Dynamics of Viscous Compressible Fluids

Feireisl, Eduard

doi:10.1093/acprof:oso/9780198528388.001.0001

Cited by 407 publications

(407 citation statements)

References 44 publications

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“…For the convenience of the reader we list some useful functional analysis results (for proofs, see, e.g., [14]). Throughout the paper we use overbars to denote weak limits, with the underlying spaces being (silently) given by the context.…”

Section: Preliminary Materialsmentioning

confidence: 99%

“…Now, to conclude the proof, we appeal to Proposition 4.2 in [14] stating that the square-integrable weak solution˜ is also a renormalized solution.…”

Section: Whereũ(t) Denotes the Extension Of U(t)mentioning

confidence: 99%

“…Consequently, we may assume that there exist functions , w, u such that [14,22]. In particular, , log , and log belong to…”

Section: Convergencementioning

confidence: 99%

“…where the last inequality follows as in [14,22], so the following relation holds: log = log a.e. in (0, T ) × Ω.…”

Section: Lemma 65 (Discrete Effective Viscous Flux)mentioning

confidence: 99%

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Convergence of a mixed method for a semi-stationary compressible Stokes system

Karlsen

Karper

2010

Math. Comp.

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Abstract. We propose and analyze a finite element method for a semistationary Stokes system modeling compressible fluid flow subject to a Navierslip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nédélec spaces of the first kind, while the continuity equation is approximated by a piecewise constant upwind discontinuous Galerkin method. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only div or curl conforming. (ii) To prove the strong convergence of the discontinuous Galerkin approximations, which is required in view of a nonlinear pressure function. Some tools involved in the analysis include a higher space-time integrability estimate for the discontinuous Galerkin approximations, an equation for the effective viscous flux, various renormalized formulations of the discontinuous Galerkin method, and weak convergence arguments.

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Section: Preliminary Materialsmentioning

confidence: 99%

“…Now, to conclude the proof, we appeal to Proposition 4.2 in [14] stating that the square-integrable weak solution˜ is also a renormalized solution.…”

Section: Whereũ(t) Denotes the Extension Of U(t)mentioning

confidence: 99%

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Convergence of a mixed method for a semi-stationary compressible Stokes system

Karlsen

Karper

2010

Math. Comp.

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“…For the convenience of the reader we collect these results in a lemma (for proofs, see, for example, [4] …”

Section: Some Preliminariesmentioning

confidence: 99%

Convergent difference schemes for the Hunter–Saxton equation

2007

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Abstract. We propose and analyze several finite difference schemes for the Hunter-Saxton equationThis equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.

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Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum

Xin

2021

Comm Pure Appl Math

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In the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one-dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than O. 1x 2 /. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi-type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

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Dynamics of Viscous Compressible Fluids

Abstract: The goal of this lecture series is to discuss the main ingredients of the mathematical theory describing the time evolution of a viscous, compressible, and heat conducting fluid. The principal topics can be specified as follows:

Cited by 407 publications

References 44 publications

Convergence of a mixed method for a semi-stationary compressible Stokes system

Convergence of a mixed method for a semi-stationary compressible Stokes system

Convergent difference schemes for the Hunter–Saxton equation

Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum

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