The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation



We study the limit of the hyperbolic–parabolic approximation
$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A} \left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx} \quad v^{\varepsilon} \in \mathbb{R}^N \cr \tilde{\rm \ss}(v^{\varepsilon} (t, \, 0)) \equiv \bar g \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$
The function \({\tilde {\ss}}\) is defined in such a way as to guarantee that the initial boundary value problem is well posed even if \({\tilde {B}}\) is not invertible. The data \({\bar {g}}\) and \({\bar {v}_{0}}\) are constant. When \({\tilde {B}}\) is invertible, the previous problem takes the simpler form
$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A}\left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx}\quad v^{\varepsilon} \in \mathbb{R}^N \cr v^{\varepsilon} (t, \, 0) \equiv \bar v_b \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$
Again, the data \({\bar {v}_b}\) and \({\bar {v}_0}\) are constant. The conservative case is included in the previous formulations. Convergence of the \({v^{\varepsilon}}\) , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of \({\tilde {A}}\) can be 0. Second, as pointed out before, we take into account the possibility that \({\tilde {B}}\) is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if this condition is not satisfied, then pathological behaviors may occur.


  1. 1.
    Amadori D.: Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4, 1–42 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amadori D., Colombo R.M.: Viscosity Solutions and Standard Riemann Semigroup for Conservation Laws with Boundary. Rend. Sem. Mat. Univ. Padova 99, 219–245 (1998)MATHMathSciNetGoogle Scholar
  3. 3.
    Ancona, F., Bianchini, S.: Vanishing viscosity solutions for general hyperbolic systems with boundary. Preprint IAC-CNR 28, 2003Google Scholar
  4. 4.
    Ancona F., Marson A.: Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 185(2), 287–340 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ball J., Kirchheim B., Kristensen J.: Regularity of quasiconvex envelopes. Calc. Var. 11(4), 333–359 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benzoni-Gavage S., Serre D., Zumbrun K.: Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32(5), 929–962 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bianchini S.: On the Riemann Problem for Non-Conservative Hyperbolic Systems. Arch. Ration. Mech. Anal. 166(1), 1–26 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bianchini S., Bressan A.: BV estimates for a class of viscous hyperbolic systems. Indiana Univ. Math. J. 49, 1673–1713 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bianchini S., Bressan A.: A case study in vanishing viscosity. Discrete Contin. Dyn. Syst. 7, 449–476 (2001)MATHMathSciNetGoogle Scholar
  10. 10.
    Bianchini S., Bressan A.: A center manifold technique for tracing viscous waves. Comm. Pure Appl. Anal. 1, 161–190 (2002)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bianchini S., Bressan A.: Vanishing viscosity solutions of non linear hyperbolic systems. Ann. Math. 161, 223–342 (2005)MATHMathSciNetGoogle Scholar
  12. 12.
    Bianchini S., Hanouzet B., Natalini R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math. 60(11), 1559–1622 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bianchini, S., Spinolo, L.V.: Invariant manifolds for a singular ordinary differential equation. Preprint SISSA 04/2008/M, 2008.
  14. 14.
    Bressan A.: Global solution to systems of conservation laws by wave-front-tracking. J. Math. Anal. Appl. 170, 414–432 (1992)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bressan A.: The unique limit of the Glimm scheme. Arch. Ration. Mech. Anal. 130, 205–230 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bressan A.: Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  17. 17.
    Bressan, A.: Lecture notes on the center manifold theorem, pp. 1–16, 2003.
  18. 18.
    Bressan A., Colombo R.M.: The semigroup generated by 2 × 2 conservation laws. Arch. Ration. Mech. Anal. 133, 1–75 (1995)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Bressan, A., Crasta, G., Piccoli, B.: Well-posedness of the Cauchy problem for n × n conservation laws. Mem. Am. Math. Soc. 694 (2000)Google Scholar
