We prove the convergence of the vanishing viscosity limit of the one-dimensional, isentropic, compressible Navier-Stokes equations to the isentropic Euler equations in the case of a general pressure law. Our strategy relies on the construction of fundamental solutions to the entropy equation that remain controlled for unbounded densities, and employs an improved reduction framework to show that measure-valued solutions constrained by the Tartar commutation relation (but with possibly unbounded support) reduce to a Dirac mass. As the Navier-Stokes equations do not admit an invariant region, we work in the finite-energy setting, where a detailed understanding of the high density regime is crucial. and the characteristic fields are genuinely non-linear under the assumption ρp ′′ (ρ) + 2p ′ (ρ) > 0.(1.5)A typical pressure law (equation of state) for a barotropic fluid is that of a gamma-law gas,

The aim of this paper is to discuss the mathematical modeling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). We explore the onset of such a transition for each model in the parameter space of occupied volume fraction and Péclet number via a linear stability analysis and numerical simulations at both the microscopic and macroscopic levels. We establish that one of the models, namely, the mean-field model which assumes long-range repulsive interactions, cannot explain the emergence of MIPS. In contrast, MIPS is observed for the remaining three models that assume short-range interactions that localize the interaction terms in space.

We show that any smooth solution (u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L 6 (R 3 ) and BM O −1 (R 3 ) must be identically zero. This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ∇u, ∇H ∈ L 2 (R 3 ), i.e., finite Dirichlet integral. Theme: Partial differential equations.

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We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift–diffusion equation for which a maximum principle is available.