbl 0864.35083 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.