Convention: The letter $C$ with or without subscripts will denote an unimportant constant, whose value may change in different places. Moreover, $A\preceq B$ means that $A\leq CB$ for some constant $C>0$, and $A\asymp B$ means that $C^{-1}B\leq A\leq CB$ for some $C>1$.

After this work was finished, I was informed by Professor Enrico Priola during “The 8th International Conference on Stochastic Analysis and its Applications" held at BIT that, very recently, Fedrizzi, Flandoli, Priola and Vovelle [@Fe-Fl-Pr-Vo] also obtained the strong well-posedness together with their flow property of SDE under the conditions $\sigma_t(z)=\mI$ and $b_t(z)=b(z)$ possessing the following regularity $$\|(\mI-\Delta_x)^{s/2}b\|_p<\infty$$ for some $s>2/3$ and $p>6d$.

Preliminaries
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For $\alpha\geq 0$ and $p\in(1,\infty)$, let $H^\alpha_p:=H^\alpha_p(\mR^d):=(\mI-\Delta)^{-\frac{\alpha}{2}}(L^p(\mR^d))$ be the usual Bessel potential space with norm $$\|f\|_{\alpha,p}:=\|(\mI-\Delta)^{\frac{\alpha}{2}} f\|_p,$$ where $\|\cdot\|_p$ is the usual $L^p$-norm, and $\Delta$ is the Laplacian. For $\alpha\in(0,2)$, let $\Delta^{\frac{\alpha}{2}}:=-(-\Delta)^{\frac{\alpha}{2}}$ be the usual fractional Laplacian. Notice that up to a constant $C(\alpha,d)>0$, an alternative definition of $\Delta^{\frac{\alpha}{2}}$ is given by $$\begin{aligned}
\label{Def1}
\Delta^{\frac{\alpha}{2}} f(x):=\lim_{\eps\downarrow 0}\int_{|y|\geq\eps}\delta_yf(x)|y|^{-d-\alpha}\dif y,\ \ \delta_yf(x):=f(x+y)-f(x).\end{aligned}$$ We will frequently use such a definition below. It is well-known that by the boundedness of Riesz’s transformation (cf. [@St]), $$\begin{aligned}
\label{Riesz}
\|\Delta^{\frac{1}{2}}f\|_p\asymp\|\nabla f\|_p,\ p>1,\end{aligned}$$ and an equivalent norm in $H^\alpha_p$ is given 