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Linear and Nonlinear Waves in Gas Dynamics

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Industrial Mathematics and Complex Systems

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Abstract

Although systems of hyperbolic conservation laws form an important model for many phenomena in fluid dynamics, including compressible flow, surface waves in shallow water, reacting fluids, magnetohydrodynamics, and multiphase flow, the underlying theory of quasilinear hyperbolic systems in more than one space variable is poorly developed. This survey outlines a few reasons for the absence of a comprehensive theory and examines some current research on multidimensional problems. When one examines the structure of the characteristics of the gas dynamics equations, it is noteworthy that they fall into two distinct types, which could be called “nonlinear” and “linear”. Each type governs some aspects of a solution, and the two types interact in complicated ways. The study of examples gives many suggestions for further research, although we are still far from a theory.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Brenner, P.: The Cauchy problem for symmetric hyperbolic systems in \(L_p\). Math. Scand. 19, 27–37 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  3. Čanić, S., Keyfitz, B.L.: Riemann problems for the two-dimensional unsteady transonic small disturbance equation. SIAM J. Appl. Math. 58, 636–665 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Čanić, S., Keyfitz, B.L., Kim, E.H.: Free boundary problems for the unsteady transonic small disturbance equation: transonic regular reflection. Methods Appl. Anal. 7, 313–336 (2000)

    MATH  MathSciNet  Google Scholar 

  5. Čanić, S., Keyfitz, B.L., Kim, E.H.: A free boundary problem for a quasilinear degenerate elliptic equation: regular reflection of weak shocks. Commun. Pure Appl. Math. LV, 71–92 (2002)

    Google Scholar 

  6. Čanić, S., Keyfitz, B.L., Kim, E.H.: Mixed hyperbolic-elliptic systems in self-similar flows. Boletim da Sociedade Brasileira de Matemática 32, 1–23 (2002)

    Google Scholar 

  7. Čanić, S., Keyfitz, B.L., Kim, E.H.: Free boundary problems for nonlinear wave systems: mach stems for interacting shocks. SIAM J. Math. Anal. 37, 1947–1977 (2005)

    MATH  MathSciNet  Google Scholar 

  8. S. Čanić, B. L. Keyfitz, and G. M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Communications on Pure and Applied Mathematics LIII (2000), 1–28

    Google Scholar 

  9. Cao, W., Huang, F., Wang, D.: Isometric immersions of surfaces with two classes of metrics and negative Gauss curvature. Arch. Ration. Mech. Anal. 218, 1431–1457 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, G.-Q., Deng, X., Xiang, W.: Shock diffraction by convex cornered wedges for the nonlinear wave system. Arch. Ration. Mech. Anal. 211, 61–112 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, G.-Q., Feldman, M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Am. Math. Soc. 16, 461–494 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, G.-Q., Feldman, M.: Global solutions of shock reflection by large-angle wedges for potential flow. Ann. Math. 171, 1067–1182 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, G.-Q., Feldman, M.: Mathematics of shock reflection-diffraction, von Neumann’s conjectures, and related analysis. University Press, Princeton (2017)

    Google Scholar 

  14. Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Wiley-Interscience, New York (1948)

    MATH  Google Scholar 

  15. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  16. De Lellis, C., Székelyhidi Jr., L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 225–260 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Elling, V.: Non-existence of strong regular reflections in self-similar potential flow. J. Differ. Eq. 252, 2085–2103 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Elling, V., Liu, T.-P.: The ellipticity principle for selfsimilar polytropic potential flow. J. Hyperb. Differ. Eq. 2, 909–917 (2005)

    Article  MATH  Google Scholar 

  19. Elling, V., Liu, T.-P.: Supersonic flow onto a solid wedge. Commun. Pure Appl. Math. 61, 1347–1448 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Garabedian, P.R.: Partial Differential Equations, 2nd edn. Chelsea, New York (1986)

    MATH  Google Scholar 

  21. Guderley, K.G.: The Theory of Transonic Flow. Pergamon Press, Oxford (1962)

    MATH  Google Scholar 

  22. Himonas, A.A., Misiołek, G.: Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics. Commun. Math. Phys. 296, 285–301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Holmes, J., Keyfitz, B.L., Tiğlay, F.: Nonuniform dependence on initial data for compressible gas dynamics: the Cauchy problem on \({\mathbb{R}}^2\). In preparation

    Google Scholar 

  24. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, New York (1983)

    MATH  Google Scholar 

  25. Jegdić, I., Jegdić, K.: Properties of solutions in semi-hyperbolic patches for the unsteady transonic small disturbance equation. Electron. J. Differ. Eq. 243, 20 (2015)

