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# Existence of global entropy solutions to the isentropic Euler equations with geometric effects

Abstract: In the paper Lu (2011) [1], the maximum principle was used to study the uniformly bounded $L^{\infty}$ estimates $z(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq B(x)$, $w(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq M(t)$ for the $\varepsilon$-viscosity and $\delta$-flux-approximation solutions $(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon})$ of the nonhomogeneous system (\ref{1.3}), where $w$, $z$ are Riemann invariants of (\ref{1.3}) and $M(t)$ depends on the bound of the nonlinear function $a(x)$, which excludes the class of discontinuous functions. In this short paper, we obtain the estimate $w(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq \beta$ when $a'(x) \geq 0$ for a suitable constant $\beta$ depending only on the bound of $a(x)$ and prove the existence of bounded entropy solutions, for the Cauchy problem of the isentropic Euler equations with geometric effects (\ref{1.1}), which extend the results of finite energy solution in [2], and weak solutions in [3] for a polytropic gas with $\gamma \in (1, \frac{5}{3}]$ to the general pressure function $P(\rho)$. $$\left\{ \label{1.1} \begin{array}{l} ( \rho a(x))_{t}+(\rho u a(x))_{x}=0, \\ ( \rho u a(x))_t+( \rho u^2 a(x))_x+ a(x) P(\rho)_x=0 \end{array}\right.\tag{1.1}$$ $$\left\{ \label{1.3} \begin{array}{l} \rho_{t}+(\rho u)_{x}= - \frac{a'(x)}{a(x)} \rho u \\ ( \rho u)_t+( \rho u^2+ P(\rho))_x= - \frac{a'(x)}{a(x)} \rho u^{2}. \end{array}\right.\tag{1.3}$$

References
[1] Y.-G. Lu, Global Existence of Resonant Isentropic Gas Dynamics, Nonlinear Analysis, Real World Applications(2011), 12 (2011), 2802–2810.
[2] P. LeFloch and M. Westdickenberg, Finite energy solutions to the isentropic Euler equations with geometric effects, Jour Math.Pures Appl. 88 (2007), 389–429.
[3] N. Tsuge, {\em Global $L^{ \infty}$ Solutions of the Compressible Euler Equations with Spherical Symmetry}, J. Math. Kyoto Univ., 46 (2006), 457–524.
Paper:
Available as PDF (226 Kbytes).
Author(s):
Yun-guang Lu
Publishing information:
Submitted
Submitted by:
; 2012-06-27.