# American Institute of Mathematical Sciences

July  2019, 39(7): 4259-4277. doi: 10.3934/dcds.2019172

## A polygonal scheme and the lower bound on density for the isentropic gas dynamics

 1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA 2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA 3 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

* Corresponding author

Received  November 2018 Published  April 2019

Fund Project: G. Chen is partially supported by National Science Foundation under grant DMS-1715012. R. Pan is partially supported by National Science Foundation under grants DMS-1516415 and DMS-1813603, and by National Natural Science Foundation of China under grant 11628103. S. Zhu is partially supported by National Natural Science Foundation of China under grants 11231006 and 11571232, Natural Science Foundation of Shanghai under grant 14ZR1423100, Australian Research Council grant DP170100630, Newton International Fellowships NF170015 and China Scholarship Council.

Positive density lower bound is one of the major obstacles toward large data theory for one dimensional isentropic compressible Euler equations, also known as p-system in Lagrangian coordinates. The explicit example first studied by Riemann shows that the lower bound of density can decay to zero as time goes to infinity of the order $O\left( {\frac{1}{{1 + t}}} \right)$, even when initial density is uniformly positive. In this paper, we establish a proof of the lower bound on density in its optimal order $O\left( {\frac{1}{{1 + t}}} \right)$ using a method of polygonal scheme.

Citation: Geng Chen, Ronghua Pan, Shengguo Zhu. A polygonal scheme and the lower bound on density for the isentropic gas dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4259-4277. doi: 10.3934/dcds.2019172
##### References:
 [1] A. Bressan, Hyperbolic Systems of Conservation Laws: The 1-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000. [2] A. Bressan, G. Chen and Q. Zhang, Lack of BV bounds for approximate solutions to the $p$-system with large data, J. Differential Equations, 256 (2014), 3067-3085.  doi: 10.1016/j.jde.2014.01.032. [3] T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [4] G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 671-690.  doi: 10.1142/S0219891611002536. [5] G. Chen and H. K. Jenssen, No TVD fields for 1-D isentropic gas flow, Comm. Partial Differential Equations, 38 (2013), 629-657.  doi: 10.1080/03605302.2012.755543. [6] 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. [7] 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. [8] G. Chen and R. Young, Shock formation and exact solutions for the compressible Euler equation, Arch. Rational Mech. Anal., 217 (2015), 1265–1293. doi: 10.1007/s00205-015-0854-1. [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948. [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1. [11] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33–41. doi: 10.1016/0022-247X(72)90114-X. [12] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 20 (1976), 187–212. doi: 10.1016/0022-0396(76)90102-9. [13] H. Holden and N. H. Risebro, Front tracking for Hyperbolic Conservation Laws, New York: Springer, 2002. doi: 10.1007/978-3-642-56139-9. [14] H. K. Jessen, On exact solutions of rarefaction-rarefaction interactions in compressible isentropic flow, J. Math. Fluid Mech., 19 (2017), 685–708. doi: 10.1007/s00021-016-0309-y. [15] P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154. [16] L. Lin, Vacuum states and equidistribution of the random sequence for Glimm's scheme, J. Math. Anal. Appl., 124 (1987), 117-126.  doi: 10.1016/0022-247X(87)90028-X. [17] L. Lin, On the vacuum state for the equations of isentropic gas dynamics, Math. Anal. Appl., 121 (1987), 406-425.  doi: 10.1016/0022-247X(87)90253-8. [18] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Appl. Math., 1 (1980), 345-359.  doi: 10.1016/0196-8858(80)90016-0. [19] T. P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1–32. [20] B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Cambridge University Press, (2014), 145–164. doi: 10.1017/CBO9781139568050.009. [21] B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), 341-363.  doi: 10.4310/MAA.2009.v16.n3.a5. [22] C. Tsikkou, Sharper total variation bounds for the $p$-system of fluid dynamics, J. Hyperbolic Differ. Equ., 8 (2011), 173-232.  doi: 10.1142/S0219891611002391. [23] D. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations, 68 (1987), 118–136. doi: 10.1016/0022-0396(87)90188-4.

