Global strong solutions of three-dimensional heat conducting incompressible magnetohydrodynamic equations with vacuum. (English) Zbl 1529.35396
Summary: This paper is devoted to studying global existence and large time behavior of strong solutions of the three-dimensional heat conducting incompressible magnetohydrodynamic flows with initial vacuum and a vacuum far field. Under the scaling invariant quantity \((1 + \|\rho_0\|_\infty)^2 (\|\sqrt{\rho_0} u_0\|_2^2 + \|b_0\|_2^2) (\|\nabla u_0\|_2^2 + \|\nabla b_0\|_2^2)\) is suitably small, the global well-posedness of strong solution to the Cauchy problem is proved. Here, the explicit dependence on the initial norms is given and the smallness depends only on the known parameters in the system. This implies that the global well-posedness result is also valid for some large initial data. Furthermore, the algebraic decay rates of the global solution are obtained.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
heat conducting incompressible magnetohydrodynamic equations; global strong solution; vacuum; decayReferences:
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