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Collapsing and Explosion Waves in Phase Transitions with Metastability, Existence, Stability and Related Riemann Problems

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Abstract

Collapsing waves were observed numerically before and were used to explain the ring formations in dynamic flows involving phase transitions with metastability. In this paper, necessary and sufficient conditions for collapsing type of waves to exist are given. The conditions are that the wave speed of the collapsing wave is not less than a number and is supersonic on both sides of the wave. Existence and non-existence conditions for the explosion waves are also found. The stability of these waves are studied numerically. Although there are infinitely many collapsing (or explosion) waves for a fixed downstream state, the collapsing (or explosion) wave appeared in the solution of Riemann problem is numerically verified to be the one with the slowest speed. Although a Riemann problem in the zero viscosity limit may have two solutions, one with, the other without, a collapsing (or explosion) wave, from the vanishing viscosity point of view, the one with a collapsing (or explosion) wave is numerically verified to be admissible.

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Authors and Affiliations

  1. Department of Mathematics, Georgetown University, Washington, DC, 20057, USA

    Haitao Fan

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    Xiao-Biao Lin

Authors
  1. Haitao Fan
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  2. Xiao-Biao Lin
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Corresponding author

Correspondence to Xiao-Biao Lin.

Additional information

Dedicated to Professor Jack Hale’s 80th birthday.

Research of Lin was supported in part by the National Science Foundation under grant DMS-0708386.

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Fan, H., Lin, XB. Collapsing and Explosion Waves in Phase Transitions with Metastability, Existence, Stability and Related Riemann Problems. J Dyn Diff Equat 22, 163–191 (2010). https://doi.org/10.1007/s10884-009-9150-9

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  • DOI: https://doi.org/10.1007/s10884-009-9150-9

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