On oscillatory solutions to the complete Euler system

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible oscillations and/or concentrations in the associated generating sequence. Unlike the conventional measure-valued solutions or rather their expected values, the dissipative solutions comply with a natural compatibility condition -they are classical solutions as long as they enjoy certain degree of smoothness.

show abstract

“…Next, for each Ω i , fix ̺ i > 0 -a constant density distribution. Similarly to [17], [21], we consider the following problem:…”

Section: Oscillatory Solutionsmentioning

confidence: 99%

Generalized solutions to models of inviscid fluids

Breit¹,

Feireisl

Hofmanová

2020

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, these solutions satisfy the entropy inequality (1.7). In addition, examples of regular initial data producing infinitely many weak solutions in the long run have been also obtained in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The recent adaptation of the method of convex integration, developed in the context of incompressible fluids by De Lellis and Székelyhidi [10], gave rise to numerous examples of ill-posedness also in the class of compressible fluids, see Chiodaroli, De Lellis, Kreml [6], Chiodaroli and Kreml [7] , Chiodaroli et al [8], among others. In particular, it was shown in [12] that the Euler system (1.1)-(1.5) is ill-posed, specifically it admits infinitely many weak solutions for a large class of initial data. Moreover, these solutions satisfy the entropy inequality (1.7).…”

Section: Introductionmentioning

confidence: 99%

Dissipative Solutions and Semiflow Selection for the Complete Euler System

Breit

Feireisl

Hofmanová

2020

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

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 solutions to the full Euler system describing the motion of a compressible inviscid fluid, our approach permits to select a system of solutions (one trajectory for every initial condition) satisfying the classical semiflow property. Moreover, the selection respects the well accepted admissibility criteria for physical solutions, namely, maximization of the entropy production rate and the weak-strong uniqueness principle. Consequently, strong solutions are always selected whenever they exist and stationary states are stable and included in the selection as well. To this end, we introduce a notion of dissipative solution, which is given by a triple of density, momentum and total entropy defined as expectations of a suitable measure-valued solution.

show abstract

“…They consider the case F = 0. The complete system has been treated in [5] and [6] -both papers focus on the strong stratification, i.e. Ma = Fr = ε as ε → 0, for well-prepared initial data.…”

Section: Introductionmentioning

confidence: 99%

“…Ma = Fr = ε as ε → 0, for well-prepared initial data. Unlike in the isentropic case, the target system for the complete system case is not uniquely determined and it depends on the choice of the initial data -the isothermal case has been studied in [6] and the isentropic case in [5].…”

Section: Introductionmentioning

confidence: 99%

Measure-valued solutions to the complete Euler system

Březina¹,

Feireisl²

2018

J. Math. Soc. Japan

Self Cite

View full text Add to dashboard Cite

The present paper takes advantage of the concept of dissipative measure-valued solutions to show the rigorous derivation of the Euler-Boussinesq (EB) system that has been successfully used in various meteorological models. In particular, we show that EB system can be obtained as a singular limit of the complete Euler system. We provide two types of result -firstly, we treat the case of well-prepared initial data for any sufficiently regular bounded domain. Secondly, we use the dispersive estimates for acoustic equation to tackle the case of the illprepared initial data on an unbounded exterior domain.

show abstract

On oscillatory solutions to the complete Euler system

Cited by 38 publications

References 64 publications

Generalized solutions to models of inviscid fluids

Generalized solutions to models of inviscid fluids

Dissipative Solutions and Semiflow Selection for the Complete Euler System

Measure-valued solutions to the complete Euler system

Contact Info

Product

Resources

About