Singularity formation for the 1D compressible Euler equations with variable damping coefficient

Sugiyama, Yuusuke

doi:10.1016/j.na.2017.12.013

Cited by 39 publications

(31 citation statements)

References 35 publications

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“…For the case −1 < λ < 1, the story comes from what follows. As clearly shown in [2,21,29,30,33] for the 1-D compressible Euler equations (the so-called p-system) with time-dependent damping  …”

Section: (13)mentioning

confidence: 98%

Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Wu¹,

Luan²

2021

CPAA

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Section: (13)mentioning

confidence: 98%

Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Wu¹,

Luan²

2021

CPAA

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“…As for the time-dependent damping model (1.1), Pan [14,15] gave the thresholds of α and λ to separate the existence and nonexistence of global smooth solutions in small data regime. Sugiyama [16] obtained the sharp upper and lower estimates of life span for some cases of λ and α. Recently, [17] gave some sufficient conditions to make all solutions blow up with monotonic initial Riemann invariants.…”

Section: Ying Sui and Huimin Yumentioning

confidence: 99%

Singularity formation for compressible Euler equations with time-dependent damping

Sui¹,

Yu²

2021

DCDS

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In this paper, we consider the compressible Euler equations with time-dependent damping α (1+t) λ u in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case λ = 1 and λ = 1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1 < γ < 3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.

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“…Therefore, the study on this problem is full of challenges and quite incomplete. Due to the practicability and strong relation with Euler equation, it has attracted extensive attention of many scholars in recent years, see [8][9][10][11][12][13][14][15][16][17] for example.…”

Section: Introductionmentioning

confidence: 99%

Vacuum and singularity formation for compressible Euler equations with time-dependent damping

Ying¹,

Wang²,

Yu³

2022

Preprint

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In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For 1 < γ ≤ 3, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all λ, which was open in [1] for some cases. Moreover, the singularity formation of the compressible Euler equations when γ = 3 is investigated, too.

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Singularity formation for the 1D compressible Euler equations with variable damping coefficient

Cited by 39 publications

References 35 publications

Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Singularity formation for compressible Euler equations with time-dependent damping

Vacuum and singularity formation for compressible Euler equations with time-dependent damping

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