    MATH  MathSciNet  Google Scholar 

  26. Jegdić, K.: Weak regular reflection for the nonlinear wave system. J. Hyperbolic Differ. Eq. 5, 399–420 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jegdić, K., Keyfitz, B.L., Čanić, S.: Transonic regular reflection for the nonlinear wave system. J. Hyperbolic Differ. Eq. 3, 443–474 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jegdić, K., Keyfitz, B.L., Čanić, S., Ying, H.: A free boundary problem for the isentropic gas dynamics equations — transonic regular reflection, submitted (2015), 40 pages

    Google Scholar 

  29. Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  30. Keyfitz, B.E.: Self-similar solutions of two-dimensional conservation laws. J. Hyperbolic Differ. Eq. 1, 445–492 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Keyfitz, B.L., Tiğlay, F.: Nonuniform dependence on initial data for compressible gas dynamics: the periodic Cauchy problem, J. Differ. Eq. arXiv:1611.05840

  32. Keyfitz, B.L., Tesdall, A., Payne, K.R., Popivanov, N.I.: The sonic line as a free boundary. Quart. Appl. Math. LXXI, 119–133 (2013)

    MATH  MathSciNet  Google Scholar 

  33. Lax, P.D.: Hyperbolic Partial Differential Equations. Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (2006)

    Book  MATH  Google Scholar 

  34. Littman, W.: The wave operator and \(L_p\) norms. J. Math. Mech. 12, 55–68 (1963)

    MATH  MathSciNet  Google Scholar 

  35. Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)

    Book  MATH  Google Scholar 

  36. Majda, A., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  37. Mizohata, S.: On the Cauchy Problem. Academic Press, San Diego (1985)

    MATH  Google Scholar 

  38. Rauch, J.: \({BV}\) estimates fail in most quasilinear hyperbolic systems in dimensions greater than one. Commun. Math. Phys. 106, 481–484 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  39. Renardy, M., Rogers, R.C.: An introduction to partial differential equations. Springer, New York (1993)

    MATH  Google Scholar 

  40. Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3, 343–401 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  41. Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50, 1261–1286 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Skews, B., Li, G., Paton, R.: Experiments on Guderley Mach reflection. Shock Waves 19, 95–102 (2009)

    Article  Google Scholar 

  43. Slemrod, M.: Lectures on the isometric embedding problem \((M^n,g)\mapsto {\mathbb{R}}^m\), \(m=\frac{n}{2}(n+1)\). Differential Geometry and Continuum Mechanics, vol. 137, pp. 77–120. Springer, Berlin (2015)

    Google Scholar 

  44. Taylor, M.E.: Partial differential equations i: basic theory. Springer, New York (1996)

    Book  MATH  Google Scholar 

  45. Tesdall, A.M., Keyfitz, B.L.: A continuous, two-way free boundary in the unsteady transonic small disturbance equations. J. Hyperbolic Differ. Eq. 7, 317–338 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. Tesdall, A.M., Sanders, R., Keyfitz, B.L.: The triple point paradox for the nonlinear wave system. SIAM J. Appl. Math. 67, 321–336 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Tesdall, A.M., Sanders, R., Keyfitz, B.L.: Self-similar solutions for the triple point paradox in gasdynamics. SIAM J. Appl. Math. 68, 1360–1377 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  48. Tesdall, A.M., Hunter, J.K.: Self-similar solutions for weak shock reflection. SIAM J. Appl. Math. 63, 42–61 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  49. Ying, H., Keyfitz, B.L.: A two-dimensional Riemann problem for scalar conservation laws. In: Bressan, A., Chen, G.-Q., Lewicka, M., Wang, D.-H. (eds.) Nonlinear Conservation Laws and Applications. IMA, vol. 153, pp. 447–455. Springer, New York (2011)

    Google Scholar 

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Acknowledgements

This paper is based on a talk at the Silver Jubilee Conference of the Indian Society of Industrial and Applied Mathematics, which took place at Sharda University in Greater Noida, Uttar Pradesh, India, January 29–31, 2016. It is a pleasure to acknowledge the hospitality and support of ISIAM and Sharda University.

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Authors and Affiliations

  1. The Ohio State University, Columbus, OH, USA

    Barbara Lee Keyfitz

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  1. Barbara Lee Keyfitz
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Correspondence to Barbara Lee Keyfitz .

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Editors and Affiliations

  1. Department of Mathematics, Guru Nanak Dev University, Amritsar, Punjab, India

    Pammy Manchanda

  2. CNRS, Dieudonné Center of Mathematics, University Côte d’Azur, Nice, France

    René Lozi

  3. School of Basic Sciences and Research, Sharda University, Greater Noida, Uttar Pradesh, India

    Abul Hasan Siddiqi

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Keyfitz, B.L. (2017). Linear and Nonlinear Waves in Gas Dynamics. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Industrial Mathematics and Complex Systems. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3758-0_1

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