Related papers: Well-posedness of the Euler system of gas dynamics
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…
To circumvent the ill-posedness issues present in various models of continuum fluid mechanics, we present a dynamical systems approach aiming at selection of physically relevant solutions. Even under the presence of infinitely many…
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…
The quest for a good solution concept for the partial differential equations (PDEs) arising in mathematical fluid dynamics is an outstanding open problem. An important notion of solutions are the measure-valued solutions. It is well known…
We consider the complete Euler system describing the time evolution of a general inviscid compressible fluid. We introduce a new concept of measure-valued solution based on the total energy balance and entropy inequality for the physical…
We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a parameterized (Young) measure and a dissipation…
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…
It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill--posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both…
We introduce the concept of stochastic measure-valued solutions to the complete Euler system describing the motion of a compressible inviscid fluid subject to stochastic forcing, where the nonlinear terms are described by defect measures.…
In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler…
The Cauchy problem for the complete Euler system is in general ill posed in the class of admissible (entropy producing) weak solutions. This suggests there might be sequences of approximate solutions that develop fine scale oscillations.…
We introduce the concept of maximal dissipative measure-valued solution to the complete Euler system. These are solutions that maximize the entropy production rate. We show that these solutions exist under fairly general hypotheses imposed…
Three very different algorithms have been proposed for solution of the Rayleigh-Taylor turbulent mixing problem. They are based upon three different physical principles governing the Euler equations for fluid flow, which serve to complete…
We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. The underlying relative energy inequality holds as an equality for classical solutions and if the additional variable vanishes, these…
The concept of renormalized dissipative measures-valued (rDMV) solutions to a complete Euler system for a perfect gas was introduced in [8] and further discussed in [9]. Moreover it was shown there that rDMV solutions satisfy the weak…
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…
We introduce the concept of a dissipative measure-valued solution to the Euler alignment system. This approach incorporates a modified total energy balance, utilizing a binary tensor Young measure. The central finding is a weak…
We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of…
The Euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles: Conservation of mass, momentum, and energy. We show that the Cauchy problem is basically ill-posed for the…
We consider the weak solutions to the Euler-Fourier system describing the motion of a compressible heat conducting gas. Employing the method of convex integration, we show that the problem admits infinitely many global-in-time weak…