Lower bound of density for Lipschitz continuous solutions in the isentropic gas dynamics

Chen, Geng; Pan, Ronghua; Zhu, Shengguo

doi:10.48550/arxiv.1410.3182

Cited by 3 publications

(5 citation statements)

References 9 publications

(16 reference statements)

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“…Although the time-dependent bound is not as sharp as that in [18], the proof is much simpler and elementary. A generalization of [18] with O(1) 1+t bound on density to general initial data for all γ > 1 has been carried out in our work [4]. This O (1) 1+t rate is optimal for generic C 1 initial data due to the example in [9,28].…”

Section: Hence This Theorem Can Be Understood As That Classical Globa...mentioning

confidence: 99%

“…One of the main contributions of this paper is to provide a good enough new time-dependent estimate on the density lower bound for generic C 1 initial data away from vacuum, when 1 < γ < 3. The idea we developed here is simple and neat, but does not offer the optimal rate 1 1+t , which is achieved through a much more complicated method in our preprint [4] for generic C 1 initial data away from vacuum.…”

Section: Introductionmentioning

confidence: 99%

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Singularity Formation for the Compressible Euler Equations

Chen¹,

Pan²,

Zhu³

2017

SIAM J. Math. Anal.

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It is well-known that singularity will develop in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including Lax [14], John [13], Liu [22], Li-Zhou-Kong [16], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field, under some structural conditions. A natural question is that: Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on density lower bound, which is known to decay to zero as time goes to infinity for certain class of solutions. In this paper, we offer a simple way to characterize the decay of density lower bound in time, and therefore successfully classify the questions on singularity formation in compressible Euler equations. For isentropic flow, we offer a complete picture on the finite time singularity formation from smooth initial data away from vacuum, which is consistent with the small data theory. For adiabatic flow, we show a striking observation that initial weak compressions do not necessarily develop singularity in finite time. Furthermore, we follow [7] to introduce the critical strength of nonlinear compression, and prove that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there are a class of initial data admitting global smooth solutions with maximum strength of compression equals to this critical value.

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Section: Hence This Theorem Can Be Understood As That Classical Globa...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singularity Formation for the Compressible Euler Equations

Chen¹,

Pan²,

Zhu³

2017

SIAM J. Math. Anal.

Self Cite

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“…In fact, the function a 2 might vanish as time tends to infinity, such as for the gas dynamic case (c.f. [4,5,17]). Combining (4.12), (4.13) and (4.14), we can complete the proof.…”

Section: Singularity Formationmentioning

confidence: 99%

Singularity formation for the compressible Euler equations with general pressure law

Zheng

2016

Journal of Mathematical Analysis and Applications

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“…By studying Riemann's construction, Lipschitz continuous examples for isentropic Euler equations (1.6)∼(1.8) were provided in Section 82 in [9], in which the function min x∈R ρ(x, t) was proved to decay to zero in an order of O(1+t) −1 as t → ∞, while the initial density is uniformly away from zero. 1 A relative detailed discussion can be found in [5], when the adiabatic constant γ = 2N +1 2N −1 with any positive integer N . Then there were many articles working on time-dependent lower bound on density for general classical solutions of isentropic Euler equations (1.6)∼(1.8) under assumption that initial density is uniformly positive.…”

Section: Introductionmentioning

confidence: 99%

“…Using this result together with Lax's decomposition in [12], Pan, Zhu and the author proved that gradient blowup of u and/or τ happens in finite time if and only if the initial data are forward or backward compressive somewhere. Next, for general Lipschitz continuous solution, Pan, Zhu and the author in [5] improved the lower bound on density from the order of O(1 + t) −4/(3−γ) to the optimal order O(1 + t) −1 by introducing a polygonal scheme. The advantage of this method is that it works for not only classical solutions but also Lipschitz continuous solutions.…”

Section: Introductionmentioning

confidence: 99%

Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

Chen¹

2017

Indiana Univ. Math. J.

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Abstract. For the compressible Euler equations, even when initial data are uniformly away from vacuum, solutions can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this paper, for the initial value problems of isentropic and full Euler equations in one space dimension, assuming the initial density has positive lower bound, we prove that density functions in classical solutions have positive lower bounds in the order of O(1 + t) −1 and O(1 + t) −1−δ for any 0 < δ ≪ 1, respectively, where t is time. The orders of these bounds are optimal or almost optimal, respectively. Furthermore, for classical solutions in Eulerian coordinates (y, t) ∈ R × [0, T ), we show velocity u satisfies that u y (y, t) is uniformly bounded from above by a constant independent of T , although u y (y, t) tends to negative infinity when gradient blowup happens, i.e. when shock forms, in finite time.2010 Mathematical Subject Classification: 76N15, 35L65, 35L67.

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Lower bound of density for Lipschitz continuous solutions in the isentropic gas dynamics

Cited by 3 publications

References 9 publications

Singularity Formation for the Compressible Euler Equations

Singularity Formation for the Compressible Euler Equations

Singularity formation for the compressible Euler equations with general pressure law

Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

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