Asymptotic behavior of solutions toward the constant state to the Cauchy problem for the non-viscous diffusive dispersive conservation law

References

[1] K. Andreiev, I. Egorova, T. L. Lange and G. Teschl, Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent, J. Differential Equations 261 (2016), no. 10, 5371–5410. 10.1016/j.jde.2016.08.009Suche in Google Scholar

[2] J. L. Bona, S. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two-dimensional theory, Differential Integral Equations 7 (1994), no. 3–4, 699–734. 10.57262/die/1370267701Suche in Google Scholar

[3] J. L. Bona and M. E. Schonbek, Travelling-wave solutions to the Korteweg–de Vries–Burgers equation, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 3–4, 207–226. 10.1017/S0308210500020783Suche in Google Scholar

[4] R. P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC, Boca Raton, 2006. 10.1201/9781420015386Suche in Google Scholar

[5] R. P. Chhabra, Non-Newtonian fluids: An introduction, Rheology of Complex Fluids, Springer, Dordrecht (2010), 3–34. 10.1007/978-1-4419-6494-6_1Suche in Google Scholar

[6] R. P. Chhabra and J. F. Richardson, Non-Newtonian flow and applied rheology, 2nd ed., Butterworth–Heinemann, Oxford, 2008. Suche in Google Scholar

[7] Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl. 155 (1991), no. 1, 21–45. 10.1016/0022-247X(91)90024-TSuche in Google Scholar

[8] R. Duan, L. Fan, J. Kim and L. Xie, Nonlinear stability of strong rarefaction waves for the generalized KdV–Burgers–Kuramoto equation with large initial perturbation, Nonlinear Anal. 73 (2010), no. 10, 3254–3267. 10.1016/j.na.2010.07.005Suche in Google Scholar

[9] R. Duan and H. Zhao, Global stability of strong rarefaction waves for the generalized KdV–Burgers equation, Nonlinear Anal. 66 (2007), no. 5, 1100–1117. 10.1016/j.na.2006.01.008Suche in Google Scholar

[10] I. Egorova, K. Grunert and G. Teschl, On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data. I. Schwartz-type perturbations, Nonlinearity 22 (2009), no. 6, 1431–1457. 10.1088/0951-7715/22/6/009Suche in Google Scholar

[11] I. Egorova and G. Teschl, On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data II. Perturbations with finite moments, J. Anal. Math. 115 (2011), 71–101. 10.1007/s11854-011-0024-9Suche in Google Scholar

[12] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1–2, 35–49. 10.1016/0025-5564(77)90062-1Suche in Google Scholar

[13] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527–536. 10.1007/BF01229452Suche in Google Scholar

[14] I. Hashimoto and A. Matsumura, Large-time behavior of solutions to an initial-boundary value problem on the half line for scalar viscous conservation law, Methods Appl. Anal. 14 (2007), no. 1, 45–59. 10.4310/MAA.2007.v14.n1.a4Suche in Google Scholar

[15] Y. Hattori and K. Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math. 8 (1991), no. 1, 85–96. 10.1007/BF03167186Suche in Google Scholar

[16] A. M. Il’in, A. S. Kalašnikov and O. A. Oleĭnik, Second-order linear equations of parabolic type (in Russian), Uspekhi Mat. Nauk 17 (1962), no. 3 (105), 3–146; translation in Russian Math. Surveys 17 (1962), 1–143. Suche in Google Scholar

[17] A. M. Il’in and O. A. Oleĭnik, Asymptotic behavior of the solutions of the Cauchy problem for some quasi-linear equations for large values of the time (in Russian), Mat. Sb. 51 (1960), 191–216. Suche in Google Scholar

[18] P. Jahangiri, R. Streblow and D. Müller, Simulation of non-Newtonian fluids using modelica, Proceedings of the 9th International Modelica Conference, RWTH Aachen University, Aachen (2012), 57–62. 10.3384/ecp1207657Suche in Google Scholar

[19] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. 10.1002/cpa.3160100406Suche in Google Scholar

[20] H. W. Liepmann and A. Roshko, Elements of Gasdynamics, John Wiley & Sons, New York, 1957. 10.1063/1.3060140Suche in Google Scholar

[21] T.-P. Liu, A. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal. 29 (1998), no. 2, 293–308. 10.1137/S0036141096306005Suche in Google Scholar

