Advertisement

A Theory of L 1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

  • Boris AndreianovEmail author
  • Kenneth Hvistendahl Karlsen
  • Nils Henrik Risebro
Open Access
Article

Abstract

We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux:
$$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL) $$
where the fluxes f l , f r are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers in SIAM J Numer Anal 38(2):681–698, 2000; Towers in SIAM J Numer Anal 39(4):1197–1218, 2001; Towers et al. in Skr K Nor Vidensk Selsk 3:1–49, 2003; Adimurthi et al. in J Math Kyoto University 43(1):27–70, 2003; Adimurthi et al. in J Hyperbolic Differ Equ 2(4):783–837, 2005; Audusse and Perthame in Proc Roy Soc Edinburgh A 135(2):253–265, 2005; Garavello et al. in Netw Heterog Media 2:159–179, 2007; Bürger et al. in SIAM J Numer Anal 47:1684–1712, 2009), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form
$$ c(x)=c^l11_{\left\{{x < 0}\right\}}+c^r11_{\left\{{x > 0}\right\}}. $$
We refer to such a family as a “germ”. It is well known that (CL) admits many different L 1 contractive semigroups, some of which reflect different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the “vanishing viscosity” germ, which is a way of expressing the “Γ-condition” of Diehl (J Hyperbolic Differ Equ 6(1):127–159, 2009). For any given germ, we formulate “germ-based” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} [in the spirit of Vol’pert (Math USSR Sbornik 2(2):225–267, 1967)] and in the form of a family of global entropy inequalities [following Kruzhkov (Math USSR Sbornik 10(2):217–243, 1970) and Carrillo (Arch Ration Mech Anal 147(4):269–361, 1999)]. We characterize those germs that lead to the L 1-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities “adapted” to the choice of a germ), or for specific germ-adapted finite volume schemes.

