**Lemma 1** Suppose that $1\leq p_1 \leq p \leq p_2\leq \infty$, and $p=p_1\theta + p_2(1-\theta)$, then we have $$ \norm{f}_{L^p}^p \leq \norm{f}_{L^{p_1}}^{p_1\theta} \norm{f}_{L^{p_2}}^{p_2(1-\theta)}.$$ //Proof:// Using the H$\ddot{\text{o}}$lder's inequality with $q=1/\theta,q'=1/(1-\theta)$, we obtain the result. **Lemma 2** Let $\Omega$ be a three-dimensional Lipschitz domain. If $f\in H^1(\Omega)$, then $$ \norm{f}_{L^\alpha} \lesssim \norm{f}_{L^2} + \norm{f}_{L^2}^{2\theta/\alpha} \norm{\nabla f}_{L^2}^{6(1-\theta)/\alpha},$$ where $2\leq \alpha \leq 6$ such that $\alpha = 2\theta + 6(1-\theta)$ for some $\theta$ between $0$ and $1$ (actually $\theta=(6-\alpha)/4$). //Proof:// This lemma is proved by the Sobolev embedding and interpolation (see **Lemma 1**), see also [PTZ08, pp. 57] for a proof in the case when $\alpha = 4$. **Theorem 1** Let $\Omega$ be a three-dimensional Lipschitz domain, and $T>0$. Then we have $$ L^2(0,T;H^1(\Omega))\cap L^\infty(0,T;L^2(\Omega)) \subset L^\beta(0,T; L^\alpha(\Omega)),$$ with the injection is continuous, where $2\leq \alpha \leq 6$ and $\beta=4\alpha/(3(\alpha-2))$. //Proof:// Using **Lemma 2**, and carefully calculating the index, we can obtain the result. === References [PTZ08] M. Petcu, R. Temam, and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of numerical analysis, vol. **14**, North-Holland, Amsterdam, 2009.