Abstract:
We study the contraction properties (up to shift) for admissible Rankine-Hugoniot discontinuities of $n\times n$ systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in [47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. All results do not involve any smallness condition on the initial perturbation, nor on the size of the shock.

Abstract:
After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.

Abstract:
From pulsar scintillations we infer the presence of sheet-like structures in the ISM; it has been suggested that these are current sheets. Current sheets probably play an important role in heating the solar corona, and there is evidence for their presence in the solar wind. Such magnetic discontinuities have been found in numerical simulations with particular boundary conditions, as well as in simulations using an incompressible equation of state. Here, I investigate their formation under more general circumstances by means of topological considerations as well as numerical simulations of the relaxation of an arbitrary smoothly-varying magnetic field. The simulations are performed with a variety of parameters and boundary conditions: in low, high and of-order-unity plasma-$\beta$ regimes, with periodic and fixed boundaries, with and without a friction force, at various resolutions and with various diffusivities. Current sheets form, over a dynamical timescale, under {\it all} conditions explored. At higher resolution they are thinner, and there is a greater number of weaker current sheets. The magnetic field eventually relaxes into a smooth minimum energy state, the energy of which depends on the magnetic helicity, as well as on the nature of the boundaries.

Abstract:
We consider the isentropic compressible Euler system in 2 space dimensions with pressure law $p({\rho}) = {\rho}^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.

Abstract:
We prove the existence of a generalization of Kelvin's circulation theorem in general relativity which is applicable to perfect isentropic magnetohydrodynamic flow. The argument is based on a new version of the Lagrangian for perfect magnetohydrodynamics. We illustrate the new conserved circulation with the example of a relativistic magnetohydrodynamic flow possessing three symmetries.

Abstract:
The quantum hydrodynamic model for charged particle systems is extended to the cases of non zero magnetic fields. In this way, quantum corrections to magnetohydrodynamics are obtained starting from the quantum hydrodynamical model with magnetic fields. The quantum magnetohydrodynamics model is analyzed in the infinite conductivity limit. The conditions for equilibrium in ideal quantum magnetohydrodynamics are established. Translationally invariant exact equilibrium solutions are obtained in the case of the ideal quantum magnetohydrodynamic model.

Abstract:
Instead of the infinitesimal extrinsic and intrinsic perturbations on strings, considered so far, we discuss the evolution and propagation of finite-amplitude perturbations. Those intrinsic perturbations may result in appearance of stable discontinuities similar to the shock waves.

Abstract:
We consider the compressible Navier-Stokes equations for isentropic dynamics with real viscosity on a bounded interval. In the case of boundary data defining an admissible shock wave for the corresponding unviscous hyperbolic system, we determine a scalar differential equation describing the motion of the internal transition layer. In particular, for small viscosity, the velocity of the motion is exponentially small. The approach is based on the construction of a one-parameter manifold of approximate solutions and on an appropriate projection of the evolution of the complete Navier-Stokes system towards such manifold.

Abstract:
When a gravitating object moves across a given mass distribution, it creates an overdense wake behind it. Here, we performed an analytical study of the structure of the flow far from object when the flow is isentropic and the object moves subsonically within it. We show that the dynamical friction force is the main drag force on the object and by using a perturbation theory, we obtain the density, velocity and pressure of the perturbed flow far from the mass. We derive the expression of the dynamical friction force in an isentropic flow and show its dependence on the Mach number of the flow and on the adiabatic index. We find that the dynamical friction force becomes lower as the adiabatic index increases. We show analytically that the wakes are less dense in our isentropic case in comparison to the isothermal ones.