Keywords

Weak Solution Boris Riemann Problem Entropy Solution Entropy Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This paper was written as part of the research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo, which took place during the academic year 2008–2009. The authors thank S. Mishra, D. Mitrovič, E. Panov, N. Seguin, and J. Towers for interesting discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Adimurthi , Ghoshal S.S., Dutta R., Veerappa Gowda G.D.: Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Comm. Pure Appl. Math. 64(1), 84–115 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adimurthi , Veerappa Gowda G.D.: Conservation laws with discontinuous flux. J. Math. Kyoto University 43(1), 27–70 (2003)MathSciNetGoogle Scholar
  3. 3.
    Adimurthi , Jaffré J., Veerappa Gowda G.D.: Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J. Numer. Anal. 42(1), 179–208 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Adimurthi , Mishra S., Veerappa Gowda G.D.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2(4), 783–837 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Alt H.W., Luckhaus S.: Quasilinear elliptic-parabolic differential equations. Mat. Z. 183, 311–341 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ammar K., Wittbold P.: Existence of renormalized solutions of degenerate elliptic-parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A 133(3), 477–496 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ammar K., Wittbold P., Carrillo J.: Scalar conservation laws with general boundary condition and continuous flux function. J. Differ. Equ. 228(1), 111–139 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Andreianov B., Bénilan P., Kruzhkov S.N.: L 1 theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171(1), 15–33 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Andreianov B., Goatin P., Seguin N.: Finite volume schemes for locally constrained conservation laws. Numer. Math. 115(4), 609–645 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of L 1-dissipative solvers for scalar conservation laws with discontinuous flux. II (in preparation)Google Scholar
  11. 11.
    Andreianov B., Karlsen K.H., Risebro N.H.: On vanishing viscosity approximation of conservation laws with discontinuous flux. Netw. Heterog. Media 5(3), 617–633 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Andreianov B., Sbihi K.: Scalar conservation laws with nonlinear boundary conditions. C. R. Acad. Sci. Paris, Ser. I 345, 431–434 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Audusse E., Perthame B.: Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. Roy. Soc. Edinburgh A 135(2), 253–265 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bachmann F., Vovelle J.: Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Comm. Partial Differ. Equ. 31, 371–395 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Baiti P., Jenssen H.K.: Well-posedness for a class of 2 × 2 conservation laws with L data. J. Differ. Equ. 140(1), 161–185 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bardos C., LeRoux A.-Y., Nédélec J.-C.: First order quasilinear equations with boundary conditions. Comm. Partial Differ. Equ. 4(9), 1017–1034 (1979)zbMATHCrossRefGoogle Scholar
  17. 17.
    Bénilan, P.: Equations d’évolution dans un espace de Banach quelconques et applications. Thèse d’état, 1972Google Scholar
  18. 18.
    Bénilan P., Carrillo J., Wittbold P.: Renormalized entropy solutions of scalar conservation laws. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(2), 313–327 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bénilan P., Kruzhkov S.N.: Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3(4), 395–419 (1996)zbMATHCrossRefGoogle Scholar
  20. 20.
    Bürger R., García A., Karlsen K.H., Towers J.D.: Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Netw. Heterog. Media 3, 1–41 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Bürger, R., Karlsen, K.H., Mishra, S., Towers, J.D.: On conservation laws with discontinuous flux. Trends in Applications of Mathematics to Mechanics (Eds. Wang Y. and Hutter K.) Shaker Verlag, Aachen, 75–84, 2005Google Scholar
  22. 22.
    Bürger R., Karlsen K.H., Towers J.: An Engquist-Osher type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47, 1684–1712 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cancès C.: Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution. SIAM J. Math. Anal. 42(2), 946–971 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Cancès C.: Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping. SIAM J. Math. Anal. 42(2), 972–995 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Cancès C.: On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Netw. Heterog. Media 5(3), 635–647 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Chechkin, G.A., Goritsky, A.Yu.: S. N. Kruzhkov’s lectures on first-order quasilinear PDEs. (Eds. Emmrich E. and Wittbold P.) Anal. Numer. Asp. Partial Differ. Equ., de Gruyter, 2009Google Scholar
  28. 28.
    Chen G.-Q., Even N., Klingenberg C.: Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems. J. Differ. Equ. 245(11), 3095–3126 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Chen G.-Q., Frid H.: Divergence-measure fields and hyperbolic conservationlaws. Arch. Ration. Mech. Anal. 147, 89–118 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Chen G.-Q., Rascle M.: Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal. 153, 205–220 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Colombo R.M., Goatin P.: A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234(2), 654–675 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Crandall M.G.: The semigroup approach to first-order quasilinear equations in several space variables. Israel J. Math. 12, 108–132 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78(3), 385–390 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Diehl S.: On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 26(6), 1425–1451 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Diehl, S.: Scalar conservation laws with discontinuous flux function. I. The viscous profile condition. II. On the stability of the viscous profiles. Commun. Math. Phys. 176(1), 23–44 and 45–71 (1996)Google Scholar
  36. 36.
    Diehl S.: A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56(2), 388–419 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Diehl S.: A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 6(1), 127–159 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII. North-Holland, Amsterdam, 713–1020, 2000Google Scholar
  39. 39.
    Gallouët T., Hubert F.: On the convergence of the parabolic approximation of a conservation law in several space dimensions. Chin. Ann. Math. Ser. B 20(1), 7–10 (1999)zbMATHCrossRefGoogle Scholar
  40. 40.
    Garavello M., Natalini R., Piccoli B., Terracina A.: Conservation laws with discontinuous flux. Netw. Heterog. Media 2, 159–179 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Gelfand, I.M.: Some problems in the theory of quasilinear equations. (Russian) Uspekhi Mat. Nauk 14(2), 87–158, 1959; English tr. in Am. Math. Soc. Transl. Ser. 29(2), 295–381 (1963)Google Scholar
  42. 42.
