Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
HTML articles powered by AMS MathViewer
- by Gui-Qiang Chen and Hermano Frid PDF
- Trans. Amer. Math. Soc. 353 (2001), 1103-1117 Request permission
Abstract:
We prove the uniqueness of Riemann solutions in the class of entropy solutions in $L^\infty \cap BV_{loc}$ for the $3\times 3$ system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global $L^2$-stability of the Riemann solutions even in the class of entropy solutions in $L^\infty$ with arbitrarily large oscillation for the $3\times 3$ system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under $L^1$ perturbation of the Riemann initial data, as long as the corresponding solutions are in $L^\infty$ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions $U(x,t)$, piecewise Lipschitz in $x$, for any $t>0$.References
- Bressan, A., Crasta, G., Piccoli, B., Well posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Memoirs Amer. Math. Soc. 146 (2000), no. 694.
- Alberto Bressan and Philippe LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rational Mech. Anal. 140 (1997), no. 4, 301–317. MR 1489317, DOI 10.1007/s002050050068
- Alberto Bressan, Tai-Ping Liu, and Tong Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1–22. MR 1723032, DOI 10.1007/s002050050165
- Gui-Qiang Chen and Hermano Frid, Large-time behavior of entropy solutions of conservation laws, J. Differential Equations 152 (1999), no. 2, 308–357. MR 1674529, DOI 10.1006/jdeq.1998.3527
- Gui-Qiang Chen and Hermano Frid, Large-time behavior of entropy solutions in $L^\infty$ for multidimensional conservation laws, Advances in nonlinear partial differential equations and related areas (Beijing, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 28–44. MR 1690821
- Gui-Qiang Chen and Hermano Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89–118. MR 1702637, DOI 10.1007/s002050050146
- C. M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 107 (1989), no. 2, 127–155. MR 996908, DOI 10.1007/BF00286497
- C. M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1640, Springer, Berlin, 1996, pp. 48–69. MR 1600904, DOI 10.1007/BFb0093706
- Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, DOI 10.1512/iumj.1979.28.28011
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- A. F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 (93) (1960), 99–128 (Russian). MR 0114016
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Conf. Board Math. Sci. Regional Conf. Ser. Appl. Math., vol. 11, SIAM, Philadelphia, 1973.
- Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR 0393870
- Tai Ping Liu, Initial-boundary value problems for gas dynamics, Arch. Rational Mech. Anal. 64 (1977), no. 2, 137–168. MR 433017, DOI 10.1007/BF00280095
- Tai-Ping Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), no. 6, 767–796. MR 499781, DOI 10.1002/cpa.3160300605
- Liu, T.-P. and Yang, T., Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 1553–1586.
- Yue-Jun Peng, Solutions faibles globales pour l’équation d’Euler d’un fluide compressible avec de grandes données initiales, Comm. Partial Differential Equations 17 (1992), no. 1-2, 161–187 (French, with English summary). MR 1151259, DOI 10.1080/03605309208820837
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146, DOI 10.1007/978-1-4684-0152-3
- J. Blake Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations 41 (1981), no. 1, 96–161. MR 626623, DOI 10.1016/0022-0396(81)90055-3
- B. Temple and R. Young, The large time stability of sound waves, Comm. Math. Phys. 179 (1996), no. 2, 417–466. MR 1400747, DOI 10.1007/BF02102596
- A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338
- David H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations 68 (1987), no. 1, 118–136. MR 885816, DOI 10.1016/0022-0396(87)90188-4
Additional Information
- Gui-Qiang Chen
- Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208
- MR Author ID: 249262
- ORCID: 0000-0001-5146-3839
- Email: gqchen@math.northwestern.edu
- Hermano Frid
- Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil
- Email: hermano@im.ufrj.br
- Received by editor(s): February 8, 1999
- Received by editor(s) in revised form: October 4, 1999
- Published electronically: September 21, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 1103-1117
- MSC (2000): Primary 35B40, 35L65; Secondary 35B35, 76N15
- DOI: https://doi.org/10.1090/S0002-9947-00-02660-X
- MathSciNet review: 1804414