有界区间上一维非线性方程组的粘性消失极限
The Vanishing Viscosity Limit for a One-Dimensional Nonlinear System of Equations on a Bounded Interval
摘要: 对于定义在有界区域上的一维非线性粘性抛物方程组，本文主要探讨其与相应无粘方程组之间解的渐近等价性。具体而言，首先利用匹配渐近展开的方法构建粘性守恒律方程组的近似解，再借助能量估计的方法进行稳定性分析，证明在远离边界层的区域中，抛物方程组的解一致收敛于相应无粘方程组的解。
Abstract: For a system of one-dimensional nonlinear viscous parabolic equations defined on a bounded domain, this paper primarily investigates the asymptotic equivalence between its solutions and those of the corresponding inviscid system. Specifically, the method of matched asymptotic expansions is first employed to construct approximate solutions for the system of viscous conservation laws; subsequently, energy estimates are utilized to conduct a stability analysis, thereby demonstrating that, in regions away from the boundary layers, the solutions of the parabolic system converge uniformly to the solutions of the corresponding inviscid system.
文章引用：肖慧, 陆正琪, 曹艺. 有界区间上一维非线性方程组的粘性消失极限[J]. 应用数学进展, 2026, 15(5): 618-643. https://doi.org/10.12677/AAM.2026.155256

参考文献

[1] Matsumura, A. and Nishida, T. (1979) The Initial Value Problem for the Equations of Motion of Compressible Viscous and Heat-Conductive Fluids. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 55, 337-342. [Google Scholar] [CrossRef] 
[2] Szepessy, A. and Xin, Z. (1993) Nonlinear Stability of Viscous Shock Waves. Archive for Rational Mechanics and Analysis, 122, 53-103. [Google Scholar] [CrossRef] 
[3] Sattinger, D.H. (1976) On the Stability of Waves of Nonlinear Parabolic Systems. Advances in Mathematics, 22, 312-355. [Google Scholar] [CrossRef] 
[4] Serre, D. (1998) Global Solutions (−∞ < t < +∞) of Parabolic Systems of Conservation Laws. Annales de l’institut Fourier, 48, 1069-1091.
[5] Grenier, E. and Guës, O. (1998) Boundary Layers for Viscous Perturbations of Noncharac- teristic Quasilinear Hyperbolic Problems. Journal of Differential Equations, 143, 110-146. [Google Scholar] [CrossRef] 
[6] Li, H., Markowich, P. and Mei, M. (2002) Asymptotic Behaviour of Solutions of the Hydro- dynamic Model of Semiconductors. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 132, 359-378. [Google Scholar] [CrossRef] 
[7] Tong, L. and Wang, J. (2016) Stability of Multiple Boundary Layers for 2D Quasilin- ear Parabolic Equations. Journal of Mathematical Analysis and Applications, 435, 349-368. [Google Scholar] [CrossRef] 
[8] Joseph, K.T. (1989) Boundary Layers in Approximate Solutions. Transactions of the American Mathematical Society, 314, 709-709. [Google Scholar] [CrossRef] 
[9] Goodman, J. and Xin, Z. (1992) Viscous Limits for Piecewise Smooth Solutions to Systems of Conservation Laws. Archive for Rational Mechanics and Analysis, 121, 235-265. [Google Scholar] [CrossRef] 
[10] Li, H., Markowich, P. and Mei, M. (2002) Asymptotic Behavior of Subsonic Entropy Solutions of the Isentropic Euler-Poisson Equations. Quarterly of Applied Mathematics, 60, 773-796. [Google Scholar] [CrossRef] 
[11] Zeng, H. (2009) Stability of a Superposition of Shock Waves with Contact Discontinuities for Systems of Viscous Conservation Laws. Journal of Differential Equations, 246, 2081-2102. [Google Scholar] [CrossRef] 
[12] Goodman, J. (1986) Nonlinear Asymptotic Stability of Viscous Shock Profiles for Conservation Laws. Archive for Rational Mechanics and Analysis, 95, 325-344. [Google Scholar] [CrossRef] 
[13] Yu, S. (1999) Zero-Dissipation Limit of Solutions with Shocks for Systems of Hyperbolic Con- servation Laws. Archive for Rational Mechanics and Analysis, 146, 275-370. [Google Scholar] [CrossRef] 
[14] Liu, T. (1976) The Entropy Condition and the Admissibility of Shocks. Journal of Mathematical Analysis and Applications, 53, 78-88. [Google Scholar] [CrossRef] 
[15] Liu, T. (1985) Nonlinear Stability of Shock Waves for Viscous Conservation Laws. Bulletin of the American Mathematical Society, 12, 233-236. [Google Scholar] [CrossRef] 
[16] Liu, T. and Xin, Z. (1992) Stability of Viscous Shock Waves Associated with a System of Non- strictly Hyperbolic Conservations Laws. Communications on Pure and Applied Mathematics, 45, 361-388. [Google Scholar] [CrossRef] 
[17] Xin, Z. (1992) On the Linearized Stability of Viscous Shock Profiles for Systems of Conservation Laws. Journal of Differential Equations, 100, 119-136. [Google Scholar] [CrossRef] 
[18] Wang, J. (2009) Boundary Layers for Parabolic Perturbations of Quasi-Linear Hyperbolic Problems. Mathematical Methods in the Applied Sciences, 32, 2416-2438. [Google Scholar] [CrossRef] 
[19] Zhen, M. (2000) Numerical Bifurcation Analysis for Reaction-Diffusion Equations. Springer.

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