Partially Regular Weak Solutions of the Navier–Stokes Equations in $\mathbb {R}^4 \times [0,\infty [$

Abstract

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.

Fractional Power of Nonnegative Smooth Function

In this appendix, we prove that certain fractional power of any nonnegative smooth function is Lipschitz continuous, which is a direct consequence of the following lemma proved by Fefferman and Phong [6].

Lemma 5.1

(Fefferman and Phong [6]; Lemma 4, Guan [12]) If $f:\mathbb {R}^n \rightarrow \mathbb {R}$ is a $C^{3,1}$ nonnegative function, with $\Vert f\Vert _{C^4} {\,\leqq \,}A$, then there is $N \in \mathbb {N}$ (only depends on n) and functions $g_1, g_2, \ldots , g_N \in C^{1,1}$, with $\Vert g_j\Vert _{C^2} {\,\leqq \,}C$, such that

$$\begin{aligned} f=\sum _{j=1}^{N} g_j^2, \end{aligned}$$

where the constant C depends on n and A.

Corollary 5.1

Suppose $f:\mathbb {R}^n \rightarrow \mathbb {R}$ is a $C^{3,1}$ nonnegative function, then $h:{=}f^{\alpha }$ is Lipschitz continuous for any $\alpha \in [\frac{1}{2},1]$.

Proof

This result follows from Lemma 5.1 and the bound

$$\begin{aligned} \begin{aligned} |h'|&= \frac{2\alpha \big |\sum _{j=1}^{N}g_jg'_j\big |}{\big (\sum _{i=1}^{N} g_i^2\big )^{1-\alpha }} \\&{\,\leqq \,}\sum _{j=1}^{N} \frac{2\alpha \big |g_j^{2\alpha -1}g'_j\big |}{\big (\sum _{i=1}^{N} (g_i/g_j)^2\big )^{1-\alpha }} \\&{\,\leqq \,}\sum _{j=1}^{N} 2\alpha \big |g_j^{2\alpha -1}g'_j\big |. \end{aligned} \end{aligned}$$

   $\square$