# THE DELTA STANDING WAVE SOLUTION FOR THE LINEAR SCALAR CONSERVATION LAW WITH DISCONTINUOUS COEFFICIENTS USING A SELF-SIMILAR VISCOUS REGULARIZATION

• LI, XIUMEI (School of Mathematics and Statistics Science Ludong University) ;
• SHEN, CHUN (School of Mathematics and Statistics Science Ludong University)
• 투고 : 2014.09.18
• 발행 : 2015.11.30
• 90 6

#### 초록

This paper is mainly concerned with the formation of delta standing wave for the scalar conservation law with a linear flux function involving discontinuous coefficients by using the self-similar viscosity vanishing method. More precisely, we use the self-similar viscosity to smooth out the discontinuous coefficient such that the existence of approximate viscous solutions to the delta standing wave for the Riemann problem is established and then the convergence to the delta standing wave solution is also obtained when the viscosity parameter tends to zero. In addition, the Riemann problem is also solved with the standard method and the instability of Riemann solutions with respect to the specific small perturbation of initial data is pointed out in some particular situations.

#### 키워드

linear flux function;discontinuous coefficient;delta standing wave;viscosity vanishing method;Riemann problem;scalar conservation law

#### 참고문헌

1. A. Ambroso, B. Boutin, F. Coquel, E. Godlewski, and P. G. LeFloch, Coupling two scalar conservation laws via Dafermos self-similar regularization, Numerical Mathematics and Advanced Applications 2008 (2008), 209-216.
2. S. Bernard, J. F. Colombeau, A. Meril, and L. Remaki, Conservation laws with discon- tinuous coefficients, J. Math. Anal. Appl. 258 (2001), no. 1, 63-86. https://doi.org/10.1006/jmaa.2000.7360
3. F. Bouchut and G. Crippa, Uniqueness, renormalization and smooth approximations for linear transport equations, SIAM J. Math. Anal. 38 (2006), no. 4, 1316-1328. https://doi.org/10.1137/06065249X
4. F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), no. 7, 891-933. https://doi.org/10.1016/S0362-546X(97)00536-1
5. B. Boutin, F. Coquel, and E. Godlewski, Dafermos regularization for interface coupling of conservation laws, Hyperbolic problems: theory, numerics, applications, 567-575, Springer, Berlin, 2008
6. G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal. 34 (2003), no. 4, 925-938. https://doi.org/10.1137/S0036141001399350
7. C. M. Dafermos, Solutions of the Riemann problem for a class hyperbolic system of conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52 (1973), 1-9. https://doi.org/10.1007/BF00249087
8. C. M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems, Arch. Rational Mech. Anal. 106 (1989), no. 3, 243-260. https://doi.org/10.1007/BF00281215
9. V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations 245 (2008), no. 12, 3704-3734. https://doi.org/10.1016/j.jde.2008.03.006
10. V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of - shock waves in conservation law systems, J. Differential Equations 221 (2005), no. 2, 333-381.
11. V. G. Danilov and V. M. Shelkovich, Delta-shock waves type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), no. 3, 401-427. https://doi.org/10.1090/S0033-569X-05-00961-8
12. G. Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. 58 (2000), no. 1, 177-199. https://doi.org/10.1090/qam/1739044
13. E. Godlewski and P. A. Raviart, The numerical interface coupling of nonlinear hyper- bolic systems of conservation laws: The scalar case, Numer. Math. 97 (2004), no. 1, 81-130. https://doi.org/10.1007/s00211-002-0438-5
14. L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp. 69 (2000), no. 231, 987- 1015. https://doi.org/10.1090/S0025-5718-00-01185-6
15. L. Guo, W. Sheng, and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal. 9 (2010), no. 2, 431-458.
16. J. Hu, A limiting viscosity approach to Riemann solutions containing Delta-shock waves for non-strictly hyperbolic conservation laws, Quart. Appl. Math. 55 (1997), no. 2, 361- 372. https://doi.org/10.1090/qam/1447583
