On the Ideal Magnetohydrodynamics in Three-Dimensional Thin Domains: Well-Posedness and Asymptotics

Abstract

We consider the ideal magnetohydrodynamics (MHD) subjected to a strong magnetic field along $x_1$ direction in three-dimensional (3D) thin domain $\Omega _\delta =\mathop {{\mathbb {R}}}\nolimits ^2\times (-\delta ,\delta )$ 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 $\Omega _\delta$ with $\delta >0$. In addition, the uniform energy estimates are obtained with respect to the parameter $\delta$.

• (ii) We justify the asymptotics of the MHD equations from the thin domain $\Omega _\delta$ to the plane $\mathop {{\mathbb {R}}}\nolimits ^2$. More precisely, we prove that the 3D Alfvén waves in $\Omega _\delta$ will converge to the 2D Alfvén waves in $\mathop {{\mathbb {R}}}\nolimits ^2$ in the limit that $\delta$ goes to zero provided that the leading part (horizontal component) of the initial 3D Alfvén waves on $\Omega _\delta$ converges to a 2D vector filed on $\mathop {{\mathbb {R}}}\nolimits ^2$ as $\delta \rightarrow 0$. This shows that Alfvén waves propagating along the horizontal direction of the 3D thin domain $\Omega _\delta$ are stable and can be approximated by the 2D Alfvén waves when $\delta$ 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 $\Omega _\delta$ 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 term null structures means things: separation of the Alfvén waves ($z_+$ and $z_-$); and no bad quadratic terms $Q(\partial _3z_-^h,\partial _3 z_+^h)$ where $z_\pm =(z_\pm ^h, z^3_\pm )$, $z_\pm ^h=(z_\pm ^1,z_\pm ^2)$ and $Q(\partial _3 z_-^h,\partial _3 z_+^h)$ is the linear combination of terms $\partial ^{\alpha }\partial _3z_-^h\partial ^{\beta }\partial _3z_+^h$ with $\alpha ,\beta \in ({\mathop {{\mathbb {Z}}}\nolimits }_{\ge 0})^2$.