论文标题
$ \ mathbb {r}^4 \ times [0，\ infty [$
Partially regular weak solutions of the Navier-Stokes equations in $\mathbb{R}^4 \times [0,\infty[$
论文作者
论文摘要
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 $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular套装的本地有限$ 2 $维抛物线霍斯多夫度量。借助抛物线浓度 - 紧密度定理，我们能够通过使用缺陷度量来克服在空间上$ 4 $维的设置中可能缺乏紧凑性，然后将其纳入部分规则性理论中。
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 $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular sets have 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.