[22] J. Málek, Some frequently used models for non-Newtonian fluids, http://www.karlin.mff.cuni.cz/malek/new/images/lecture4.pdf. Suche in Google Scholar

[23] J. Málek, D. Pražák and M. Steinhauer, On the existence and regularity of solutions for degenerate power-law fluids, Differential Integral Equations 19 (2006), no. 4, 449–462. 10.57262/die/1356050508Suche in Google Scholar

[24] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 3 (1986), no. 1, 1–13. 10.1007/BF03167088Suche in Google Scholar

[25] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994), no. 1, 83–96. 10.1007/BF02099739Suche in Google Scholar

[26] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of Burgers’ equation with nonlinear degenerate viscosity, Nonlinear Anal. 23 (1994), no. 5, 605–614. 10.1016/0362-546X(94)90239-9Suche in Google Scholar

[27] A. Matsumura and N. Yoshida, Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux, SIAM J. Math. Anal. 44 (2012), no. 4, 2526–2544. 10.1137/110839448Suche in Google Scholar

[28] A. Matsumura and N. Yoshida, Global asymptotics toward the rarefaction waves for solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosity, Osaka J. Math. 57 (2020), no. 1, 187–205. Suche in Google Scholar

[29] K. Nishihara and S. V. Rajopadhye, Asymptotic behaviour of solutions to the Korteweg–de Vries-Burgers equation, Differential Integral Equations 11 (1998), no. 1, 85–93. 10.57262/die/1367414136Suche in Google Scholar

[30] S. V. Rajopadhye, Decay rates for the solutions of model equations for bore propagation, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 2, 371–398. 10.1017/S0308210500028080Suche in Google Scholar

[31] L. Ruan, W. Gao and J. Chen, Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation, Nonlinear Anal. 68 (2008), no. 2, 402–411. 10.1016/j.na.2006.11.006Suche in Google Scholar

[32] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren Math. Wiss. 258, Springer, New York, 1983. 10.1007/978-1-4684-0152-3Suche in Google Scholar

[33] T. Sochi, Pore-scale modeling of non-Newtonian flow in porous media, PhD thesis, Imperial College, London, 2007. Suche in Google Scholar

[34] Z. Wang and C. Zhu, Stability of the rarefaction wave for the generalized KdV–Burgers equation, Acta Math. Sci. Ser. B (Engl. Ed.) 22 (2002), no. 3, 319–328. 10.1016/S0252-9602(17)30301-6Suche in Google Scholar

[35] N. Yoshida, Decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux, Nonlinear Anal. 96 (2014), 189–210. 10.1016/j.na.2013.08.014Suche in Google Scholar

[36] N. Yoshida, Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity, Nonlinear Anal. 128 (2015), 48–76. 10.1016/j.na.2015.07.019Suche in Google Scholar

[37] N. Yoshida, Asymptotic behavior of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with the Ostwald–de Waele-type viscosity, SIAM J. Math. Anal. 49 (2017), no. 3, 2009–2036. 10.1137/16M1090491Suche in Google Scholar

[38] N. Yoshida, Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity, J. Differential Equations 263 (2017), no. 11, 7513–7558. 10.1016/j.jde.2017.08.008Suche in Google Scholar

[39] N. Yoshida, Asymptotic behavior of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear flux and viscosity, SIAM J. Math. Anal. 50 (2018), no. 1, 891–932. 10.1137/17M1118798Suche in Google Scholar

[40] N. Yoshida, Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the scalar diffusive dispersive conservation laws, Nonlinear Anal. 189 (2019), Article ID 111573. 10.1016/j.na.2019.111573Suche in Google Scholar

[41] N. Yoshida, Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity, J. Differential Equations 269 (2020), no. 11, 10350–10394. 10.1016/j.jde.2020.07.010Suche in Google Scholar

[42] N. Yoshida, Asymptotic behavior of solutions toward a multiwave pattern to the Cauchy problem for the dissipative wave equation with partially linearly degenerate flux, Funkcial. Ekvac. 64 (2021), no. 1, 49–73. 10.1619/fesi.64.49Suche in Google Scholar

[43] N. Yoshida, Asymptotic behavior of solutions to the rarefaction problem for the generalized Korteweg–de Vries Burgers equation, Bull. Fac. Edu. 32 (2022), 135–150. Suche in Google Scholar