Electron. J. Diff. Equ., Vol. 2015 (2015), No. 32, pp. 1-18.

Existence of solutions to the Riemann problem for a model of two-phase flows

Mai Duc Thanh, Dao Huy Cuong

Abstract:
We study the existence of solutions of the Riemann problem for a model of two-phase flows. The model has the form of a nonconservative hyperbolic system of balance laws. Based on a phase decomposition approach, we obtain all the wave curves. By developing an analytic method, we can establish a system of nonlinear algebraic equations for each solution of the Riemann problem. The system is under-determined and can be parameterized by the volume fraction in one phase. Therefore, an argument relying on the Implicit-Function Theorem leads us to the existence of solutions of the Riemann problem for the model for sufficiently large initial data. Furthermore, the structure of the Riemann solutions obtained by this method can also be obtained.

Submitted November 19, 2014. Published February 5, 2015.
Math Subject Classifications: 35L65, 35L67, 76T10, 76N10.
Key Words: Two-phase flow; nonconservative; source term; jump relation; shock; Riemann problem.

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Mai Duc Thanh
Department of Mathematics, International University
Vietnam National University - HCMC, Quarter 6
Linh Trung Ward, Thu Duc District
Ho Chi Minh City, Vietnam
email: mdthanh@hcmiu.edu.vn
Dao Huy Cuong
Nguyen Huu Cau High School, 07 Nguyen Anh Thu
Trung Chanh Ward, Hoc Mon District
Ho Chi Minh City, Vietnam
email: cuongnhc82@gmail.com

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