2017/07/09 by Li Xu, Xu, Li
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1707.02544
openalex publication_date 2017/07/09 · openalex created_date 2017/07/21 · openalex updated_date 2026/07/28
We consider the ideal magnetohydrodynamics (MHD) subjected to a strong magnetic field along x1 direction in three-dimensional thin domains Ωδ=ℝ2×(-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 Ωδ 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 Ωδ to the plane ℝ2. More precisely, we prove that the 3D Alfvén waves in Ωδ will converge to the Alfvén waves in ℝ2 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 Ωδ 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(∂3 z-h, ∂3 z+h) where z_±=(z_±h, z3_±) and Q(∂3 z-h,∂3 z+h) is the linear combination of terms ∂α∂3 z-h∂β∂3 z+h with α, β∈ (ℤ≥0)3.