independent of $T$ in the next sections.

[**Proposition 3.1**]{} (uniform estimates I)[**.**]{}

Furthermore, we have the $L^{p+1}$-energy estimate for $\partial _x u$ as follows.

[**Proposition 3.2**]{} (uniform estimates II)[**.**]{}

[**Proof of Lemma 3.2.**]{} We first note $$\begin{aligned}
\left\{\begin{array}{ll}
\displaystyle{ \| \, \phi_0^{\epsilon} \, \| _{L^q} } 
\leq \| \, \phi_0 \, \| _{L^q} 
\qquad \qquad \qquad \qquad \qquad \quad \: \: \, \bigl(1 \leq q \leq \infty \bigr), \\[5pt]
\displaystyle{ \| \, \partial_x u_0^{\epsilon} \, \| _{L^{r+1}} } 
\leq \| \, \partial_x u_0 \, \| _{L^{r+1}} 
\qquad \qquad \qquad \qquad \quad \; \: \: \, \, \, (r \geq 1), \\[5pt]
\displaystyle{ 
\sup_{- \widetilde{C}  \leq u \leq  \widetilde{C} } 
\bigl| \, D^k f^{\epsilon} (u) \, \bigr| } 
\leq \displaystyle{ 
     \sup_{0 \leq u \leq \widetilde{C} +1 } 
     \bigl| \, D^k f (u) \, \bigr| } 
\quad \quad \; \, (k=0,1,2), 
\end{array} 
\right.\,\end{aligned}$$ where the positive constant $\widetilde{C}$ is defined in Lemma 3.1. By using Lemma 1.2.1, we can get

[**Lemma 3.5.**]{}

On the other hand, we easily have $$\begin{aligned}
\begin{aligned}
|\, U^{\epsilon} (t,x) \,|
&= \int_{-\infty}^{\infty} 
   \rho_{\epsilon} (y) \, 
   |\, U (t,x-y) \,|\, \mathrm{d}y \\
&\leq \int_{-\infty}^{\infty} 
      \rho_{\epsilon} (y) \, \mathrm{d}y \, 
      \| \, U       (t) \, \| _{L^{\infty}} 
      \leq | 