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Wave-front tracking for the equations of non-isentropic gas dynamics—basic lemmas

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Abstract

In the random choice and its alternative wave-front tracking methods, approximate solutions are constructed by solving exactly or approximately the Riemann problem in each neighborhood of a certain finite set of jump points depending on the time. In order to obtain global in time BV solutions, one has to get a priori estimates for the total variation in x at the time t of approximate solutions, which is, roughly speaking, the summation of amplitudes of waves at t constituting the solutions to the Riemann problems. Since amplitudes may increase through the interaction of neighboring waves, the crucial point is to estimate the amplitudes of outgoing waves by those of incoming waves in a single Riemann solution, which is called the local interaction estimates.

The aim of this note is to provide a detailed description of the Riemann problem to the equations of polytropic gas dynamics and a complete proof of the basic lemmas on which the local interaction estimates are based. Although all of them, except for Lemmas 4.2 and 5.1, are presented in T.-P. Liu (Indiana Univ. Math. J. 26:147–177, 1977), that paper is difficult and not well understood even at the present day in spite of its importance. For the sake of completeness, this note includes proofs of the local interaction estimates.

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Notes

  1. [λ j ]=[x][t]−1=ML −2 T −1 (j=1,2).

  2. [w]=[z]=LT −1.

  3. [D]=L −1 T, [D 1]=1, [D 2]=[D 2]=L −2 T 2.

  4. The next expression shows that the difference q−q 0 is a dimensionless quantity.

References

  1. Asakura, F., Corli, A.: Wave-front tracking for the equations of non-isentropic gas dynamics. Preprint, 2012

  2. Bressan, A.: Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170, 414–432 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bressan, A.: Hyperbolic Systems of Conservation Laws. Oxford University Press, London (2000)

    MATH  Google Scholar 

  4. Dafermos, C.: Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu, T.-P.: Solutions in the large for the equations of nonisentropic gas dynamics. Indiana Univ. Math. J. 26, 147–177 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu, T.-P.: Initial-boundary value problems for gas dynamics. Arch. Ration. Mech. Anal. 64, 137–168 (1977)

    Article  MATH  Google Scholar 

  8. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61(1), 75–130 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Publ. Math. No. 79-02. Univ. Paris-Sud, Orsay (1978)

  10. Risebro, N.H.: A front-tracking alternative to the random choice method. Proc. Am. Math. Soc. 117, 1125–1139 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Smoller, J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer, New York (1994)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Science, Osaka Electro-Communication University, 18-8 Hatucho, Neyagawa, Osaka, 572-8530, Japan

    Fumioki Asakura

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  1. Fumioki Asakura
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Correspondence to Fumioki Asakura.

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Asakura, F. Wave-front tracking for the equations of non-isentropic gas dynamics—basic lemmas. Acta Math Vietnam. 38, 487–516 (2013). https://doi.org/10.1007/s40306-013-0030-3

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