Plane one-dimensional MHD flows: symmetries and conservation laws

Abstract

The paper considers the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The inviscid, thermally non-conducting medium is modeled by a polytropic gas. The equations are examined for symmetries and conservation laws. For the case of the finite electric conductivity we establish Lie group classification, i.e. we describe all cases of the conductivity $\sigma (\rho ,p)$ for which there are symmetry extensions. The conservation laws are derived by the direct computation. For the case of the infinite electrical conductivity the equations can be brought into a variational form in the Lagrangian coordinates. Lie group classification is performed for the entropy function as an arbitrary element. Using the variational structure, we employ the Noether theorem for obtaining conservation laws. The conservation laws are also given in the physical variables.

References in corpus (4)

• One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: symmetry classification, conservation laws, difference schemes
• Natural curvilinear coordinates for ideal magnetohydrodynamics equations. Solutions with constant total pressure
• Group analysis of the Fourier transform of the spatially homogeneous and isotropic Boltzmann equation with a source term
• Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation

Cited by in corpus (4)

• One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws
• Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations
• Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates
• Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation