# 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, 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

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##### 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
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