This note aims at providing a rather informal and hopefully accessible overview of the fairly long and technical work [F. Ancona et al. ArXiv: 2401.14865], where new global-in-time existence results for admissible solutions of nonlinear systems of conservation laws defined in domains with boundaries are established. The main novelty of that work is that the solution is constructed by taking into account the underlying viscous mechanism, which is relevant because, in the case of initial-boundary value problems, different viscous approximations yield in general different limits. In the present note we will frame the analysis of the paper mentioned in the relevant context, compare the main result with the previous existing literature, and touch upon the most innovative technical points of the proof.

This paper establishes the weak convergence of global solutions for the Navier-Stokes-Korteweg equations under the weak Kolmogorov hypothesis in the three-dimensional periodic domain. Specifically, the weak Kolmogorov hypothesis offers uniform bounds for weak solutions, ensuring their weak stability under vanishing viscosity. With compactness arguments, we show that the solutions of the Navier-Stokes-Korteweg equations converge to a global weak solution of the Euler-Korteweg equations.

In this paper, two examples of sonic interfaces ([2–6]) are presented. The first example shows the case of sonic interfaces as weak discontinuities in self-similar shock configurations of unsteady Euler system. The second example shows the case of sonic interfaces as regular interfaces in accelerating transonic flows governed by the steady Euler-Poisson system with self-generated electric forces. And, we discuss analytic differences of the two examples, and introduce an open problem on decelerating transonic solution to the steady Euler-Poisson system.

In this paper we show that, as the viscosity is properly small, there exists a viscous transonic shock solution for the steady 1-D Navier-Stokes system with prescribed pressure at the exit, and it converges to a transonic shock solution to the 1-D steady Euler system as the viscosity goes to zero. Moreover, the position of the shock front is also derived. The key step is to reduce the pressure condition at the exit into a nonlinear boundary condition on the velocity, such that the boundary value problem for the Navier-Stokes system can be reformulated as a boundary value problem for an ODE with an unknown parameter.

In this note, we demonstrate that solutions to scalar balance laws, in one space dimension, which exhibit bounded variation, must be functions of special bounded variation (SBV). This case study illustrates the strategy applied in [Ancona et al., preprint, University of Padova, 2024] to systems of balance laws, extending the methodologies developed in pioneering previous works by several authors. While for a single balance law a more general work is already available, generalizing the first breakthrough related to a conservation law, the case of 1$D$ systems presents new behaviors that require a different strategy. This is why in this note we make the effort to introduce the notation and tools that are required for the case of more equations. When the flux presents linear degeneracies, it is known that entropy solutions can really present nasty fractal Cantor-like behaviours, although $f ′ (u)$ is still SBV: We thus discuss SBV-like regularity, as SBV-regularity fails.

We prove the $L^2$ stability for weak solutions of non-isentropic Euler equations in one space dimension whose initial data are perturbed from a small BV data under the $L^2$ distance. Using this result, we can show the uniqueness of small BV solutions among a large family of weak solutions.