Forward Self-similar Solutions to the MHD Type System in the Half Space

Abstract

We study the existence of forward self-similar solutions to the magnetohydrodynamic type (MHD type) system in the half space \(\mathbb {R}^{3}_{+}\). For any self-similar initial data belonging to \(L^{\infty }_{loc}(\overline{\mathbb {R}^{3}_{+}}\setminus \{0\})\), we employ a compactness argument and Leray–Schauder theorem to construct a forward self-similar solution to the MHD type system that is smooth in \(\mathbb {R}^{3}_{+}\times (0,\infty )\). Moreover, our results extend the existence theorem of Korobkov and Tsai [25] for the Navier–Stokes equations by relaxing the regularity requirement on the initial data from \(C^{1}_{loc}(\overline{\mathbb {R}^{3}_{+}}\setminus \{0\})\) to \(L^{\infty }_{loc}(\overline{\mathbb {R}^{3}_{+}}\setminus \{0\})\).