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doi: 10.3934/dcds.2021062

Singularity formation for compressible Euler equations with time-dependent damping

Ying Sui and  Huimin Yu ,

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

* Corresponding author: hmyu@sdnu.edu.cn

Received  September 2020 Revised  January 2021 Published  March 2021

Fund Project: This work is supported in part by the National Natural Science Foundation of China (Grant No. 11671237)

In this paper, we consider the compressible Euler equations with time-dependent damping $ \frac{{\alpha}}{(1+t)^\lambda}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 $ \lambda\neq1 $ and $ \lambda = 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<\gamma<3 $ we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.

Keywords: Singularity formation, compressible Euler equations, time-dependent damping, shock wave.
Mathematics Subject Classification: Primary: 35Q31, 76N10; Secondary: 76L05.
Citation: Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021062
References:
[1]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd ed., Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[2]

P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154.  Google Scholar

[3]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.  Google Scholar

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M. Slemrod, Damped conservation laws in continuum mechanics, Nonlinear Analysis and Mechanics, Vol. Ⅲ, Pitman, New York, 30 (1978), 135–173.  Google Scholar

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G. Chen, R. Pan and S. Zhu, Singularity formation for the compressible Euler equations, SIAM J. Math. Anal., 49 (2017), 2591-2614.  doi: 10.1137/16M1062818.  Google Scholar

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L. Hsiao and T.-P. Liu, Convergence to diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599-605.  doi: 10.1007/BF02099268.  Google Scholar

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P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differ. Equ., 84 (1990), 129-147.  doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

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P. Marcati and M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Q. Appl. Math., 58 (2000), 763-784.  doi: 10.1090/qam/1788427.  Google Scholar

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P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), S224–S240. doi: 10.1007/s00021-005-0155-9.  Google Scholar

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K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differ. Equ., 131 (1996), 171-188.  doi: 10.1006/jdeq.1996.0159.  Google Scholar

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K. Nishihara, W. K. Wang and T. Yang, Lp-convergence rates to nonlinear diffusion waves for p-system with damping, J. Differ. Equ., 161 (2000), 191-218.  doi: 10.1006/jdeq.1999.3703.  Google Scholar

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T. Sideris, B. Thomases and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Commun. Partial Differ. Equ., 28 (2003), 795-816.  doi: 10.1081/PDE-120020497.  Google Scholar

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X. Pan, Blow up of solutions to 1-d Euler equations with time-dependent damping, J. Math. Anal. Appl., 442 (2016), 435-445.  doi: 10.1016/j.jmaa.2016.04.075.  Google Scholar

[15]

X. Pan, Global existence of solutions to 1-d Euler equations with time-dependent damping, Nonlinear Anal., 132 (2016), 327-336.  doi: 10.1016/j.na.2015.11.022.  Google Scholar

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Y. Sugiyama, Singularity formation for the 1D compressible Euler equations with variable damping coefficient, Nonlinear Anal., 170 (2018), 70-87.  doi: 10.1016/j.na.2017.12.013.  Google Scholar

[17]

S. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-up solutions for compressible Euler equations with time-dependent damping, J. Differential Equations, 268 (2020), 5033-5077.  doi: 10.1016/j.jde.2019.11.002.  Google Scholar

[18]

F. Hou, I. Witt and H. Yin, Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping, Pac. J. Math., 292 (2018), 389-426.  doi: 10.2140/pjm.2018.292.389.  Google Scholar

[19]

F. Hou and H. Yin, On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping, Nonlinearity, 30 (2017), 2485-2517.  doi: 10.1088/1361-6544/aa6d93.  Google Scholar

[20]

H. Cui, H. Yin, J. Zhang and C. Zhu, Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending damping, J. Differ. Equ., 264 (2018), 4564-4602.  doi: 10.1016/j.jde.2017.12.012.  Google Scholar

[21]

G. Chen, R. Young and Q. Zhang, Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10 (2013), 149-172.  doi: 10.1142/S0219891613500069.  Google Scholar

[22]

F. Huang, R. Pan and Z. Wang, $L^1$ Convergence to the Barenblatt solution for compressible Euler equations with damping, Archive for Rational Mechanics and Analysis, 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.  Google Scholar

[23]