    Gimse, T., Risebro, N.H.: Riemann problems with a discontinuous flux function. Proceedings of Third International Conference on Hyperbolic Problems, Vol. I, II Uppsala, 1990, 488–502, Studentlitteratur, Lund, 1991Google Scholar
  43. 43.
    Gimse T., Risebro N.H.: Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23(3), 635–648 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. In: Applied Mathematical Sciences, vol. 152. Springer, New York, 2002Google Scholar
  45. 45.
    Hopf E.: The partial differential equation u t + uu x = μ u xx. Comm. Pure Appl. Math. 3(3), 201–230 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Jimenez J.: Some scalar conservation laws with discontinuous flux. Int. J. Evol. Equ. 2(3), 297–315 (2007)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Jimenez J., Lévi L.: Entropy formulations for a class of scalar conservations laws with space-discontinuous flux functions in a bounded domain. J. Engrg. Math. 60(3–4), 319–335 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Kaasschieter E.F.: Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium. Comput. Geosci. 3, 23–48 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Karlsen K.H., Towers J.D.: Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chin. Ann. Math. Ser. B. 25(3), 287–318 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Karlsen, K.H., Risebro, N.H., Towers, J.D.: On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. Electron J. Differ. Equ., pages No. 93, 23 pp. (electronic), (2002)Google Scholar
  51. 51.
    Karlsen K.H., Risebro N.H., Towers J.D.: Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22, 623–664 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Karlsen K.H., Risebro N.H., Towers J.D.: L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3, 1–49 (2003)MathSciNetGoogle Scholar
  53. 53.
    Klingenberg C., Risebro N.H.: Convex conservation laws with discontinuous coefficients: Existence, uniqueness and asymptotic behavior. Comm. Partial Differ. Equ. 20(11–12), 1959–1990 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Kruzhkov S.N.: First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10(2), 217–243 (1970)zbMATHCrossRefGoogle Scholar
  55. 55.
    Kruzhkov S.N., Hildebrand F.: The Cauchy problem for first order quasilinear equations in the case when the domain of dependence on the initial data is infinite (Russian). Vestnik Moskov. Univ. Ser. I Mat. Meh. 29, 93–100 (1974)MathSciNetGoogle Scholar
  56. 56.
    Kruzhkov, S.N., Panov, E.Yu.: First-order quasilinear conservation laws with infinite initial data dependence area (Russian). Dokl. Akad. Nauk URSS 314(1), 79–84, 1990; English tr. in Sov. Math. Dokl. 42(2), 316–321 (1991)Google Scholar
  57. 57.
    Kwon Y.-S., Vasseur A.: Strong traces for solutions to scalar conservation laws with general flux. Arch. Ration. Mech. Anal. 185(3), 495–513 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    LeFloch, P.G.: Hyperbolic systems of conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. The theory of classical and nonclassical shock wavesGoogle Scholar
  59. 59.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. (French) Dunod, Paris, 1969Google Scholar
  60. 60.
    Lions P.-L., Perthame B., Tadmor E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Málek J., Nečas J., Rokyta M., Ružička M.: Weak and Measure-valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)zbMATHGoogle Scholar
  62. 62.
    Maliki M., Touré H.: Uniqueness of entropy solutions for nonlinear degenerate parabolic problems. J. Evol. Equ. 3(4), 603–622 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Mishra, S.: Analysis and Numerical Approximation of Conservation Laws with Discontinuous Coefficients. PhD thesis, Indian Institute of Science, Bangalore, India, 2005Google Scholar
  64. 64.
    Mitrovič D.: Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks Het. Media 5(1), 163–188 (2010)CrossRefGoogle Scholar
  65. 65.
    Olenik O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi–linear equation. Amer. Math. Soc Transl. Ser. 2 33, 285–290 (1963)Google Scholar
  66. 66.
    Ostrov D.N.: Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence. J. Differ. Equ. 182(1), 51–77 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Otto F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Otto F.: L 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131(1), 20–38 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Panov E.Yu.: Strong measure-valued solutions of the Cauchy problem for a first-order quasilinear equation with a bounded measure-valued initial function. Moscow Univ. Math. Bull. 48(1), 18–21 (1993)MathSciNetGoogle Scholar
  70. 70.
    Panov, E.Yu.: On sequences of measure valued solutions for a first order quasilinear equation (Russian). Mat. Sb. 185(2), 87–106 1994; Engl. tr. in Russian Acad. Sci. Sb. Math. 81(1), 211–227 (1995)Google Scholar
  71. 71.
    Panov E.Yu.: Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Panov E.Yu.: Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4(4), 729–770 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Panov E.Yu.: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Arch. Ration. Mech. Anal. 195(2), 643–673 (2009)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Panov E.Yu.: On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux. J. Hyperbolic Differ. Equ. 6(3), 525–548 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Perthame, B.: Kinetic formulation of conservation laws, volume 21 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002Google Scholar
  76. 76.
    Quinn (Keyfitz), B.: Solutions with shocks, an example of an L 1-contractive semigroup. Comm. Pure Appl. Math. 24(1), 125–132 (1971)Google Scholar
  77. 77.
    Seguin N., Vovelle J.: Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13(2), 221–257 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Szepessy A.: Measure-valued solution of scalar conservation laws with boundary conditions. Arch. Ration. Mech. Anal. 107(2), 182–193 (1989)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Temple B.: Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3(3), 335–375 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Towers J.D.: Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38(2), 681–698 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Towers J.D.: A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39(4), 1197–1218 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Vallet G.: Dirichlet problem for a nonlinear conservation law. Rev. Math. Comput. 13(1), 231–250 (2000)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Vasseur A.: Strong traces of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 160(3), 181–193 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Vol’pert A.I.: The spaces BV and quasi-linear equations. Math. USSR Sbornik 2(2), 225–267 (1967)zbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Boris Andreianov
    • 1
    Email author
  • Kenneth Hvistendahl Karlsen
    • 2
  • Nils Henrik Risebro
    • 2
  1. 1.Laboratoire de Mathématiques CNRS UMR 6623Université de Franche-ComtéBesançonFrance
  2. 2.Centre of Mathematics for ApplicationsUniversity of OsloOsloNorway

Personalised recommendations