Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas

Kazhikhov, A. V.; Shelukhin, V. V.

doi:10.1016/0021-8928(77)90011-9

Cited by 492 publications

(457 citation statements)

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“…It follows from (2.1) and (1.11) that for any α ∈ [2,3], 15) which together with Holder's inequality yields that 16) where in the second inequality we have used (2.1) and the following simple fact that for any w ∈ H 1 (Ω),…”

Section: Lemma 21 It Holds Thatmentioning

confidence: 97%

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Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

Liang

2015

Arch Rational Mech Anal

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This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heatconducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

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Section: Lemma 21 It Holds Thatmentioning

confidence: 97%

“…Finally, it follows from the proof in [1,16] that there exists some constant c > 2 such that for all (x, t) ∈ Ω × [0, ∞) c −1 e −ct ≤ θ(x, t), which together with (2.36) implies that for all (x, t) ∈ Ω × [0, ∞) c −1 e −cT 0 ≤ θ(x, t).…”

Section: Lemma 24 It Holds Thatmentioning

confidence: 98%

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

Liang

2015

Arch Rational Mech Anal

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“…In Sect. 3 we establish the upper and lower bounds for the specific volume u, adapting the methods by Kazhikhov-Shelukhin [11,12] and Jiang [8,9]. Next, in Sect.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…A key ingredient of the proof is the following "localisation trick" in [11,12] by Kazhikhov-Shelukhin. For self-containedness we include the proof below:…”

Section: Uniform Bounds For the Specific Volume Umentioning

confidence: 99%

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On one-dimensional compressible Navier–Stokes equations for a reacting mixture in unbounded domains

2017

Z. Angew. Math. Phys.

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Abstract. In this paper we consider the one-dimensional Navier-Stokes system for a heat-conducting, compressible reacting mixture which describes the dynamic combustion of fluids of mixed kinds on unbounded domains. This model has been discussed on bounded domains by Chen (SIAM J Math Anal 23:609-634, 1992) and Chen-Hoff-Trivisa (Arch Ration Mech Anal 166:321-358, 2003), among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius' law of thermodynamics. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in H 1 . Moreover, the large-time behaviour of the weak solution is identified. In particular, the uniform-in-time bounds for the temperature and specific volume have been established via energy estimates. For this purpose we utilise techniques developed by Kazhikhov-Shelukhin (cf. Kazhikhov in Siber Math J 23:44-49, 1982; Solonnikov and Kazhikhov in Annu Rev Fluid Mech 13:79-95, 1981) and refined by Jiang (Commun Math Phys 200:181-193, 1999, Proc R Soc Edinb Sect A 132:627-638, 2002, as well as a crucial estimate in the recent work by Li-Liang (Arch Ration Mech Anal 220:1195-1208. Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.Mathematics Subject Classification. Primary 35Q30, 35Q35, 35Q79; Secondary 76N10, 76N15.

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Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

Huang

Xin

2011

Comm Pure Appl Math

452

398

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We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions which may have large oscillations and can contain vacuum states.

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Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas

Cited by 492 publications

References 1 publication

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

On one-dimensional compressible Navier–Stokes equations for a reacting mixture in unbounded domains

Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

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