also prove the following lemma. Because the proof is given in the same way as the above lemma, we omit the proof.

[**Lemma 3.4**]{} (a priori estimates III)[**.**]{} Once the Lemmas 3.2, 3.3 and 3.4 are proved, taking the limit $\epsilon \searrow 0$ as in the arguments in [@matsu-nishi2], we have Theorem 3.3.

[**Theorem 3.3.**]{} Indeed, for the initial data which satisfies $$\phi_0 \in L^2, \qquad \partial_x \phi_0 \in L^2 \cap L^{p+1},$$ we can take a subsequence of $\bigl\{ u_{\epsilon } \bigr\}$ (write $\bigl\{ u_{\epsilon } \bigr\}$ again for simplicity) and a limit function $\phi$ (correspondingly $u := \phi + U + U^r$) by Lemmas 3.2, 3.3 and 3.4, such that $$\begin{aligned}
 \left\{\begin{array}{ll}
 \displaystyle{\operatorname{s-\lim}\displaylimits_{\epsilon \rightarrow 0}} \, 
 \phi_{\epsilon} = \phi 
 \in C^{0}\bigl(\, [\, 0,T\, ] \, ;L^{2} \bigr), \\[10pt]
  \displaystyle{\operatorname{w^{*}-\lim}\displaylimits_{\epsilon \rightarrow 0}} \, 
 \partial_x \phi_{\epsilon} = \partial_x \phi
 \in L^{\infty}\bigl( \, 0,T \, ;L^{p+1} \bigr), \\[10pt]
 \displaystyle{\operatorname{w^{*}-\lim}\displaylimits_{\epsilon \rightarrow 0}} \, 
 \partial_x u_{\epsilon} = \partial_x u 
 \in L^{\infty}\bigl( \, 0,T \, ;L^{p+1} \bigr), \\[10pt]
  \displaystyle{\operatorname{w-\lim}\displaylimits_{\epsilon \rightarrow 0}} \, 
 \partial_x \left( \, 
 \left| \, \partial_x u_{\epsilon} \, \right|^{p-1} \partial_x u_{\epsilon} \, 
 \right)
 =\partial_x \left( \, 
  \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
  \right)
 \in L^{2}\bigl( \, 0,T \, ;L^{2} \bigr), \\[10pt]
 \displaystyle{\operatorname{w-\lim}\displaylimits_{\epsilon 