Warianty tytułu
Języki publikacji
Abstrakty
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation law where the far field constant state is prescribed. We prove that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.
Czasopismo
Rocznik
Tom
Strony
251--259
Opis fizyczny
Bibliogr. 43 poz.
Twórcy
autor
- Graduate Faculty of Interdisciplinary Research Faculty of Education, University of Yamanashi, Kofu, Yamanashi 400-8510, Japan, 14v00067@gmail.com
Bibliografia
- [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.
- [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.
- [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.
- [4] R. P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC, Boca Raton, 2006.
- [5] R. P. Chhabra, Non-Newtonian fluids: An introduction, in: Rheology of Complex Fluids, Springer, Dordrecht (2010), 3-34.
- [6] R. P. Chhabra and J. F. Richardson, Non-Newtonian flow and applied rheology, 2nd ed., Butterworth-Heinemann, Oxford, 2008.
- [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.
- [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.
- [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] 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.
- [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.
- [12] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1-2, 35-49.
- [13] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527-536.
- [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.
- [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.
- [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.
- [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.
- [18] P. Jahangiri, R. Streblow and D. Müller, Simulation of non-Newtonian fluids using modelica, in: Proceedings of the 9th International Modelica Conference, RWTH Aachen University, Aachen (2012), 57-62.
- [19] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566.
- [20] H. W. Liepmann and A. Roshko, Elements of Gasdynamics, John Wiley & Sons, New York, 1957.
- [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.
- [22] J. Málek, Some frequently used models for non-Newtonian fluids, http://www.karlin.mff.cuni.cz/malek/new/images/lecture4.pdf.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [32] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren Math. Wiss. 258, Springer, New York, 1983.
- [33] T. Sochi, Pore-scale modeling of non-Newtonian flow in porous media, PhD thesis, Imperial College, London, 2007.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
- [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.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.baztech-49183134-73d8-48c1-9425-5e2384a92eb7