这是一份GLA格拉斯哥大学MATHS4102_1作业代写的成功案例
at least in a neighborhood of the static state $(\bar{\varrho}, \bar{\vartheta})$, we conclude, in agreement with the formal asymptotic expansion discussed that the quantities
$$
\varrho_{0, \varepsilon}^{(1)}=\frac{\rho(0, \cdot)-\bar{\varrho}}{\varepsilon} \text { and } \vartheta_{0, \varepsilon}^{(1)}=\frac{\vartheta(0, \cdot)-\bar{\vartheta}}{\varepsilon}, \text { and } \mathbf{u}_{0, \varepsilon}=\mathbf{u}(0, \cdot)
$$
have to be bounded uniformly for $\varepsilon \rightarrow 0$, or, in the terminology introduced the initial data must be at least ill-prepared.
As a direct consequence of the structural properties of $H_{\bar{v}}$ established it is not difficult to deduce from
$$
\varrho^{(1)}(t, \cdot)=\frac{\varrho(t, \cdot)-\bar{g}}{\varepsilon} \text { and } \vartheta^{(1)}=\frac{\vartheta(t, \cdot)-\bar{\vartheta}}{\varepsilon}
$$
remain bounded, at least in $L^{1}(\Omega)$, uniformly for $t \in[0, T]$ and $\varepsilon \rightarrow 0$.
MATHS4102_1 COURSE NOTES ：
$$
\vartheta \mapsto H_{\bar{\vartheta}}(\varrho, \vartheta)-(\varrho-\bar{\varrho}) \frac{\partial H_{\bar{\vartheta}}(\bar{\varrho}, \bar{\vartheta})}{\partial \varrho}-H_{\bar{\vartheta}}(\bar{\varrho}, \bar{\vartheta})
$$
is decreasing for $\vartheta<\bar{\vartheta}$ and increasing whenever $\vartheta>\bar{\vartheta}$; whence $(5.39)$ follows. Finally, as $\mathcal{F}$ is strictly convex, we have
$$
H_{\vec{\vartheta}}(\varrho, \vartheta)-(\varrho-\bar{\varrho}) \frac{\partial H_{\bar{v}}(\bar{\varrho}, \bar{\vartheta})}{\partial \varrho}-H_{\bar{\vartheta}}(\bar{\varrho}, \bar{\vartheta}) \geq c(\bar{\varrho}, \bar{\vartheta}) \varrho \text { whenever } \varrho \geq 2 \bar{\varrho}
$$