On the local and global existence of unique solutions to the Lüst model

Abstract

The Magnetohydrodynamics equations (MHD) provide a macroscopic description of a plasma and it provides a relevant description many physical phenomena. However, MHD is deficient in many respect and some of these deficiencies are accounted for by extending Ohm’s law with the momentum equations being usually unchanged. But, this approach does not give conservation of the energy for some cases. In this paper, we take a model proposed by Lüst who appears to be the first to correct the generalized Ohm’s law to guarantee conservation of the energy. We first show the existence of unique solutions and establish a blow-up criterion to the Lüst model with initial data in H3. We then prove that this solution is defined globally-in-time and decays algebraically when the initial data is sufficiently small.