We show that for any given solenoidal initial data in \(L^2\) and any solenoidal external force in \(L_{\text {loc}}^q \bigcap L^{3/2}\) with \(q>3\), there exist partially regular weak solutions of the Navier–Stokes equations in \(\mathbb {R}^4 \times [0,\infty [\) which satisfy certain local energy inequalities and whose singular sets have a locally finite 2-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to overcome the possible lack of compactness arising in the spatially 4-dimensional setting by using defect measures, which we then incorporate into the partial regularity theory.

我们证明对于\（L ^ 2 \）中的任何给定螺线管初始数据和\（L _ {\ text {loc}} ^ q \ bigcap L ^ {3/2} \）中的任何螺线管外力都带有\（q > 3 \），\（\ mathbb {R} ^ 4 \ times [0，\ infty [\）中存在满足某些局部能量不等式且其奇异集具有局部局部性的Navier–Stokes方程的部分正则弱解有限二维抛物型Hausdorff测度。借助抛物线浓度紧致定理，我们能够通过使用缺陷度量来克服在空间4维设置中可能出现的紧致性不足，然后将其纳入部分规则性理论中。