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June 2021 The Riemann problem for van der Waals fluids with nonclassical phase transitions
Mai Duc THANH, Duong Xuan VINH
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Hokkaido Math. J. 50(2): 263-295 (June 2021). DOI: 10.14492/hokmj/2019-115


We consider the Riemann problem for fluids of van der Waals type with phase transitions involving nonclassical shocks. The model is elliptic-hyperbolic and the pressure function admits two inflection points. First, a unique classical Riemann solver is constructed, which is based on rarefaction waves, classical shocks and zero-speed shocks. Second, we investigate nonclassical Riemann solvers, which involve nonclassical shocks. Nonclassical shocks are shock waves which violate the Liu entropy condition and satisfy a kinetic relation. It can be shown that then two wave curves always intersect either once or twice at different phases. Consequently, the Riemann problem always admit one or two solutions in the class of classical and nonclassical shocks, zero-speed shocks, and rarefaction waves.


The authors would like to thank the Reviewers for their very constructive comments and helpful suggestions. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2018-28-01.


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Mai Duc THANH. Duong Xuan VINH. "The Riemann problem for van der Waals fluids with nonclassical phase transitions." Hokkaido Math. J. 50 (2) 263 - 295, June 2021.


Received: 18 February 2019; Revised: 15 October 2019; Published: June 2021
First available in Project Euclid: 27 August 2021

Digital Object Identifier: 10.14492/hokmj/2019-115

Primary: 35L40 , 35L65 , 76L05 , 76N10 , 76T10

Keywords: Lax shock inequalities , Nonclassical shock , phase transition , Riemann problem , van der Waals fluid

Rights: Copyright c 2021 Hokkaido University, Department of Mathematics


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Vol.50 • No. 2 • June 2021
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