Rousset, F. Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems. (English) Zbl 1022.35022 Trans. Am. Math. Soc. 355, No. 7, 2991-3008 (2003). An initial-boundary value problem for a one-dimensional system of conservation laws with a small parameter \( \varepsilon > 0,\) is considered \[ u^{\varepsilon}_t + F(u^{\varepsilon})_x = \varepsilon (B(u^{\varepsilon})u^{\varepsilon}_x)_x , \quad x,t > 0, \] where \(u^{\varepsilon} \in \mathbb{R}^n\), \(F: U\rightarrow \mathbb{R}^n\), \(B: U\rightarrow \mathbb{R}^{n\times n}\); \(F\), \(B\) are smooth. An initial condition and some dissipative boundary conditions are added to this system. The eigenvalues of \(B \) have nonnegative real parts and the rank of \(B\) does not depend on \(U.\) The author proves that if a boundary layer is weak, then \(u^{\varepsilon}\) converges to an inviscid solution as \( \varepsilon \) tends to zero. An application of this result to the isentropic gas dynamics is given. Reviewer: Nickolaj A.Lar’kin (Maringa) Cited in 11 Documents MSC: 35K65 Degenerate parabolic equations 35L65 Hyperbolic conservation laws Keywords:stability; boundary layer; hyperbolic-parabolic systems PDFBibTeX XMLCite \textit{F. Rousset}, Trans. Am. Math. Soc. 355, No. 7, 2991--3008 (2003; Zbl 1022.35022) Full Text: DOI References: [1] Marguerite Gisclon, Étude des conditions aux limites pour un système strictement hyperbolique, via l’approximation parabolique, J. Math. Pures Appl. (9) 75 (1996), no. 5, 485 – 508 (French, with English and French summaries). · Zbl 0869.35061 [2] Marguerite Gisclon and Denis Serre, Étude des conditions aux limites pour un système strictement hyberbolique via l’approximation parabolique, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 377 – 382 (French, with English and French summaries). · Zbl 0808.35075 [3] Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325 – 344. · Zbl 0631.35058 · doi:10.1007/BF00276840 [4] Emmanuel Grenier and Olivier Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations 143 (1998), no. 1, 110 – 146. · Zbl 0896.35078 · doi:10.1006/jdeq.1997.3364 [5] Emmanuel Grenier and Frédéric Rousset, Stability of one-dimensional boundary layers by using Green’s functions, Comm. Pure Appl. Math. 54 (2001), no. 11, 1343 – 1385. · Zbl 1026.35015 · doi:10.1002/cpa.10006.abs [6] S. Kawashima, Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics, Ph.D. thesis, Kyoto University (1983). [7] Gunilla Kreiss and Heinz-Otto Kreiss, Stability of systems of viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1397 – 1424. , https://doi.org/10.1002/(SICI)1097-0312(199811/12)51:11/123.3.CO;2-O · Zbl 0935.35013 [8] Ta Tsien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, V, Duke University, Mathematics Department, Durham, NC, 1985. · Zbl 0627.35001 [9] C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems, Preprint (2001). · Zbl 1058.35160 [10] Akitaka Matsumura and Kenji Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys. 222 (2001), no. 3, 449 – 474. · Zbl 1018.76038 · doi:10.1007/s002200100517 [11] F. Rousset, The boundary conditions coming from the real vanishing viscosity method, Discrete Continuous Dynamical Systems (to appear). · Zbl 1012.35055 [12] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Commun. Math. Phys. 221 (2001), 267-292. · Zbl 0988.35028 [13] Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249 – 275. · Zbl 0587.35046 · doi:10.14492/hokmj/1381757663 [14] Kevin Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser Boston, Boston, MA, 2001, pp. 307 – 516. · Zbl 0989.35089 [15] Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741 – 871. · Zbl 0928.35018 · doi:10.1512/iumj.1998.47.1604 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.