On the ideal magnetohydrodynamics in three-dimensional thin domains: well-posedness and asymptotics
We consider the ideal magnetohydrodynamics (MHD) subjected to a strong magnetic field along $x_1$ direction in three-dimensional thin domains $Ω_δ=\mathbb{R}^2\times(-δ,δ)$ with slip boundary conditions. It is well-known that in this situation the system will generate Alfvén waves. Our results are summarized as follows: (i).\, We construct the global solutions (Alfvén waves) to MHD in the thin domain $Ω_δ$ with $δ>0$. In addition, the uniform energy estimates are obtained with respected to the parameter $δ$. (ii). We justify the asymptotics of the MHD equations from the thin domain $Ω_δ$ to the plane $\mathbb{R}^2$. More precisely, we prove that the 3D Alfvén waves in $Ω_δ$ will converge to the Alfvén waves in $\mathbb{R}^2$ in the limit that $δ$ goes to zero. This shows that Alfvén waves propagating along the horizontal direction of the (3D) strip are stable and can be approximated by the (2D) Alfvén waves when $δ$ is sufficiently small. Moreover, the control of the (2D) Alfvén waves can be obtained from the control of (3D) Alfvén waves in the thin domain $Ω_δ$ with aid of the uniform bounds. The proofs of main results rely on the design of the proper energy functional and the null structures of the nonlinear terms. Here the null structures means two aspects: separation of the Alfvén waves ($z_+$ and $z_-$) and no bad quadratic terms $Q(\partial_3 z_-^h, \partial_3 z_+^h)$ where $z_\pm=(z_\pm^h, z^3_\pm)$ and $Q(\partial_3 z_-^h,\partial_3 z_+^h)$ is the linear combination of terms $\partial^α\partial_3 z_-^h\partial^β\partial_3 z_+^h$ with $α, β\in (\mathbb{Z}_{\geq0})^3$.