Existence and uniqueness of a solution to the nonstationary Navier–Stokes equations having a prescribed flow rate (flux) in the infinite cylinder \(\Pi =\{x=(x^\prime , x_n)\in {{\mathbb {R}}}^n:\; x^\prime \in \sigma \subset {{\mathbb {R}}}^{n-1},\; -\infty<x_n<\infty ,\, n=2,3\}\) are proved. It is assumed that the flow rate \(F\in L^2(0, T)\) and the initial data \(\mathbf{u}_0=\big (0,\ldots ,0, u_{0n}\big )\in L^2(\sigma )\). The nonstationary Poiseuille solution has the form \(\mathbf{u}(x,t)=\big (0,\ldots ,0, U(x^\prime , t)\big ), \; p(x,t)=-q(t)x_n+p_0(t)\), where \((U(x',t), q(t))\) is a solution of an inverse problem for the heat equation with a specific over-determination condition.

在无穷圆柱体\（\ Pi = \ {x =（x ^ \ prime，x_n）\ in {{\ mathbb {R中，具有规定流量（通量）的非平稳Navier–Stokes方程的解的存在性和唯一性}}} ^ n：\; x ^ \ prime \ in \ sigma \ subset {{\ mathbb {R}}} ^ {n-1}，\;-\ infty <x_n <\ infty，\，n = 2 ，3 \} \）被证明。假设流量\（F \ in L ^ 2（0，T）\）和初始数据\（\ mathbf {u} _0 = \ big（0，\ ldots，0，u_ {0n} \大）\ in L ^ 2（\ sigma）\）。非平稳Poiseuille解的格式为\（\ mathbf {u}（x，t）= \ big（0，\ ldots，0，U（x ^ \ prime，t）\ big），\; p（x，t ）=-q（t）x_n + p_0（t）\），其中\（（U（x'，t），q（t））\）是热方程的反问题的解，具有特定的测定条件。