# Preprint 2012-014

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

## Yun-guang Lu

**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.