Global asymptotic stability of the rarefaction waves to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws

Natsumi Yoshida

doi:10.3934/cpaa.2023011

Article Contents

2023, Volume 22, Issue 3: 825-850. Doi: 10.3934/cpaa.2023011

This issue Previous Article Homogenization theory of elliptic system with lower order terms for dimension two Next Article Semilinear Dirichlet problem for subordinate spectral Laplacian

Global asymptotic stability of the rarefaction waves to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws

Natsumi Yoshida^,

Graduate Faculty of Interdisciplinary Research, Faculty of Education, University of Yamanashi, Takeda 4-4-37, Kofu, Yamanashi, 400-8510 Japan

Received: May 2022

Revised: November 2022

Early access: December 19, 2022

Published online: December 19, 2022

The first author is supported by [Grant-in-Aid for Scientific Research (C) (No. 22K03371), JSPS.].

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Abstract

The present paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for the scalar diffusive dispersive conservation laws where the far field states are prescribed. Especially, we deal with the case where the dispersive flux does not consist of viscous flux, that is, non-viscous diffusive flux. If the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, then the solution of the Cauchy problem is proved to tends toward the rarefaction wave as time goes to infinity. The proof is given by a technical energy method.

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Mathematics Subject Classification: Primary: 35B40, 35G20, 35G25, 35L65, 35Q35; Secondary: 35Q53.

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References

[1]	K. Andreiev, I. Egorova, T. L. Lange and G. Teschl, Rarefaction waves of the Korteweg-de Vries equation via nonlinear steepest deecent, J. Differ. Equ., 10 (2016), 5371-5410. doi: 10.1016/j.jde.2016.08.009.
[2]	J. L. Bona and M. E. Schonbek, Travelling-wave solutions to the Korteweg-deVries-Burgers equation, Proc. Roy. Soc. Edinb., 101A (1985), 207-226. doi: 10.1017/S0308210500020783.
[3]	J. L. Bona, S.V. Rajopadhye and M. E. Schonbek, Models for the propagation of bores I. Two dimensional theory, Differ. Integral Equ., 7 (1994), 699-734.
[4]	A. de Waele, Viscometry and plastometry, J. Oil Colour Chem. Assoc., 6 (1923), 33-69.
[5]	Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl., 155 (1991), 21-45. doi: 10.1016/0022-247X(91)90024-T.
[6]	R. Duan, L. L. Fan, J. S. Kim and L. Q. Xie, Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation, Nonlinear Anal. TMA, 73 (2010), 3254-3267. doi: 10.1016/j.na.2010.07.005.
[7]	R. Duan and H. J. Zhao, Global stability of strong rarefaction waves for the generalized KdV-Burgers equation, Nonlinear Anal. TMA, 66 (2007), 1100-1117. doi: 10.1016/j.na.2006.01.008.
[8]	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), 1431-1457. doi: 10.1088/0951-7715/22/6/009.
[9]	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. doi: 10.1007/s11854-011-0024-9.
[10]	E. Harabetian, Rarefaction and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys., 114 (1988), 527-536.
[11]	Y. Hattori and K. Nishihara, A note on the stability of rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math., 8 (1991), 85-96. doi: 10.1007/BF03167186.
[12]	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, Mat. Sb., 51 (1960), 191-216 (in Russian).
[13]	O. A. Ladyženskaja, New Equations for the Description of the Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, in Boundary Value Problems of Mathematical Physics V, Amer. Math. Soc., Providence, Rhode. Island, 1970.
[14]	P. D. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.
[15]	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), 293-308. doi: 10.1137/S0036141096306005.
[16]	A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction wave of solutions of a one-dimensional model system for compressible viscous gas, Japan, J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088.
[17]	A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of Burgers' equation with nonlinear degenerate viscosity, Nonlinear Anal. TMA, 23 (1994), 605-614. doi: 10.1016/0362-546X(94)90239-9.
[18]	A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96.
[19]	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), 2526-2544. doi: 10.1137/110839448.
[20]	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), 187-205.
[21]	K. Nishihara and S. V. Rajopadhye, Asymptotic behavior of solutions to the Korteweg-deVries-Burgers equation, Differ. Integral Equ., 11 (1998), 85-93.
[22]	W. Ostwald, Über die Geschwindigkeitsfunktion der Viskositat disperser Systeme, I. Colloid Polym. Sci., 36 (1925), 99-117 (in German).
[23]	S. V. Rajopadhye, Decay rates for the solutions of model equations for bore propagation, Proc. Roy. Soc. Edinb., 125A (1995), 371-398. doi: 10.1017/S0308210500028080.
[24]	L. Z. Ruan, W. L. Gao and J. Chen, Asymptotic stability of the rarefaction wave for the generalized KdV-Burgers-Kuramoto equation, Nonlinear Anal. TMA, 68 (2008), 402-411. doi: 10.1016/j.na.2006.11.006.
[25]	J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer-Verlag, New York-Berlin, 1983.
[26]	Z. A. Wang and C. J. Zhu, Stability of the rarefaction wave for the generalized KdV-Burgers equation, Acta Math. Sci., 22B (2002), 319-328. doi: 10.1016/S0252-9602(17)30301-6.
[27]	N. Yoshida, Decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux, Nonlinear Anal. TMA, 96 (2014), 189-210. doi: 10.1016/j.na.2013.08.014.
[28]	N. Yoshida, Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity, Nonlinear Anal. TMA, 128 (2015), 48-76. doi: 10.1016/j.na.2015.07.019.
[29]	N. Yoshida, Asymptotic behavior of solutions toward a multiwave pattern for the scalar conservation law with the Ostwald-de Waele-type viscosity, SIAM J. Math. Anal., 49 (2017), 2009-2036. doi: 10.1137/16M1090491.
[30]	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. Differ. Equ., 263 (2017), 7513-7558. doi: 10.1016/j.jde.2017.08.008.
[31]	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), 891-932. doi: 10.1137/17M1118798.
[32]	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), 1-19. doi: 10.1016/j.na.2019.111573.
[33]	N. Yoshida, Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity, J. Differ. Equ., 269 (2020), 10350-10394. doi: 10.1016/j.jde.2020.07.010.
[34]	N. Yoshida, Asymptotic behavior of solutions toward a multiwave pattern to the Cauchy problem for the dissipative wave equation with partially linearly degenerate flux, Funkc. Ekvacioj, 64 (2021), 49-73. doi: 10.1619/fesi.64.49.
[35]	N. Yoshida, Asymptotic behavior of solutions to the rarefaction problem for the generalized Korteweg-de Vries Burgers equation, Bull. Fac. Edu., Univ. Yamanashi, 32 (2022), 135-150.
[36]	N. Yoshida, Asymptotic behavior of solutions toward the constant state to the Cauchy problem for the non-viscous diffusive dispersive conservation law, arXiv: 2107.07874.