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Interaction of two centered rarefaction waves
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On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability
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 |
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.
Mathematics Subject Classification:
Primary: 76N15, 35L65; Secondary: 35L67.
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
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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. |
Figure 2.
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
Figure 3.
Proof of Lemma 3.4. The picture is on the $ (x,t) $ -plane
Figure 4.
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
Figure 5.
Left: Modify blocks into diamonds; Right: the definition of $ t = L^{(n)}_0(x) $
Figure 6.
Proof of Lemma 4.2. $ l_2 $ and $ l'_2 $ are parallel with each other. The picture is on the $ (x,t) $ -plane
Figure 7.
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
Figure 8.
The proof of (24) on a diamond satisfying (25). The picture is on the $ (x,t) $ -plane
Figure 9.
The center diamond is a $R_rR_r$ block. The picture is on the $(x,t)$-plane
Figure 10.
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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