Boundary layers for self-similar viscous approximations of nonlinear hyperbolic systems
Authors:
Cleopatra Christoforou and Laura V. Spinolo
Journal:
Quart. Appl. Math. 71 (2013), 433-453
MSC (2010):
Primary 35L65, 35L50, 35K51
DOI:
https://doi.org/10.1090/S0033-569X-2013-01284-6
Published electronically:
May 16, 2013
MathSciNet review:
3112822
Full-text PDF Free Access
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Additional Information
Abstract: We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and nonconservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields.
References
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References
- F. Ancona and S. Bianchini, Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary, “WASCOM 2005”—13th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2006, pp. 13–21. MR 2231261 (2006m:35228)
- B. P. Andreianov, The Riemann problem for $p$-systems with continuous flux function, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 3, 353–367. MR 1751170 (2001b:35200)
- S. Bianchini, On the Riemann problem for non-conservative hyperbolic systems, Arch. Ration. Mech. Anal. 166 (2003), no. 1, 1–26. MR 1952077 (2003j:35206)
- S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223–342. MR 2150387 (2007i:35160)
- S. Bianchini and L.V. Spinolo, The boundary Riemann solver coming from the real vanishing viscosity approximation, Arch. Ration. Mech. Anal. 191 (2009), no. 1, 1–96. MR 2457066 (2009m:35306)
- A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. MR 1816648 (2002d:35002)
- A. Bressan, D. Serre, M. Williams, and K. Zumbrun, Hyperbolic systems of balance laws, Lecture Notes in Mathematics, vol. 1911, Springer, Berlin, 2007, Lectures given at the C.I.M.E. Summer School held in Cetraro, July 14–21, 2003. Edited and with a preface by Pierangelo Marcati. MR 2348933 (2008d:35123)
- C. Christoforou and L.V. Spinolo, A uniqueness criterion for viscous limits of boundary Riemann problems, J. Hyperbolic Differ. Equ. 8 (2011), 507–544. MR 2831272
- C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52 (1973), 1–9. MR 0340837 (49:5587)
- ---, Hyperbolic conservation laws in continuum physics, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- G. Dal Maso, P. G. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258 (97b:46052)
- M. 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. MR 1411161 (97f:35129)
- M. Gisclon and D. Serre, Étude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 377–382. MR 1289315 (95e:35119)
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770 (33:2976)
- H. Holden and N. H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206 (2003e:35001)
- K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions of hyperbolic conservation laws. II. Self-similar vanishing diffusion limits, Commun. Pure Appl. Anal. 1 (2002), no. 1, 51–76. MR 1877666 (2002k:35191)
- ---, Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions, C. R. Math. Acad. Sci. Paris 344 (2007), no. 1, 59–64. MR 2286590 (2007h:35218)
- A. S. Kalašnikov, Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Akad. Nauk SSSR 127 (1959), 27–30. MR 0108651 (21:7366)
- A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055)
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653 (20:176)
- P. G. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal. 30 (1999), no. 6, 1309–1342 (electronic). MR 1718304 (2001e:35113)
- T. P. Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218–234. MR 0369939 (51:6168)
- ---, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 0387830 (52:8669)
- L. Perko, Differential equations and dynamical systems, third ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001. MR 1801796 (2001k:34001)
- D. Serre, Systems of conservation laws. 1 and 2, Cambridge University Press, Cambridge, 1999, Translated from the 1996 French original by I. N. Sneddon. MR 1707279 (2000g:35142)
- V. A. Tupčiev, The problem of decomposition of an arbitrary discontinuity for a system of quasi-linear equations without the convexity condition, Z̆. Vyčisl. Mat. i Mat. Fiz. 6 (1966), 527–547. MR 0203203 (34:3056)
- A. E. Tzavaras, Elastic as limit of viscoelastic response, in a context of self-similar viscous limits, J. Differential Equations 123 (1995), no. 1, 305–341. MR 1359921 (97j:73033)
- ---, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Rational Mech. Anal. 135 (1996), no. 1, 1–60. MR 1414293 (97h:35149)
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Additional Information
Cleopatra Christoforou
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus
Email:
Christoforou.Cleopatra@ucy.ac.cy
Laura V. Spinolo
Affiliation:
IMATI-CNR, via Ferrata 1, 27100 Pavia, Italy
Email:
spinolo@imati.cnr.it
Keywords:
Hyperbolic systems,
boundary Riemann problem,
self-similar viscous approximation,
vanishing viscosity,
boundary layer.
Received by editor(s):
June 15, 2011
Published electronically:
May 16, 2013
Dedicated:
Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday
Article copyright:
© Copyright 2013
Brown University