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The Riemann problem for isentropic compressible Euler equations with discontinuous flux

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Abstract

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x = x(t). We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity x = x(t) is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x = x(t) + 0, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yinzheng Sun & Aifang Qu

  2. School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

    Hairong Yuan

Authors
  1. Yinzheng Sun
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  2. Aifang Qu
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  3. Hairong Yuan
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Correspondence to Aifang Qu.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

This work was supported by the National Natural Science Foundation of China (11871218, 12071298), and in part by the Science and Technology Commission of Shanghai Municipality (21JC1402500, 22DZ2229014).

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Sun, Y., Qu, A. & Yuan, H. The Riemann problem for isentropic compressible Euler equations with discontinuous flux. Acta Math Sci 44, 37–77 (2024). https://doi.org/10.1007/s10473-024-0102-6

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  • DOI: https://doi.org/10.1007/s10473-024-0102-6

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