The Cauchy problem of the Navier–Stokes equations in \(\mathbb {R}^n\) with the initial data a in the Besov space \({B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)\) for \(n<p<\infty \) and \(1 \le q \le \infty \) is considered. We construct the local solution in \(L^{\alpha ,q}(0,T;{B}^{0}_{{r},{1}}(\mathbb {R}^n))\) for \(p \le r< \infty \) satisfying \(\frac{2}{\alpha }+\frac{n}{r}=1\) with the initial data \(a \in {B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)\), where \(L^{\alpha ,q}\) denotes the Lorentz space. Conversely, if the solution belongs to \(L^{\alpha ,q}(0,T;L^{r}(\mathbb {R}^n))\) with \(\frac{2}{\alpha }+\frac{n}{r}=1\), then the initial data a necessarily belong to \({B}^{-1+\frac{n}{r}}_{{r},{q}}(\mathbb {R}^n)\). It implies that the initial data in the Besov space \(B^{-1+\frac{n}{p}}_{p,q} (\mathbb {R}^n)\) are a necessary and sufficient condition for the existence of solutions in the Serrin class.

\（\ mathbb {R} ^ n \）中带有初始数据a的Besov空间\（{B} ^ {-1+ \ frac {n} {p}} _中的Navier–Stokes方程的柯西问题对于（n <p <\ infty \）和\（1 \ le q \ le \ infty \）的{{p}，{q}}（\ mathbb {R} ^ n）\）被考虑。我们在\（L ^ {\ alpha，q}（0，T; {B} ^ {0} _ {{r}，{1}}（\ mathbb {R} ^ n））\）中构造局部解满足\（p \ le r <\ infty \）满足\（\ frac {2} {\ alpha} + \ frac {n} {r} = 1 \）和初始数据\（a \ in {B} ^ {-1+ \ frac {n} {p}} _ {{p}，{q}}（\ mathbb {R} ^ n）\），其中\（L ^ {\ alpha，q} \）表示洛伦兹空间。相反，如果解决方案属于\（L ^ {\ alpha，q}（0，T; L ^ {r}（\ mathbb {R} ^ n））\）与\（\ frac {2} {\ alpha} + \ frac {n} {r} = 1 \），则初始数据a必须属于\（{B} ^ {-1+ \ frac {n} {r}} _ {{r}，{q}}（\ mathbb {R } ^ n）\）。这意味着Besov空间\（B ^ {-1+ \ frac {n} {p}} _ {p，q}（\ mathbb {R} ^ n）\）中的初始数据是必要条件和充分条件Serrin类中解决方案的存在。