Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms
Abstract
:1. Introduction
2. Traveling Wave and Main Results
3. Proof to Theorem 3 on Nonlinear Stability
4. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Affouf, M.; Caflsch, R.E. A numerical study of Riemann problem solutions and stability for a system of viscous conservation laws of mixed type. SIAM J. Appl. Math. 1991, 51, 605–634. [Google Scholar] [CrossRef] [Green Version]
- Korteweg, D.J. Sur la forme que prennent lea équation des mouvements des fluids si l’on tient compte des forces capillarires par des variations de densité. Arch. Neerl. Sci. Exactes Nat. Ser. II 1901, 6, 1–24. [Google Scholar]
- Hattori, H.; Li, D.N. Solutions for two dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 1994, 25, 85–98. [Google Scholar] [CrossRef]
- Hattori, H.; Li, D.N. The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 1996, 9, 323–342. [Google Scholar]
- Haspot, B. Existence of global weak solution for compressible fluid models of Korteweg type. J. Math. Fluid Mech. 2011, 13, 223–249. [Google Scholar] [CrossRef] [Green Version]
- Chen, Z.Z. Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type. J. Math. Anal. Appl. 2012, 394, 438–448. [Google Scholar] [CrossRef] [Green Version]
- Chen, Z.; He, L.; Zhao, H. Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type. J. Math. Anal. Appl. 2015, 422, 1213–1234. [Google Scholar] [CrossRef]
- Slemrod, M. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 1983, 81, 301–315. [Google Scholar] [CrossRef] [Green Version]
- Slemrod, M. Dynamic phase transitions in a Van der Waalsfluid. J. Differ. Equ. 1984, 52, 1–23. [Google Scholar] [CrossRef]
- Glimm, J. The continuous structure of discontinuities. Lect. Notes Phys. 1986, 344, 175–186. [Google Scholar]
- Goodman, J. Nonlinear asymptotic stability of viscous shock profiles for conservations laws. Arch. Rational. Mech. Anal. 1986, 95, 325–344. [Google Scholar] [CrossRef]
- Pego, R.L. Remarks on the stability of shock profiles for conservation laws with dissipation. Trans. Amer. Math. Soc. 1985, 291, 353–361. [Google Scholar] [CrossRef]
- Matsumura, A.; Nishihara, K. On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous Gas. Jpn. J. Appl. Math. 1985, 2, 17–25. [Google Scholar] [CrossRef]
- Kawashima, S.; Matsumura, A. Asymptotic Stability of Traveling Wave Solutions of Systems for One-dimensional Gas Motion. Commun. Math. Phys. 1985, 1, 97–127. [Google Scholar] [CrossRef]
- Matsumura, A.; Nishihara, K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun. Math. Phys. 1992, 144, 325–335. [Google Scholar] [CrossRef]
- Zhang, W.; Li, X.; Yong, Y. Asymptotic stability of monotone increasing traveling wave solutions for viscous compressible fluid equations with capillarity term. J. Math. Anal. Appl. 2016, 434, 401–412. [Google Scholar] [CrossRef]
- Jones, C.; Gardner, R.; Kapitula, T. Stability of travelling waves for non-convex scalar viscous conservation laws. Commun. Pure Appl. Math. 1993, 46, 505–526. [Google Scholar] [CrossRef]
- Matsumura, A.; Nishihara, K. Asymptotic stability of traveling waves for scale viscous conservation laws with non-convex nonlinearity. Comm. Math. Phys. 1994, 165, 83–96. [Google Scholar] [CrossRef]
- Mei, M. Stability of shock profiles for non-convex scalar viscous conservation laws. Math. Model. Methods Appl. Sci. 1995, 5, 279–296. [Google Scholar] [CrossRef]
- Kawashima, S.; Matsumura, A. Stability of shock profiles in viscoelasticity with non-convex constitutive relations. Commun. Pure Appl. Math. 1994, 47, 1547–1569. [Google Scholar] [CrossRef]
- Matsumura, A.; Mei, M. Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity. Osaka J. Math. 1997, 34, 589–603. [Google Scholar]
- Matsumura, A.; Mei, M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Rational Mech. Anal. 1999, 146, 1–22. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Z.F.; Ding, T.R.; Huang, W.Z.; Dong, Z.X. Qualitative Theory of Differential Equations, Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1992. [Google Scholar]
- Sarma, R.; Mondal, P.K. Marangoni instability in a heated viscoelastic liquid film: Longwave versus shortwave perturbations. Phys. Rev. E 2019, 100, 1–14. [Google Scholar] [CrossRef]
- Sarma, R.; Mondal, P.K. Thermosolutal Marangoni instability in a viscoelastic liquid film: Effect of heating from the free surface. J. Fluid Mech. 2020, 909, 1–24. [Google Scholar] [CrossRef]
- Sarma, R.; Mondal, P.K. Marangoni instability in a viscoelastic binary film with cross-diffusive effect. J. Fluid Mech. 2021, 910, 1–34. [Google Scholar] [CrossRef]
- Jin, H.Y.; Wang, Z.A. Boundedness, blowup and critical mass phenomenon in competing chemotaxis. J. Differ. Equ. 2016, 260, 162–196. [Google Scholar] [CrossRef]
- Xu, S.; Dai, H.; Li, F.; Chen, H.; Chai, Y.; Zheng, W.X. Fault Estimation for Switched Interconnected Nonlinear Systems with External Disturbances via Variable Weighted Iterative Learning. IEEE Trans. Circuits Syst. II Express Briefs 2023, 70. [Google Scholar] [CrossRef]
- Sun, L.; Hou, J.; Xing, C.; Fang, Z. A Robust Hammerstein-Wiener Model Identification Method for Highly Nonlinear Systems. Processes 2022, 10, 2664. [Google Scholar] [CrossRef]
- Ye, R.; Liu, P.; Shi, K.B.; Yan, B. State Damping Control: A Novel Simple Method of Rotor UAV with High Performance. Access 2020, 8, 214346–214357. [Google Scholar] [CrossRef]
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Li, X.; Zhang, W.; Jin, H. Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms. Mathematics 2023, 11, 1734. https://doi.org/10.3390/math11071734
Li X, Zhang W, Jin H. Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms. Mathematics. 2023; 11(7):1734. https://doi.org/10.3390/math11071734
Chicago/Turabian StyleLi, Xiang, Weiguo Zhang, and Haipeng Jin. 2023. "Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms" Mathematics 11, no. 7: 1734. https://doi.org/10.3390/math11071734
APA StyleLi, X., Zhang, W., & Jin, H. (2023). Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms. Mathematics, 11(7), 1734. https://doi.org/10.3390/math11071734