Abstract
In this paper, we consider a steady MHD fluid model of non-Newtonian type in a smooth bounded domain \(\Omega \in {\mathbb {R}}^3\). Using the iterative method, under the condition that the external force is small in a suitable sense, we proved the existence of \(C^{1,\gamma }({\bar{\Omega }})\times W^{2,r}(\Omega )\) solutions of the systems for the exponent \(1<p<2\) and we show that this solution is unique in case \(\frac{6}{5}<p<2\). Moreover, we also proved the higher regularity properties of this solution.
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References
Biskamp, D.: Magnetohydrodynamical Turbulence. Cambridge University Press, Cambridge (2003)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981)
Ladyzhenskaya, O.A., Solonnikov, V.: Solution of some nonstationary magnetohydrodynamical problems for incompressible fluid. Trudy Steklov Math. Inst. 69, 115–173 (1960)
Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)
Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)
Chen, Q., Miao, C., Zhang, Z.: The Beale–Kato–Majda criterion for the 3D magneto-hydrodynamics equations. Commun. Math. Phys. 275, 861–872 (2007)
He, C., Xin, Z.: Partial regularity of suitable weak solutions to the incompressible magne-tohydrodynamic equations. J. Funct. Anal. 227, 113–152 (2005)
Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226, 1803–1822 (2011)
Lin, F., Zhang, P.: Global small solutions to an MHD-type system: the three-dimensional case. Commun. Pure Appl. Math. 67, 531–580 (2014)
Gerbeau, J.-F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, New York (2006)
Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)
He, C., Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)
He, C., Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213, 235–254 (2005)
Ladyzhensakya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
Ladyzhensakya, O.A.: New equations for description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Seminar in Mathematics V. A. Steklov Mathematical Institute, Vol. 102, Boundary value problems of mathematical physics, Part V, Providence, Rhode Island, AMS (1970)
Samokhin, V.N.: On a system of equations in the magnetohydrodynamics of nonlinearly viscous media. Differ. Equ. 27, 628–636 (1991)
Gunzburger, M.D., Ladyzhenskaya, O.A., Peterson, J.S.: On the global unique solvability of initial-boundary value problems for the coupled modified Navier–Stokes and Maxwell equations. J. Math. Fluid Mech. 6, 462–482 (2004)
Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)
Parés, C.: Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids. Appl. Anal. 43(3–4), 245–296 (1992)
Crispo, F., Grisanti, C.: On the existence, uniqueness and \(C^{1,\gamma }({\bar{\Omega }})\cap W^{2,2}(\Omega )\) regularity for a class of shear-thinning fluids. J. Math. Fluid Mech. 10, 455–487 (2008)
Ebmeyer, E.: Regularity in Sobolev spaces of steady flows of fluids with shear dependent viscosity. Math. Methods Appl. Sci. 29, 1687–1707 (2006)
Crispo, F., Grisanti, C.R.: On the \(C^{1,\gamma }({\bar{\Omega }})\cap W^{2,2}(\Omega )\) regularity for a class of electro-rheological fluids. J. Math. Anal. Appl. 356, 119–132 (2009)
Kufner, A., John, O., Fučk, S.: Function spaces. In: Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Prague (1977)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I, p. 38. Springer, Berlin (1994)
Acknowledgements
This work was supported by the fund of the “Thirteen Five” Scientific and Technological Research Planning Project of the Department of Education of Jilin Province(Grant no. JJKH20200727KJ ).
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Yang, H., Wang, C. On the Existence, Uniqueness and Regularity of Solutions for a Class of MHD Equations of Non-Newtonian Type. Mediterr. J. Math. 17, 144 (2020). https://doi.org/10.1007/s00009-020-01570-y
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DOI: https://doi.org/10.1007/s00009-020-01570-y