From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an “s-d” potential term, including the Schrödinger--Landau--Lifshitz--Maxwell system, the Pauli--Landau--Lifshitz system, and the Schrödinger--Landau--Lifshitz system with successive simplifications. For the latter two systems, we propose using the time splitting spectral method for the quantum spin current and the Gauss--Seidel projection method for the magnetization. Accuracy of the time splitting spectral method applied to the Pauli equation is analyzed and verified by numerous examples. Moreover, behaviors of the Schrödinger--Landau--Lifshitz system in different “s-d” coupling regimes are explored numerically.

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In this addendum note we fill in the gap left in the description of 2D homogeneous solutions to the stationary Euler system initiated in the previous publication. This completes the classification of all homogeneous stationary solutions. The note includes updated classification tables.

We study the vanishing viscosity limit of the compressible NavierStokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible NavierStokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the NavierStokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

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