| Abstract: |
| In this talk, we want to present the isothermal limit of entropy solutions in $L^\infty$, containing the vacuum states, of the Euler equations for isentropic gas dynamics. First, We want to start with the explicit asymptotic analysis of the Riemann solutions containing the vacuum states. Then, we want to show the entropy solutions in $L^\infty$ of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent $\gamma \rightarrow 1$. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent $\gamma\to 1$. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to $\gamma>1$. This is the joint work with Gui-Qiang G. Chen, and Fei-Min Huang. |
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