Global Existence and Stability of Solutions to River Flow System

doi:10.1007/s42967-022-00198-x

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (3): 1247-1255.doi: 10.1007/s42967-022-00198-x

• ORIGINAL PAPER • Previous Articles Next Articles

Global Existence and Stability of Solutions to River Flow System

Xian-ting Wang¹, Yun-guang Lu², Naoki Tsuge³

1. Wuxi Institute of Technology, Wuxi, 214121, Jiangsu, China;
2. K. K. Chen Institute for Advanced Studies, Hangzhou Normal University, Hangzhou, 311121, Zhejiang, China;
3. Department of Mathematics, Faculty of Education, Gifu University, Gifu, Japan

Received:2021-11-16 Revised:2022-05-02 Online:2023-09-20 Published:2023-08-29
Contact: Yun-guang Lu,E-mail:ylu2005@ustc.edu.cn E-mail:wangxt@wxit.edu.cn;ylu2005@ustc.edu.cn;tuge@gifu-u.ac.jp
Supported by:
This paper is dedicated, with respect, to Prof. Tong Zhang on the occasion of his 90th birthday. The first author is supported by the Zhejiang Natural Science Foundation of China (Grant No. LY17A010019); the second author is supported by the Zhejiang Natural Science Foundation of China (Grant No. LY20A010023) and the National Natural Science Foundation of China (Grant No. 12071106) and the third author is supported by the Grant-in-Aid for Scientific Research (C) 17K05315, Japan.

Abstract

Abstract: In this short note, we are concerned with the global existence and stability of solutions to the river flow system. We introduce a new technique to set up a relation between the Riemann invariants and the finite mass to obtain a time-independent, bounded solution for any adiabatic exponent. The global existence of solutions was known long ago [Klingenberg and Lu in Commun. Math. Phys. 187: 327–340, 1997]. However, since the uncertainty of the function b(x), which corresponds physically to the slope of the topography, the L^∞ estimates growed larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. By employing a suitable mathematical transformation to control the slope of the topography by the friction and the finite mass, we prove the uniformly bounded estimate with respect to the time variable. This means that our solutions are stable.

Key words: Stability, River flow system, Time-independent estimate, Viscosity approximation, Maximum principle

CLC Number:

Xian-ting Wang, Yun-guang Lu, Naoki Tsuge. Global Existence and Stability of Solutions to River Flow System[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1247-1255.

TrendMD

References

[1] Ding, X.-X., Chen, G.-Q., Luo, P.-Z.: Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Commun. Math. Phys. 121, 63–84 (1989)
[2] DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)
[3] Fang, X., Yu, H.: Uniform boundedness in weak solutions to a specific dissipative system. J. Math. Anal. Appl. 461, 1153–1164 (2018)
[4] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 95–105 (1965)
[5] Hu, Y.-B., Klingenberg, C., Lu, Y.-G.: Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors. Math. Ann. 382, 1031–1046 (2022)
[6] Hu, Y.-B., Lu, Y.-G., Tsuge, N.: Global existence and stability to the polytropic gas dynamics with an outer force. Appl. Math. Lett. 95, 36–40 (2019)
[7] Huang, F.M., Wang, Z.: Convergence of viscosity solutions for isothermal gas dynamics. SIAM J. Math. Anal. 34, 595–610 (2003)
[8] Klingenberg, C., Lu, Y.-G.: Existence of solutions to hyperbolic conservation laws with a source. Commun. Math. Phys. 187, 327–340 (1997)
[9] LeRoux, A.Y.: Numerical stability for some equations of gas dynamics. Math. Comput. 37, 435–446 (1981)
[10] Lions, P.L., Perthame, B., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, 599–638 (1996)
[11] Lions, P.L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415–431 (1994)
[12] Lu, Y.-G.: Some results on general system of isentropic gas dynamics. Differ. Equ. 43, 130–138 (2007)
[13] Lu, Y.-G.: Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductors. arXiv: 2003.01375 (2020)
[14] Lu, Y.-G.: Hyperbolic Conservation Laws and the Compensated Compactness Method, vol. 128. Chapman and Hall, CRC Press, New York (2002)
[15] Lu, Y.-G., Tsuge, N.: Uniformly time-independent L^∞ estimate for a one-dimensional hydrodynamic model of semiconductors. Front. Math. China (2021). https://doi.org/10.1007/s11464-021-0990-x
[16] Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa 5, 489–507 (1978)
[17] Tartar, T.: Compensated compactness and applications to partial differential equations. In: Knops, R.J. (ed) Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, vol. 4. Pitman Press, London (1979)
[18] Tsuge, N.: Existence and stability of solutions to the compressible Euler equations with an outer force. Nonlinear Anal. Real World Appl. 27, 203–220 (2016)
[19] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1973)

Global Existence and Stability of Solutions to River Flow System

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan. On the Vortex Sheets of Compressible Flows [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 967-986.
[2]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 573-595.
[3]	Linjin Li, Jingmei Qiu, Giovanni Russo. A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 170-198.
[4]	F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi. P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1293-1312.
[5]	Shima Baharlouei, Reza Mokhtari, Nabi Chegini. A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1351-1373.
[6]	Hendrik Ranocha, Gregor J. Gassner. Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 880-903.
[7]	Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.
[8]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.
[9]	Giacomo Albi, Lorenzo Pareschi. High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 701-718.
[10]	Leilei Wei, Shuying Zhai, Xindong Zhang. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 429-444.
[11]	Le Zhao, Can Li, Fengqun Zhao. Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 1-40.
[12]	Wenhui Guan, Xuenian Cao. A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 41-59.
[13]	Min Zhang, Yang Liu, Hong Li. High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 613-640.
[14]	Mostafa Abbaszadeh, Hanieh Amjadian. Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 653-669.
[15]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.