On the well-posedness of strong solution to ideal magnetohydrodynamic equations

ABSTRACT

In this paper, we study the N-dimensional incompressible flow governed by the ideal magnetohydrodynamic (MHD) equations combining Euler equation (for the fluid velocity) and Maxwell's equation (for the magnetic field). In a bounded domain with the smooth boundary, as the initial data $({\mathit{u}}_0,{\mathit{B}}_0)\in ({\left(H^m\right(\mathrm{\Omega }\left)\right)}^N\times {\left(H^m\right(\mathrm{\Omega }\left)\right)}^N)$, the existence of the strong solution $\left(\mathit{u}\right(\cdot ,t),\mathit{B}(\cdot ,t\left)\right)\in ({\left(H^m\right(\mathrm{\Omega }\left)\right)}^N\times {\left(H^m\right(\mathrm{\Omega }\left)\right)}^N)$ to the ideal MHD equations is obtained by Galerkin method. Moreover, based on specially dealing with the priori estimates to those nonlinear terms in the MHD equations, we prove that the strong solution to the equations is unique and depends continuously on the initial data in the spaces ${\left(L^2\right(\mathrm{\Omega }\left)\right)}^N$ and ${\left(H^{m-1}\right(\mathrm{\Omega }\left)\right)}^N$.

KEYWORDS:

• Ideal magnetohydrodynamic equations
• incompressible flow
• strong solution
• well-posedness
• Galerkin method

2010 AMS SUBJECT CLASSFICATIONS:

• 35Q35
• 76B03
• 76E25

Disclosure statement

No potential conflict of interest was reported by the authors.