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On the attainable set for a class of triangular systems of conservation laws

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Abstract

We explore attainability for a special class of triangular systems of conservation laws, not necessarily strictly hyperbolic, which includes the system of multi-component chromatography. Roughly speaking, such systems consist of linear continuity equations coupled with a scalar genuinely nonlinear conservation law. The classical Keyfitz–Kranzer system is also included, with minor modifications. We prove that the backward solutions we construct are appropriate solutions of the system in view of the classical theories for general conservation laws. In particular, we get isentropic solutions whenever nontrivial entropies for the system are defined. We give numerical examples of the isentropic backward resolution of such systems for attainable target data.

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References

  1. Adimurthi, S. S. Ghoshal, and G. D. Veerappa Gowda. Exact controllability of scalar conservation law with strict convex flux. Math. Control Relat. Fields 4(4) (2014), 401–449.

  2. Adimurthi, S.S. Ghoshal, and G. D. Veerappa Gowda. Optimal controllability for scalar conservation laws with strict convex flux. J. Hyperbolic Differ. Equ, 11(3) (2014) 477–491.

  3. L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. Invent. Math., 158(2) (2004) 227–260.

  4. L. Ambrosio, F. Bouchut, and C. De Lellis. Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm. Partial Differential Equations, 29(9-10) (2004) 1635–1651.

  5. Ambrosio L., Crippa G., Figalli A., Spinolo L. V.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J. Math. Anal., 41(5), 1890–1920 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Ambrosio and C. De Lellis. Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. Int. Math. Res. Not., 41 (2003), 2205–2220.

  7. Ancona F., Coclite G. M.: On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43(6), 2166–2190 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ancona F., Marson A.: On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim., 36(1), 290–312 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Andreianov, C. Donadello, S. S. Ghoshal and U. Razafison. HAL preprint hal.archives-ouvertes.fr/hal-00967600

  10. Bressan A., Coclite G. M.: On the boundary control of systems of conservation laws. SIAM J. Control Optim., 41(2), 607–622 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Bouchut and F. James. Équations de transport unidimensionnelles à coefficients discontinus. C. R. Acad. Sci. Paris Sér. I Math., 320(9) (1995), 1097–1102.

  12. Bouchut F., James F.: One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal., 32(7), 891–933 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Bressan. Hyperbolic systems of conservation laws, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem.

  14. Coron J.-M.: Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems, 5(3), 295–312 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. J.-M. Coron, G. Bastin, B. d’Andréa-Novel. Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems. SIAM J. Control Optim. 47(3) (2008) 1460–1498.

  16. J.-M. Coron, O. Glass, Z. Q. Wang. Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed. SIAM J. Control and Optimization, 48(5) (2009) 3105–3122.

  17. Crippa G., Spinolo L. V.: An overview on some results concerning the transport equation and its applications to conservation laws. Commun. Pure Appl. Anal., 9(5), 1283–1293 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dafermos C.M.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J., 26(6), 1097–1119 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd ed., 2010.

  20. C. De Lellis. Notes on hyperbolic systems of conservation laws and transport equations. In Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., pages 277–382. Elsevier/North-Holland, Amsterdam, 2007.

  21. R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3) (1989) 511–547.

  22. Fatemi E., Engquist B., Osher S.: Numerical solution of the high frequency asymptotic expansion for the scalar wave equation. J. Comput. Phys., 120(1), 145–155 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Freistühler H.: On the Cauchy problem for a class of hyperbolic systems of conservation laws. J. Differential Equations, 112(1), 170–178 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Frid H.: Asymptotic stability of non-planar Riemann solutions for multi-D systems of conservation laws with symmetric nonlinearities. J. Hyperbolic Differ. Equ., 1(3), 567–579 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. S. S. Ghoshal. Finer analysis of characteristic curve and its application to exact, optimal controllability, structure of the entropy solution of a scalar conservation law with convex flux. PhD Thesis. TIFR CAM, Bangalore, 2012.

  26. O. Glass. A remark on the controllability of a system of conservation laws in the context of entropy solutions. Series in Contemporary Applied Mathematics Vol. 15 (2010), 332–343.

  27. Glass O., Guerrero S.: On the uniform controllability of the Burgers equation. SIAM J. Control Optim., 46(4), 1211–1238 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.

  29. Gosse L., James F.: Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comp., 69(231), 987–1015 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Guerrero and O. Yu. Imanuvilov. Remarks on global controllability for the Burgers equation with two control forces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24(6) (2007) 897–906.

  31. Horsin T.: On the controllability of the Burgers equation. ESAIM Control Optim. Calc. Var., 3, 83–95 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. B. L. Keyfitz and H. C. Kranzer. A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal., 72(3) (1979/80) 219–241.

  33. M. V. Korobkov and E. Yu. Panov. On isentropic solutions of first-order quasilinear equations. Mat. Sb., 197(5) (2006) 99–124.

  34. S. N. Kruzhkov. First order quasilinear equations with several independent variables. (Russian) Mat. Sb. (N.S.) 81 (197) 228–255.

  35. Lax P.D.: Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math., 10, 537–566 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  36. Léautaud M.: Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control and Optimization, 50(3), 1661–1699 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.

  38. Li T.-T.: Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals. Math. Meth. Appl. Sci. 27(9), 1089–1114 (2004)

    Article  MATH  Google Scholar 

  39. Li T.-T., Rao B.-P.: Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim. 41(6), 1748–1755 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  40. O. A. Oleinik. Discontinuous solutions of non-linear differential equations. (Russian) Uspehi Mat. Nauk (N.S.) 12 (1957), 3–73, English transl. in Amer. Math. Soc. Transl. 26 (1963) 95–172.

  41. E. Yu. Panov. On the theory of entropy solutions of the Cauchy problem for a class of nonstrictly hyperbolic systems of conservation laws. Mat. Sb., 191(1) (2000) 127–157.

  42. E. Yu. Panov. Generalized solutions of the Cauchy problem for a transport equation with discontinuous coefficients. In Instability in models connected with fluid flows. II, volume 7 of Int. Math. Ser. (N. Y.), pages 23–84. Springer, New York, 2008.

  43. D. Serre. Systems of conservation laws. I. Hyperbolicity, entropies, shock waves. Cambridge University Press, Cambridge, 1999.

  44. J. Smoller. Shock waves and reaction-diffusion equations, volume 258 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, 1994.

  45. Yu Lixin. Global exact boundary observability for first-order quasilinear hyperbolic systems of diagonal form. Math. Methods Appl. Sci. 35(13) (2012), 1505–1517.

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Besançon, CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030, Besançon, France

    Boris Andreianov, Carlotta Donadello, Shyam Sundar Ghoshal & Ulrich Razafison

  2. Institut für Mathematik, Technische Universität Berlin, 136 Straße des 17. Juni, Berlin, Germany

    Boris Andreianov

  3. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse Cedex 9, France

    Shyam Sundar Ghoshal

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  1. Boris Andreianov
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  2. Carlotta Donadello
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  3. Shyam Sundar Ghoshal
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  4. Ulrich Razafison
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Correspondence to Carlotta Donadello.

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Andreianov, B., Donadello, C., Ghoshal, S.S. et al. On the attainable set for a class of triangular systems of conservation laws. J. Evol. Equ. 15, 503–532 (2015). https://doi.org/10.1007/s00028-014-0267-x

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