Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws
Authors:
Anupam Sen, T. Raja Sekhar and V. D. Sharma
Journal:
Quart. Appl. Math. 75 (2017), 539-554
MSC (2010):
Primary 35L45, 35L65, 58J45; Secondary 35Q35, 35L67
DOI:
https://doi.org/10.1090/qam/1466
Published electronically:
March 15, 2017
MathSciNet review:
3636168
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Additional Information
Abstract: In this article, we study the interaction of delta shock waves for the one-dimensional strictly hyperbolic system of conservation laws with split delta function. We prove that Riemann solutions are stable under local small perturbations of the Riemann initial data. The global structure and large time asymptotic behaviour of the perturbed Riemann solutions are constructed and analyzed case by case.
References
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- Xiao Chen and Yuxi Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J. 59 (2010), no. 1, 231–256. MR 2666479, DOI https://doi.org/10.1512/iumj.2010.59.3752
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- Grey Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. 58 (2000), no. 1, 177–199. MR 1739044, DOI https://doi.org/10.1090/qam/1739044
- V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), no. 3, 401–427. MR 2169026, DOI https://doi.org/10.1090/S0033-569X-05-00961-8
- V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of $\delta $-shock waves in conservation law systems, J. Differential Equations 211 (2005), no. 2, 333–381. MR 2125546, DOI https://doi.org/10.1016/j.jde.2004.12.011
- Brian T. Hayes and Philippe G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), no. 6, 1547–1563. MR 1419460, DOI https://doi.org/10.1088/0951-7715/9/6/009
- Barbara Lee Keyfitz, Conservation laws, delta-shocks and singular shocks, Nonlinear theory of generalized functions (Vienna, 1997) Chapman & Hall/CRC Res. Notes Math., vol. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 99–111. MR 1699874
- Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080
- Marko Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Methods Appl. Sci. 27 (2004), no. 8, 931–955. MR 2055283, DOI https://doi.org/10.1002/mma.480
- Marko Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math. 66 (2008), no. 2, 281–302. MR 2416774, DOI https://doi.org/10.1090/S0033-569X-08-01109-5
- M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008), no. 2, 1143–1157. MR 2426339, DOI https://doi.org/10.1016/j.jmaa.2008.03.040
- Vishnu D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 142, CRC Press, Boca Raton, FL, 2010. MR 2668539
- Chun Shen and Meina Sun, Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws, Nonlinear Anal. 73 (2010), no. 10, 3284–3294. MR 2680022, DOI https://doi.org/10.1016/j.na.2010.07.008
- Meina Sun, Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett. 26 (2013), no. 6, 631–637. MR 3028067, DOI https://doi.org/10.1016/j.aml.2013.01.002
- De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
- E. Yu. Panov and V. M. Shelkovich, $\delta ’$-shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations 228 (2006), no. 1, 49–86. MR 2254184, DOI https://doi.org/10.1016/j.jde.2006.04.004
References
- Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
- Xiao Chen and Yuxi Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J. 59 (2010), no. 1, 231–256. MR 2666479, DOI https://doi.org/10.1512/iumj.2010.59.3752
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- Grey Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. 58 (2000), no. 1, 177–199. MR 1739044, DOI https://doi.org/10.1090/qam/1739044
- V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), no. 3, 401–427. MR 2169026, DOI https://doi.org/10.1090/S0033-569X-05-00961-8
- V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of $\delta$-shock waves in conservation law systems, J. Differential Equations 211 (2005), no. 2, 333–381. MR 2125546, DOI https://doi.org/10.1016/j.jde.2004.12.011
- Brian T. Hayes and Philippe G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), no. 6, 1547–1563. MR 1419460, DOI https://doi.org/10.1088/0951-7715/9/6/009
- Barbara Lee Keyfitz, Conservation laws, delta-shocks and singular shocks, Nonlinear theory of generalized functions (Vienna, 1997) Chapman & Hall/CRC Res. Notes Math., vol. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 99–111. MR 1699874
- Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080
- Marko Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Methods Appl. Sci. 27 (2004), no. 8, 931–955. MR 2055283, DOI https://doi.org/10.1002/mma.480
- Marko Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math. 66 (2008), no. 2, 281–302. MR 2416774, DOI https://doi.org/10.1007/s00205-009-0281-2
- M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008), no. 2, 1143–1157. MR 2426339, DOI https://doi.org/10.1016/j.jmaa.2008.03.040
- Vishnu D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 142, CRC Press, Boca Raton, FL, 2010. MR 2668539
- Chun Shen and Meina Sun, Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws, Nonlinear Anal. 73 (2010), no. 10, 3284–3294. MR 2680022, DOI https://doi.org/10.1016/j.na.2010.07.008
- Meina Sun, Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett. 26 (2013), no. 6, 631–637. MR 3028067, DOI https://doi.org/10.1016/j.aml.2013.01.002
- De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
- E. Yu. Panov and V. M. Shelkovich, $\delta ’$-shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations 228 (2006), no. 1, 49–86. MR 2254184, DOI https://doi.org/10.1016/j.jde.2006.04.004
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Additional Information
Anupam Sen
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India
Email:
sen.anupam123@gmail.com
T. Raja Sekhar
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India
MR Author ID:
831418
Email:
trajasekhar@maths.iitkgp.ernet.in
V. D. Sharma
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai-76, India
Email:
vsharma@math.iitb.ac.in
Keywords:
Delta shock wave,
delta contact discontinuity,
wave interactions,
Riemann problem,
split delta function
Received by editor(s):
October 6, 2016
Received by editor(s) in revised form:
February 5, 2017
Published electronically:
March 15, 2017
Article copyright:
© Copyright 2017
Brown University