show all references

##### References:
 [1] A. Bressan, Hyperbolic Systems of Conservation Laws: The 1-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000. [2] A. Bressan, G. Chen and Q. Zhang, Lack of BV bounds for approximate solutions to the $p$-system with large data, J. Differential Equations, 256 (2014), 3067-3085.  doi: 10.1016/j.jde.2014.01.032. [3] T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [4] G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 671-690.  doi: 10.1142/S0219891611002536. [5] G. Chen and H. K. Jenssen, No TVD fields for 1-D isentropic gas flow, Comm. Partial Differential Equations, 38 (2013), 629-657.  doi: 10.1080/03605302.2012.755543. [6] 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. [7] 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. [8] G. Chen and R. Young, Shock formation and exact solutions for the compressible Euler equation, Arch. Rational Mech. Anal., 217 (2015), 1265–1293. doi: 10.1007/s00205-015-0854-1. [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948. [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1. [11] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33–41. doi: 10.1016/0022-247X(72)90114-X. [12] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 20 (1976), 187–212. doi: 10.1016/0022-0396(76)90102-9. [13] H. Holden and N. H. Risebro, Front tracking for Hyperbolic Conservation Laws, New York: Springer, 2002. doi: 10.1007/978-3-642-56139-9. [14] H. K. Jessen, On exact solutions of rarefaction-rarefaction interactions in compressible isentropic flow, J. Math. Fluid Mech., 19 (2017), 685–708. doi: 10.1007/s00021-016-0309-y. [15] P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154. [16] L. Lin, Vacuum states and equidistribution of the random sequence for Glimm's scheme, J. Math. Anal. Appl., 124 (1987), 117-126.  doi: 10.1016/0022-247X(87)90028-X. [17] L. Lin, On the vacuum state for the equations of isentropic gas dynamics, Math. Anal. Appl., 121 (1987), 406-425.  doi: 10.1016/0022-247X(87)90253-8. [18] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Appl. Math., 1 (1980), 345-359.  doi: 10.1016/0196-8858(80)90016-0. [19] T. P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1–32. [20] B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Cambridge University Press, (2014), 145–164. doi: 10.1017/CBO9781139568050.009. [21] B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), 341-363.  doi: 10.4310/MAA.2009.v16.n3.a5. [22] C. Tsikkou, Sharper total variation bounds for the $p$-system of fluid dynamics, J. Hyperbolic Differ. Equ., 8 (2011), 173-232.  doi: 10.1142/S0219891611002391. [23] D. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations, 68 (1987), 118–136. doi: 10.1016/0022-0396(87)90188-4.
Interaction of two centered rarefaction waves
Definition of forward (resp. backward) R/C characters for the jump edge $l_1$ (resp. $l_2$). The picture is on the $(x,t)$-plane
Proof of Lemma 3.4. The picture is on the $(x,t)$-plane
Modify blocks into diamonds: interior block 3 (pentagon whose lower boundary is on initial line); boundary block 1 or 6 which includes one intersection point between jump edges. After modifications, diamonds 1~6 are all called complete diamonds
Left: Modify blocks into diamonds; Right: the definition of $t = L^{(n)}_0(x)$
Proof of Lemma 4.2. $l_2$ and $l'_2$ are parallel with each other. The picture is on the $(x,t)$-plane
From left to right: the middle diamonds are: $R_rC_c$; $R_r C_r$; and $R_rR_c$ diamonds, respectively, where forward character always goes first. The picture $(x,t)$-plane
The proof of (24) on a diamond satisfying (25). The picture is on the $(x,t)$-plane
The center diamond is a $R_rR_r$ block. The picture is on the $(x,t)$-plane
Bound on ${v}^{(n)}$ in a $R_rR_r$ district. In the figure, we omit the subscript ${(n)}$ for convenience. The picture is on the $(x,t)$-plane
 [1] Feimin Huang, Yeping Li. Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 455-470. doi: 10.3934/dcds.2009.24.455 [2] Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575 [3] Bernard Ducomet, Alexander Zlotnik. On a regularization of the magnetic gas dynamics system of equations. Kinetic and Related Models, 2013, 6 (3) : 533-543. doi: 10.3934/krm.2013.6.533 [4] Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic and Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041 [5] Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021296 [6] Weixia Zhao. The expansion of gas from a wedge with small angle into a vacuum. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2319-2330. doi: 10.3934/cpaa.2013.12.2319 [7] Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27 [8] Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292 [9] Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549 [10] Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 [11] Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062 [12] Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021 [13] Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111 [14] Renhui Wan. The 3D liquid crystal system with Cannone type initial data and large vertical velocity. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5521-5539. doi: 10.3934/dcds.2017240 [15] Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041 [16] Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477 [17] Walter A. Strauss, Masahiro Suzuki. Large amplitude stationary solutions of the Morrow model of gas ionization. Kinetic and Related Models, 2019, 12 (6) : 1297-1312. doi: 10.3934/krm.2019050 [18] Yongqin Liu, Shuichi Kawashima. Asymptotic behavior of solutions to a model system of a radiating gas. Communications on Pure and Applied Analysis, 2011, 10 (1) : 209-223. doi: 10.3934/cpaa.2011.10.209 [19] Chun Liu, Jan-Eric Sulzbach. The Brinkman-Fourier system with ideal gas equilibrium. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 425-462. doi: 10.3934/dcds.2021123 [20] Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic and Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001

2020 Impact Factor: 1.392