 \leq e^{(1-\alpha-\beta)\beta^{-1}\tau}\left( {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{B_1(0)} \ast \theta\|}_q + {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)} \ast \theta\|}_q \right) \\
& \leq  e^{(1-\alpha-\beta)\beta^{-1}\tau}\left( {\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{B_1(0)}\|}_1{\|\theta\|}_q + {\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)}\|}_qM\right). \end{aligned}$$ Since ${\|\theta(\tau)\|}_q \lesssim 1$, by and we have, $$\begin{aligned}
{\|\vec{v}\|}_q & \lesssim e^{(1-\alpha-\beta)\beta^{-1}\tau}\left(1 + e^{(d-\gamma)\tau} \right) \\
& \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau} + e^{(1-\beta - \gamma\beta)\beta^{-1}\tau}.  \end{aligned}$$ Since $1 - \beta - \gamma\beta \leq 0$ and $1 - \alpha \leq 0$, by Morrey’s inequality we may conclude $\vec{v} \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$. Similarly if $\gamma = d$, then by the same reasoning as above, and imply, $$\begin{aligned}
{\|\vec{v}\|}_q & \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau}\left(1 + \tau + e^{(d-\gamma)\tau} \right) \\
& \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau}\left(1 + \tau\right) + e^{(1-\beta - \gamma\beta)\beta^{-1}\tau}.  \end{aligned}$$ Since $1 - \alpha - \beta < 0$, we may conclude also in this case that $\vec{v} \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$. Therefore Lemma \[lem:rescaled\_inftybdd\] applies with the hypotheses of Theorem \[thm:IA2\]. Re-writing in terms of $x$ and $t$, this implies . A similar proof with Lemma \[lem:finite\_p\_bounded\_unifint\] in place of Lemma \[lem:finite\_p\_bounded\] also proves *(ii)*

We now prove Theorem \[thm:IA2\].

(Theorem \[thm:IA2\]: **Intermediate Asymptotics II**) To complete the proof of Theorem \[thm:IA2\], we estimate the decay of the relative entropy . The proof of Theorem \[thm:IA\] used the estimate . Here we use the bound ${\|\theta(\tau)\|}_\infty \lesssim 1$ and to imply, if $\ga