Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space

Abstract

Abstract In this paper, we introduce a new metric space called the mixed-norm Lebesgue space, which allows its norm decay to zero with different rates as ∣ x ∣ → ∞ | x| \to \infty in different spatial directions. Then we study the well posedness for the system of magnetohydrodynamic equations in 3D mixed-norm Lebesgue spaces. By using some fundamental analysis theories in mixed-norm Lebesgue space such as Young’s inequality, time decaying of solutions for heat equations, and the boundedness of the Helmholtz-Leray projection, we prove local well posedness and global well posedness of the solutions.