Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 35–52 | Cite as

An Inverse Piston Problem with Small BV Initial Data

  • Libin WangEmail author
  • Youren Wang


In this paper, we consider an inverse piston problem with small BV initial data for the system of one-dimensional adiabatic flow. Suppose that the original state of the gas on the right side of the piston and the position of the forward shock are known, then we can globally solve the inverse piston problem and estimate the speed of the piston in a unique manner.


Quasilinear hyperbolic system Bounded variation norm Inverse piston problem One-dimensional adiabatic flow 

Mathematics Subject Classification (2000)

35L45 35L60 35R30 



The authors would like to thank Prof. Tatsien Li for his constant support and encouragement. This work is supported by the National Natural Science Foundation of China, No. 11371095 and No. 11771091.


  1. 1.
    Bressan, A.: Contractive metrics for nonlinear hyperbolic systems. Indiana Univ. Math. J. 37, 409–420 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bressan, A.: Hyperbolic Systems of Conservation Laws, the One-Dimensional Cauchy Problem. Oxford University Press, London (2000) zbMATHGoogle Scholar
  3. 3.
    Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948) zbMATHGoogle Scholar
  4. 4.
    Dai, W., Kong, D.: Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields. J. Differ. Equ. 235, 127–165 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hörmander, L.: The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1256, pp. 214–280. Springer, Berlin (1987) zbMATHGoogle Scholar
  6. 6.
    Li, T.: Global Classical Solutions for Quasilinear Hyperbolic Systems. Research in Applied Mathematics, vol. 32. Wiley/Masson, New York (1994) zbMATHGoogle Scholar
  7. 7.
    Li, T.: Exact shock reconstruction. Inverse Probl. 21, 673–684 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, T., Wang, L.: Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems. Nonlinear Anal. 56, 961–974 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, T., Wang, L.: Global exact shock reconstruction. Discrete Contin. Dyn. Syst. 15, 597–609 (2006) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, T., Wang, L.: Existence and uniqueness of global solution to inverse piston problem. Inverse Probl. 23, 683–694 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, T., Wang, L.: Global Propagation of Regular Nonlinear Hyperbolic Waves. Progress in Nonlinear Differential Equations and Their Applications Birkäuser/Springer, Basel/Berlin (2009) zbMATHGoogle Scholar
  12. 12.
    Li, T., Yu, W.: Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke University Mathematics Series V (1985) zbMATHGoogle Scholar
  13. 13.
    Li, T., Zhou, Y., Kong, D.: Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems. Commun. Partial Differ. Equ. 19, 1263–1317 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, L.: Direct problem and inverse problem for the supersonic plane flow past a curved wedge. Math. Methods Appl. Sci. 34, 2291–2302 (2011) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, L.: An inverse piston problem for the system of one-dimensional adiabatic flow. Inverse Probl. 30, 085009 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhou, Y.: Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy. Chin. Ann. Math. 25B, 37–56 (2004) MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory for Contemporary Applied MathematicsShanghaiChina
  3. 3.Laboratory of Mathematics for Nonlinear ScienceShanghaiChina

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