17. J. Hu, The Riemann problem for a resonant nonlinear system of conservation laws with Dirac-measure solutions, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 1, 81-94. https://doi.org/10.1017/S0308210500027165
18. F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys. 222 (2001), no. 1, 117-146. https://doi.org/10.1007/s002200100506
19. E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 52 (1992), no. 5, 1260-1278. https://doi.org/10.1137/0152073
20. S. Jin and P. Qi, $l^1$-error estimates on the immersed interface upwind scheme for linear convertion equations with piecewise constant coefficients: a simple proof, Science China: Math. 56 (2013), 2773-2782. https://doi.org/10.1007/s11425-013-4738-2
21. H. Kalisch and D. Mitrovic, Singular solutions of a fully nonlinear 2 $\times$ 2 system of conservation laws, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 3, 711-729. https://doi.org/10.1017/S0013091512000065
22. Y. J. Kim, A self-similar viscosity approach for the Riemann problem in isentropic gas dynamics and the structure of the solutions, Quart. Appl. Math. 59 (2001), no. 4, 637-665. https://doi.org/10.1090/qam/1866552
23. G. Lai, W. Sheng, and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst. 31 (2011), no. 2, 489-523. https://doi.org/10.3934/dcds.2011.31.489
24. J. Li and H. Yang, Delta-shocks as limits of vanishing viscosity for multidimensional zero-presure gas dynamics, Quart. Appl. Math. 59 (2001), no. 2, 315-342. https://doi.org/10.1090/qam/1827367
25. M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 487-537.
26. E. Yu. Panov and V. M. Shelkovich, $\delta$'-shock waves as a new type of solutions to system of conservation laws, J. Differential Equations 228 (2006), no. 1, 49-86. https://doi.org/10.1016/j.jde.2006.04.004
27. C. Shen, Structural stability of solutions to the Riemann problem for a scalar conservation law, J. Math. Anal. Appl. 389 (2012), no. 2, 1105-1116. https://doi.org/10.1016/j.jmaa.2011.12.044
28. C. Shen, On a regularization of a scalar conservation law with discontinuous coefficients, J. Math. Phys. 55 (2014), no. 3, 031502, 15 pp. https://doi.org/10.1063/1.4867624
29. C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model, J. Differential Equations 249 (2010), no. 12, 3024-3051. https://doi.org/10.1016/j.jde.2010.09.004
30. C. Shen and M. Sun, Instability of Riemann solutions to a scalar conservation law with discontinuous flux, Z. Angew. Math. Phys. 66 (2015), no. 3, 499-515. https://doi.org/10.1007/s00033-014-0411-z
31. W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137 (1999), no. 654, viii+77 pp.
32. M. Slemrod and A. E. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Indiana Univ. Math. J. 38 (1989), no. 4, 1047-1074. https://doi.org/10.1512/iumj.1989.38.38048
33. M. Sun, Delta shock waves for the chromatography equations as self-similar viscosity limits, Quart. Appl. Math. 69 (2011), no. 3, 425-443. https://doi.org/10.1090/S0033-569X-2011-01207-3
34. M. Sun, Formation of delta standing wave for a scalar conservation law with a linear flux function involving discontinuous coefficients, J. Nonlinear Math. Phys. 20 (2013), no. 2, 229-244. https://doi.org/10.1080/14029251.2013.805573
35. D. Tan, T. Zhang, and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1-32. https://doi.org/10.1006/jdeq.1994.1093
36. G. Wang, One-dimensional non-linear chromatography system and $\delta$-shock waves, Z. Angew. Math. Phys. 64 (2013), no. 5, 1451-1469. https://doi.org/10.1007/s00033-013-0300-x
37. H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations 252 (2012), no. 11, 5951-5993. https://doi.org/10.1016/j.jde.2012.02.015
38. G. Yin and K. Song, Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas, J. Math. Anal. Appl. 411 (2014), no. 2, 506-521. https://doi.org/10.1016/j.jmaa.2013.09.050