X. Fang and H. Yu, Uniform boundedness in time of weak solutions to a kind of dissipative system, J. Math. Anal. Appl., 461 (2018), 1153-1164.  doi: 10.1016/j.jmaa.2018.01.047.  Google Scholar

[24]

P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-system, Comm. Math. Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar

show all references

References:
[1]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd ed., Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[2]

P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154.  Google Scholar

[3]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.  Google Scholar

[4]

M. Slemrod, Damped conservation laws in continuum mechanics, Nonlinear Analysis and Mechanics, Vol. Ⅲ, Pitman, New York, 30 (1978), 135–173.  Google Scholar

[5]

G. Chen, R. Pan and S. Zhu, Singularity formation for the compressible Euler equations, SIAM J. Math. Anal., 49 (2017), 2591-2614.  doi: 10.1137/16M1062818.  Google Scholar

[6]

L. Lin, Existence and non-existence of global smooth solutions for quasilinear hyperbolic systems, Chin. Ann. of math., 9 (1988), 372-377.   Google Scholar

[7]

L. Hsiao and T.-P. Liu, Convergence to diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599-605.  doi: 10.1007/BF02099268.  Google Scholar

[8]

P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differ. Equ., 84 (1990), 129-147.  doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

[9]

P. Marcati and M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Q. Appl. Math., 58 (2000), 763-784.  doi: 10.1090/qam/1788427.  Google Scholar

[10]

P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), S224–S240. doi: 10.1007/s00021-005-0155-9.  Google Scholar

[11]

K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differ. Equ., 131 (1996), 171-188.  doi: 10.1006/jdeq.1996.0159.  Google Scholar

[12]

K. Nishihara, W. K. Wang and T. Yang, Lp-convergence rates to nonlinear diffusion waves for p-system with damping, J. Differ. Equ., 161 (2000), 191-218.  doi: 10.1006/jdeq.1999.3703.  Google Scholar

[13]

T. Sideris, B. Thomases and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Commun. Partial Differ. Equ., 28 (2003), 795-816.  doi: 10.1081/PDE-120020497.  Google Scholar

[14]

X. Pan, Blow up of solutions to 1-d Euler equations with time-dependent damping, J. Math. Anal. Appl., 442 (2016), 435-445.  doi: 10.1016/j.jmaa.2016.04.075.  Google Scholar

[15]

X. Pan, Global existence of solutions to 1-d Euler equations with time-dependent damping, Nonlinear Anal., 132 (2016), 327-336.  doi: 10.1016/j.na.2015.11.022.  Google Scholar

[16]

Y. Sugiyama, Singularity formation for the 1D compressible Euler equations with variable damping coefficient, Nonlinear Anal., 170 (2018), 70-87.  doi: 10.1016/j.na.2017.12.013.  Google Scholar

[17]

S. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-up solutions for compressible Euler equations with time-dependent damping, J. Differential Equations, 268 (2020), 5033-5077.  doi: 10.1016/j.jde.2019.11.002.  Google Scholar

[18]

F. Hou, I. Witt and H. Yin, Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping, Pac. J. Math., 292 (2018), 389-426.  doi: 10.2140/pjm.2018.292.389.  Google Scholar

[19]

F. Hou and H. Yin, On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping, Nonlinearity, 30 (2017), 2485-2517.  doi: 10.1088/1361-6544/aa6d93.  Google Scholar

[20]

H. Cui, H. Yin, J. Zhang and C. Zhu, Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending damping, J. Differ. Equ., 264 (2018), 4564-4602.  doi: 10.1016/j.jde.2017.12.012.  Google Scholar

[21]

G. Chen, R. Young and Q. Zhang, Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10 (2013), 149-172.  doi: 10.1142/S0219891613500069.  Google Scholar

[22]

F. Huang, R. Pan and Z. Wang, $L^1$ Convergence to the Barenblatt solution for compressible Euler equations with damping, Archive for Rational Mechanics and Analysis, 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.  Google Scholar

[23]

X. Fang and H. Yu, Uniform boundedness in time of weak solutions to a kind of dissipative system, J. Math. Anal. Appl., 461 (2018), 1153-1164.  doi: 10.1016/j.jmaa.2018.01.047.  Google Scholar

[24]

P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-system, Comm. Math. Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar

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