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 & Continuous Dynamical Systems - A, 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948.  Google Scholar [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] T. P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1–32.  Google Scholar [20] B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Cambridge University Press, (2014), 145–164. doi: 10.1017/CBO9781139568050.009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

References:
 [1] A. Bressan, Hyperbolic Systems of Conservation Laws: The 1-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948.  Google Scholar [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] T. P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1–32.  Google Scholar [20] B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Cambridge University Press, (2014), 145–164. doi: 10.1017/CBO9781139568050.009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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] Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021 [2] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [3] Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 [4] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [5] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403 [6] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [7] Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298 [8] Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 [9] Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 [10] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [11] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [12] Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048 [13] Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262 [14] Jintai Ding, Zheng Zhang, Joshua Deaton. The singularity attack to the multivariate signature scheme HIMQ-3. Advances in Mathematics of Communications, 2021, 15 (1) : 65-72. doi: 10.3934/amc.2020043 [15] Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 [16] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [17] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [18] Paul E. Anderson, Timothy P. Chartier, Amy N. Langville, Kathryn E. Pedings-Behling. The rankability of weighted data from pairwise comparisons. Foundations of Data Science, 2021  doi: 10.3934/fods.2021002 [19] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [20] Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $p$-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

2019 Impact Factor: 1.338