This paper gives the analytical solution to the one-dimensional compressible MHD equations without magnetic diffusion in half space. The interesting result is that the solution to one-dimensional compressible isentropic MHD equations without magnetic diffusion on $ (x, t)\in[0, +\infty)\times R_{+} $ can be expressed as the initial density and boundary function of velocity strictly if and only if the solution to the Riccati differential equation of the boundary condition function exists under some assumptions. When the magnetic field is zero, the compressible MHD equations become compressible Navier–Stokes equations. The analytical solution was studied by Dou & Zhao 2021 on the 1D Navier–Stokes equations, which is a special case in this paper. The magnetic field and the coupling term of the magnetic field and velocity result in more complicated computations and estimates. And, we can prove that the analytical solution to compressible MHD equations without magnetic diffusion depends continuously on the initial value. The existence of a steady solution to compressible MHD equations is given, and the large-time behavior and asymptotic limit theorem of the analytical solution are also shown when the time tends to infinity and the initial data of magnetic field goes to zero. Finally, we give several examples to show that the set of this kind of solution is not empty.
Citation: Changsheng Dou, Hongtian Zhang, Tengzhe Zhao. Analytical solution and asymptotic limit to one-dimensional compressible magnetohydrodynamic equations without magnetic diffusion[J]. AIMS Mathematics, 2025, 10(12): 28908-28925. doi: 10.3934/math.20251272
This paper gives the analytical solution to the one-dimensional compressible MHD equations without magnetic diffusion in half space. The interesting result is that the solution to one-dimensional compressible isentropic MHD equations without magnetic diffusion on $ (x, t)\in[0, +\infty)\times R_{+} $ can be expressed as the initial density and boundary function of velocity strictly if and only if the solution to the Riccati differential equation of the boundary condition function exists under some assumptions. When the magnetic field is zero, the compressible MHD equations become compressible Navier–Stokes equations. The analytical solution was studied by Dou & Zhao 2021 on the 1D Navier–Stokes equations, which is a special case in this paper. The magnetic field and the coupling term of the magnetic field and velocity result in more complicated computations and estimates. And, we can prove that the analytical solution to compressible MHD equations without magnetic diffusion depends continuously on the initial value. The existence of a steady solution to compressible MHD equations is given, and the large-time behavior and asymptotic limit theorem of the analytical solution are also shown when the time tends to infinity and the initial data of magnetic field goes to zero. Finally, we give several examples to show that the set of this kind of solution is not empty.
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Changsheng Dou, Hongtian Zhang, Tengzhe Zhao. Analytical solution and asymptotic limit to one-dimensional compressible magnetohydrodynamic equations without magnetic diffusion[J]. AIMS Mathematics, 2025, 10(12): 28908-28925. doi: 10.3934/math.20251272
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