  20. 20.
    Bressan A., Goatin P.: Olenik type estimates and uniqueness for n × n conservation laws. J. Differ. Equ. 156, 26–49 (1999)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Bressan A., LeFloch P.: Uniqueness of weak solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 140, 301–317 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Bressan A., Lewicka M.: A uniqueness condition for hyperbolic systems of conservation laws. Discrete Contin. Dyn. Syst. 6, 673–682 (2000)MATHMathSciNetGoogle Scholar
  23. 23.
    Bressan A., Liu T.P., Yang T.: L 1 stability estimates for n × n conservation laws. Arch. Ration. Mech. Anal. 149, 1–22 (1999)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, London (1983)MATHGoogle Scholar
  25. 25.
    Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Springer, Berlin (2005)MATHGoogle Scholar
  26. 26.
    Donadello C., Marson A.: Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws. NoDEA 14, 569–592 (2007)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Dubois F., Le Floch P.: Boundary conditions for Nonlinear Hyperbolic Systems of Conservation Laws. J. Differ. Equ. 71, 93–122 (1988)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Gisclon M.: Etude des conditions aux limites pour un système hyperbolique, via l’approximation parabolique. J. Maths. Pures Appl. 75, 485–508 (1996)MATHMathSciNetGoogle Scholar
  29. 29.
    Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Godlewski E., Raviart P.: Numerical Approximations of Hyperbolic Systems of Conservation Laws. Springer, New York (1996)Google Scholar
  31. 31.
    Goodman, J.: Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. Ph.D. Thesis, California University, 1982Google Scholar
  32. 32.
    Griewank A., Rabier P.J.: On the smoothness of convex envelopes. Trans. AMS 322(2), 691–709 (1990)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge, 1995Google Scholar
  34. 34.
    Kawashima, S.: Systems of hyperbolic-parabolic type, with applications to the equations of magnetohydrodynamics. Ph.D. Thesis, Kyoto University, 1983Google Scholar
  35. 35.
    Kawashima S.: Large-time behavior of solutions to hyperbolic–parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edinburgh Sect. A 106, 169–194 (1987)MATHMathSciNetGoogle Scholar
  36. 36.
    Kawashima S., Shizuta Y.: Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14(2), 249–275 (1985)MATHMathSciNetGoogle Scholar
  37. 37.
    Kawashima S., Shizuta Y.: On the normal form of the symmetric hyperbolic–parabolic systems associated with the conservation laws. Tôhoku Math. J. 40, 449–464 (1988)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Knopp K.: Theory of Functions, Parts I and II. Dover, New York (1947)Google Scholar
  39. 39.
    Lax P.: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Liu T.P.: The Riemann Problem for General Systems of Conservation Laws. J. Differ. Equ. 18, 218–234 (1975)MATHCrossRefGoogle Scholar
  41. 41.
    Rousset, F.: Navier–Stokes equation and block linear degeneracy, Personal Communication Google Scholar
  42. 42.
    Rousset F.: Inviscid boundary conditions and stability of viscous boundary layers. Asymptot. Anal. 26, 285–306 (2001)MATHMathSciNetGoogle Scholar
  43. 43.
    Rousset F.: The residual boundary conditions coming from the real vanishing viscosity method. Discrete Contin. Dyn. Syst. 8(3), 605–625 (2002)MATHMathSciNetGoogle Scholar
  44. 44.
    Rousset F.: Characteristic boundary layers in real vanishing viscosity limits. J. Differ. Equ. 210, 25–64 (2005)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Sablé-Tougeron M.: Méthode de Glimm et problème mixte. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10(4), 423–443 (1993)MATHGoogle Scholar
  46. 46.
    Serre D.: Systems of Conservation Laws, I, II. Cambridge University Press, Cambridge (2000)Google Scholar
  47. 47.
    Serre D., Zumbrun K.: Boundary Layer Stability in Real Vanishing Viscosity Limit. Comm. Math. Phys. 221, 267–292 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Spinolo L.V.: Vanishing viscosity solutions of a 2 × 2 triangular hyperbolic system with Dirichlet conditions on two boundaries. Indiana Univ. Math. J. 56, 279–364 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.SISSA-ISASTriesteItaly
  2. 2.Northwestern UniversityEvanstonUSA

Personalised recommendations