Long time behavior of solutions to 3D generalized MHD equations

References

[1] M. Ainsworth and Z. Mao, Analysis and approximation of a fractional Cahn–Hilliard equation, SIAM J. Numer. Anal. 55 (2017), no. 4, 1689–1718. 10.1137/16M1075302Suche in Google Scholar

[2] M. Cannone, Q. Chen and C. Miao, A losing estimate for the ideal MHD equations with application to blow-up criterion, SIAM J. Math. Anal. 38 (2007), no. 6, 1847–1859. 10.1137/060652002Suche in Google Scholar

[3] D. Catania, Global attractor and determining modes for a hyperbolic MHD turbulence model, J. Turbul. 12 (2011), Paper No. 40. 10.1080/14685248.2011.619986Suche in Google Scholar

[4] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. 10.1007/s00220-004-1055-1Suche in Google Scholar

[5] Y. Dai, Z. Tan and J. Wu, A class of global large solutions to the magnetohydrodynamic equations with fractional dissipation, Z. Angew. Math. Phys. 70 (2019), no. 5, Article ID 153. 10.1007/s00033-019-1193-0Suche in Google Scholar

[6] B. Dong, Y. Jia, J. Wu, Q. Xie, Z. Ye and Z. Zhang, On large time behavior of solutions to the three-dimensional generalized Navier–Stokes equations, preprint. Suche in Google Scholar

[7] B.-Q. Dong, Y. Jia, J. Li and J. Wu, Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion, J. Math. Fluid Mech. 20 (2018), no. 4, 1541–1565. 10.1007/s00021-018-0376-3Suche in Google Scholar

[8] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972), 241–279. 10.1007/BF00250512Suche in Google Scholar

[9] J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier–Stokes and related equations, Kinet. Relat. Models 6 (2013), no. 3, 545–556. 10.3934/krm.2013.6.545Suche in Google Scholar

[10] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. Suche in Google Scholar

[11] C. He and Z. Xin, On the regularity of solutions to the magnetohydrodynamic equations, J. Differential Equations 213 (2005), 235–254. 10.1016/j.jde.2004.07.002Suche in Google Scholar

[12] W. Huo and A. Huang, The global attractor of the 2D Boussinesq equations with fractional Laplacian in subcritical case, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 8, 2531–2550. 10.3934/dcdsb.2016059Suche in Google Scholar

[13] Z. Jiang, Y. Wang and Y. Zhou, On regularity criteria for the 2D generalized MHD system, J. Math. Fluid Mech. 18 (2016), no. 2, 331–341. 10.1007/s00021-015-0235-4Suche in Google Scholar

[14] Q. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier–Stokes equations, Asymptot. Anal. 94 (2015), no. 1–2, 105–124. 10.3233/ASY-151307Suche in Google Scholar

[15] N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys. 255 (2005), no. 1, 161–181. 10.1007/s00220-004-1256-7Suche in Google Scholar

[16] T. Kato, Strong Lp-solutions of the Navier–Stokes equation in 𝐑m, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. 10.1007/BF01174182Suche in Google Scholar

[17] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. 10.1002/cpa.3160410704Suche in Google Scholar

[18] N. H. Katz and N. Pavlović, A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation, Geom. Funct. Anal. 12 (2002), no. 2, 355–379. 10.1007/s00039-002-8250-zSuche in Google Scholar

[19] H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. 10.1006/aima.2000.1937Suche in Google Scholar

[20] H. Kozono and T. Ogawa, Two-dimensional Navier–Stokes flow in unbounded domains, Math. Ann. 297 (1993), no. 1, 1–31. 10.1007/BF01459486Suche in Google Scholar

[21] Z. Lei and Y. Zhou, BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst. 25 (2009), no. 2, 575–583. 10.3934/dcds.2009.25.575Suche in Google Scholar

[22] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248. 10.1007/BF02547354Suche in Google Scholar

[23] P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces, J. Funct. Anal. 259 (2010), no. 10, 2457–2519. 10.1016/j.jfa.2010.07.013Suche in Google Scholar

[24] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. Suche in Google Scholar

[25] C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. 68 (2008), no. 3, 461–484. 10.1016/j.na.2006.11.011Suche in Google Scholar

[26] W. Ren, Y. Wang and G. Wu, Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations, Commun. Contemp. Math. 18 (2016), no. 6, Article ID 1650018. 10.1142/S0219199716500188Suche in Google Scholar

[27] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), no. 5, 635–664. 10.1002/cpa.3160360506Suche in Google Scholar

[28] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990), no. 5, 1093–1117. 10.1137/0521061Suche in Google Scholar

[29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Suche in Google Scholar

[30] W. A. Strauss, Decay and asymptotics for c⁢m⁢u=F⁢(u), J. Funct. Anal. 2 (1968), 409–457. 10.1016/0022-1236(68)90004-9Suche in Google Scholar

[31] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1988. 10.1007/978-1-4684-0313-8Suche in Google Scholar

[32] G. Wu, Regularity criteria for the 3D generalized MHD equations in terms of vorticity, Nonlinear Anal. 71 (2009), no. 9, 4251–4258. 10.1016/j.na.2009.02.115Suche in Google Scholar

[33] J. Wu, Dissipative quasi-geostrophic equations with Lp data, Electron. J. Differential Equations 2001 (2001), Paper No. 56. Suche in Google Scholar

[34] J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), no. 2, 284–312. 10.1016/j.jde.2003.07.007Suche in Google Scholar

[35] J. Wu, The generalized incompressible Navier–Stokes equations in Besov spaces, Dyn. Partial Differ. Equ. 1 (2004), no. 4, 381–400. 10.4310/DPDE.2004.v1.n4.a2Suche in Google Scholar

[36] J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces, Comm. Math. Phys. 263 (2006), no. 3, 803–831. 10.1007/s00220-005-1483-6Suche in Google Scholar

[37] J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations 33 (2008), no. 1–3, 285–306. 10.1080/03605300701382530Suche in Google Scholar

[38] K. Yamazaki, On the global well-posedness of N-dimensional generalized MHD system in anisotropic spaces, Adv. Differential Equations 19 (2014), no. 3–4, 201–224. 10.57262/ade/1391109084Suche in Google Scholar

[39] Z. Ye, Global well-posedness and decay results to 3D generalized viscousmagnetohydrodynamic equations, Ann. Mat. Pura Appl. (4) 195 (2016), no. 4, 1111–1121. 10.1007/s10231-015-0507-xSuche in Google Scholar

[40] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 3, 491–505. 10.1016/j.anihpc.2006.03.014Suche in Google Scholar