Long-time behavior of solutions to the $ M_1 $ model with boundary effect

Nangao Zhang; Changjiang Zhu

doi:10.3934/dcds.2022180

Article Contents

2023, Volume 43, Issue 5: 1824-1859. Doi: 10.3934/dcds.2022180

This issue Previous Article The Boussinesq systems on the background of a line solitary wave Next Article Non-autonomous stochastic lattice systems with Markovian switching

Long-time behavior of solutions to the $ M_1 $ model with boundary effect

Nangao Zhang^1, and
Changjiang Zhu^1, ,

1.
School of Mathematics, South China University of Technology, Guangzhou 510641, China

^*Corresponding author: Changjiang Zhu
^*Corresponding author: Changjiang Zhu

Received: December 2021

Revised: December 2022

Early access: December 2022

Published: May 2023

The second author is supported by the National Natural Science Foundation of China #12171160, 11771150, 11831003 and the Guangdong Provincial Key Laboratory of Human Digital Twin #2022B1212010004.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we are concerned with the asymptotic behavior of solutions of $ M_1 $ model on quadrant $ (x,t) \in \mathbb{R}^{+} \times \mathbb{R}^{+} $. From this model, combined with damped compressible Euler equations, a more general system is introduced. We show that the solutions to the initial boundary value problem of this system globally exist and tend time-asymptotically to the corresponding nonlinear parabolic equation governed by the related Darcy's law. Compared with previous results on compressible Euler equations with damping obtained by Nishihara and Yang in [26], and Marcati, Mei and Rubino in [17], the better convergence rates are obtained. The approach adopted is based on the technical time-weighted energy estimates together with the Green's function method.

Keywords:

Mathematics Subject Classification: Primary: 85A25, 35L65; Secondary: 35B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. Berthon, P. Charrier and B. Dubroca, An asymptotic preserving relaxation scheme for a moment model of radiative transfer, C. R. Math. Acad. Sci. Paris, 344 (2007), 467-472. doi: 10.1016/j.crma.2007.02.004.
[2]	C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the $M_{1}$ model of radiative transfer in two space dimensions, J. Scie. Comput., 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6.
[3]	C. Berthon, J. Dubois, B. Dubroca, T.-H. Nguyen-Bui and R. Turpault, A free streaming contact preserving scheme for the $M_{1}$ Model, Adv. Appl. Math. Mech., 2 (2010), 259-285. doi: 10.4208/aamm.09-m09105.
[4]	C. Buet and S. Cordier, An asymptotic preserving scheme for hydrodynamics radiative transfer models: Numerics for radiative transfer, Numer. Math., 108 (2007), 199-221. doi: 10.1007/s00211-007-0094-x.
[5]	H. Cui, H. Yin, C. Zhu and L. Zhu, Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant, Sci. China Math., 62 (2019), 33-62. doi: 10.1007/s11425-017-9271-x.
[6]	S. Geng and L. Zhang, Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 959-978. doi: 10.1017/S0308210515000219.
[7]	T. Goudona and C. Lin, Analysis of the $M_{1}$ model: Well-posedness and diffusion asymptotics, J. Math. Anal. Appl., 402 (2013), 579-593. doi: 10.1016/j.jmaa.2013.01.042.
[8]	L. Hsiao and T.-P. Liu, Convergence to diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599-605. doi: 10.1007/BF02099268.
[9]	L. Hsiao and T.-P. Liu, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chin. Ann. Math. Ser. B, 14 (1993), 465-480.
[10]	L. Hsiao and T. Luo, Nonlinear diffusion phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations, 125 (1996), 329-365. doi: 10.1006/jdeq.1996.0034.
[11]	F. Huang and R. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5.
[12]	F. Huang and R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations, 220 (2006), 207-233. doi: 10.1016/j.jde.2005.03.012.
[13]	M. Jiang and C. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations, 246 (2009), 50-77. doi: 10.1016/j.jde.2008.03.033.
[14]	C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model., Ser. B, 1 (2010), 70-92.
[15]	H. Ma and M. Mei, Best asymptotic profile for linear damped p-system with boundary effect, J. Differential Equations, 249 (2010), 446-484. doi: 10.1016/j.jde.2010.04.008.
[16]	P. Marcati and M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial-boundary problem to the hyperbolic conservation laws with damping, Quart. Appl. Math., 58 (2000), 763-784. doi: 10.1090/qam/1788427.
[17]	P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), 224-240. doi: 10.1007/s00021-005-0155-9.
[18]	A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first order dissipation, Publ. RIMS Kyoto Univ., 13 (1977), 349-379. doi: 10.2977/prims/1195189813.
[19]	M. Mei, Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping, J. Differential Equations, 247 (2009), 1275-1296. doi: 10.1016/j.jde.2009.04.004.
[20]	M. Mei, Best asymptotic profile for hyperbolic p-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.
[21]	D. Mihalas and B. W. Mihalas, Foundation of Radiation Hydrodynamics, Oxford University Press, Oxford, 1984.
[22]	K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188. doi: 10.1006/jdeq.1996.0159.
[23]	K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, 137 (1997), 384-395. doi: 10.1006/jdeq.1997.3268.
[24]	K. Nishihara, Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 177-196. doi: 10.1017/S0308210500002341.
[25]	K. Nishihara, W. Wang and T. Yang, $L^{p}$ convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations, 161 (2000), 191-218. doi: 10.1006/jdeq.1999.3703.
[26]	K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the p-system with linear damping, J. Differential Equations, 156 (1999), 439-458. doi: 10.1006/jdeq.1998.3598.
[27]	M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132.
[28]	G. C. Pomraning, The Equations of Radiation Hydrodynamics, Sciences Application, Pergamon Press, Oxford, 1973.
[29]	W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937.
[30]	N. Zhang and C. Zhu, Convergence to nonlinear diffusion waves for solutions of $M_{1}$ model, preprint, 2021, arXiv: 2111.11035. doi: 10.1016/j.jde.2022.02.061.
[31]	H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations, 174 (2001), 200-236. doi: 10.1006/jdeq.2000.3936.
[32]	H. Zhao, Asymptotic behaviors of solutions of quasilinear hyperbolic equations with linear damping Ⅱ, J. Differential Equations, 167 (2000), 467-494. doi: 10.1006/jdeq.2000.3793.
[33]	C. Zhu and M. Jiang, $L^{p}$-decay rates to nonlinear diffusion waves for p-system with nonlinear damping, Sci. China Math., Ser. A, 49 (2006), 721-739. doi: 10.1007/s11425-006-0721-5.