This is a survey highlighting several recent results concerning well/ill posedness of the Euler system of gas dynamics. Solutions of the system are identified as limits of consistent approximations generated either by physically more complex problems, notably the Navier- Stokes-Fourier system, or by the approximate schemes in numerical experiments. The role of the fundamental principles encoded in the First and Second law of thermodynamics in identifying a unique physically admissible solution is examined.

We consider the (complete) Euler system describing the motion of a compressible perfect fluid. We propose a platform suitable for constructing the statistical solutions. The main ingredients of our approach include: 1. The concept of dissipative (measure{valued) solution to the Euler system. 2. A single step selection procedure based on minimizing the Bregman divergence of a given solution to the maximal entropy equilibrium. 3. A construction of a Markov semigroup via push forward measures.

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.

We establish various results concerning the uniqueness of zero velocity solutions for the static barotropic Navier--Stokes system. Some of them can be seen as Liouville-type theorems for problems in unbounded physical space.

Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically relevant solution can be selected by maximizing the entropy production rate. Firstly, we present an example of Riemann initial data in 2-D, for which the standard self-similar solution fails to satisfy the maximal entropy production principle. Hence, maximizing the entropy production rate rules out the 1-D self-similar solution which intuitively seems to be the physically relevant solution in this context. Secondly, we show for a large class of initial data that there exist entropy admissible weak solutions with an arbitrary (non-decreasing) total entropy profile.

We consider a continuous data assimilation method for the barotropic Navier–Stokes system. The observed solution is supposed to be bounded on the whole time period of observation, while the synchronized solution, usually provided by a numerical method, belongs to the class of dissipative solutions that is considerably larger than the class of conventional weak solutions. A complete synchronization is shown on any compact prediction interval provided the nudging parameters are chosen appropriately.

We show convergence of a continuous data assimilation method for the Oberbeck-Boussinesq system in the dimension d=2,3. Our working hypothesis is boundedness of the reference solution, while the synchronized solution satisfies the equations in a weak sense. The main tool is the relative energy inequality for stochastic problems.

We consider a general compressible MHD system, where the magnetic field propagates in a heterogeneous medium. Using suitable penalization in terms of the transport coefficients we perform several singular limits. As a result we obtain: 1. A rigorous justification of physically grounded boundary conditions for the compressible MHD system on a bounded domain. 2. Existence of weak solutions for arbitrary finite energy initial data in the situation the Maxwell induction equation holds also outside the fluid domain. 3. A suitable theoretical platform for numerical experiments on domains with geometrically complicated boundaries.

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global--in--time dissipative solutions always exist as long as the rigid bodies are connected compact sets. In addition, in the absence of external driving forces, the system always tends to a static equilibrium as time goes to infinity. The results hold independently of possible collisions of rigid bodies and for any finite energy initial data.

We show several results on the convergence of the Monte Carlo method applied to a family of consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our approach is based on a combination of several new ideas developed recently in the context of the Euler system: The theoretical results are illustrated by a series of numerical simulations obtained by a viscosity finite volume scheme combined with the Monte Carlo method.

We show that any dissipative (measure-valued) solution of the compressible Euler system that complies with Dafermos' criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos' criterion yielding a unique solution of the problem for any finite energy initial data.

We study the compressible Navier-Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.

We consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well–posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet–Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain:

The method of Convex Integration has revealed a number of rather disturbing facts concerning well-posedness of the Euler system of gas dynamics. In particular, there is a dense set of "wild" initial data, for which the problem admits infinitely many physically admissible (entropy) weak solutions. We identify the class of initial data enjoying the following properties: (a) they give rise to a family of weak solutions with increasing entropy profiles; (b) the solutions are "discrete", meaning they attain only a finite number of constant states; (c) the solutions reach a prescribed terminal entropy profile when time goes to infinity.

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.

We develop a new approach to the problem of the motion of a large number of rigid bodies immersed in a viscous fluid. The leading idea is the concept of cluster - a collection of individual rigid objects that may be grouped or even connected in such a way that their collective impact on the bulk motion of the system is similar to that of a single body. The applications of the new approach include: 1. Improving the critical value of the number of balls of small radius such that their cloud has no impact on the limit system represented by the incompressible Navier–Stokes equations. 2. The balls follow the fluid flow in the asymptotic limit of vanishing radius and increasing number even if a gravitational force is imposed.

In contrast with a large variety of conventional models of thermally driven fluids, we show that the standard Oberbeck-Boussinesq approximation cannot be obtained as a singular limit of the Navier-Stokes-Fourier system in the rotational coordinate system, with the buoyancy force proportional to the sum of the gravitational and centrifugal forces multiplied by the temperature variation.

We propose a new two-step selection criterion applicable to the dissipative measure–valued solutions of the Euler system of gas dynamics. The process consists of a successive maximisation of the entropy production rate and the total energy defect, i.e. maximisation of the turbulent energy. If the selected solution is a weak solution of the Euler system, then it is identified in the first step. Solutions selected in the second step are truly measure–valued maximising the energy defect. Accordingly, they are called turbulent solutions. The energy defect of turbulent solutions vanishes with growing time. The selected solutions depend in a Borel–measurable way on the initial data. In particular, they are almost continuously dependent on the initial data.