Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system

The Navier–Stokes system \(\begin{aligned} \left\{ \begin{array}{l} u_t + (u\cdot \nabla ) u =\Delta u+\nabla P + f(x,t), \\ \nabla \cdot u=0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t+(u·\mathrm{\nabla })u=\mathrm{\Delta }u+\mathrm{\nabla }P+f(x,t),\hfill \\ \mathrm{\nabla }·u=0,\hfill \end{array}\right)\end{array}$$ is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain \(\Omega \) $\mathrm{\Omega }$ . It is firstly, inter alia, observed that if \(T>0\) $T>0$ and \(\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot \ln ^\frac{1}{2} \big (|f(x,t)|+1\big ) dx \bigg \}^2 dt <\infty , \end{aligned}\) $$\begin{array}{c}\hfill {\int }_0^T\{{\int }_{\mathrm{\Omega }}|f(x,t)|·{ln}^{\frac{1}{2}}(|f(x,t)|+1)dx{\}}^{2}dt<\infty ,\end{array}$$ then for all divergence-free \(u_0\in L^2(\Omega ;{\mathbb {R}}^2)\) $u_0\in L^2(\mathrm{\Omega };{\mathbb{R}}^2)$ , a corresponding initial-boundary value problem admits a weak solution u with \(u|_{t=0}=u_0\) ${u|}_{t=0}=u_0$ . For any positive and nondecreasing \(L\in C^0([0,\infty ))\) $L\in C^0([0,\infty ))$ such that \(\begin{aligned} \frac{L(\xi )}{\ln ^\frac{1}{2} \xi } \rightarrow 0 \qquad \text{ as } \xi \rightarrow \infty , \end{aligned}\) $$\begin{array}{c}\hfill \frac{L(\xi )}{{ln}^{\frac{1}{2}}\xi }\to 0\text{as}\xi \to \infty ,\end{array}$$ this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed \(f:\Omega \times (0,T)\rightarrow {\mathbb {R}}^2\) $f:\mathrm{\Omega }\times (0,T)\to {\mathbb{R}}^2$ fulfilling \(\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot L\big (|f(x,t)|\big ) dx \bigg \}^2 dt < \infty . \end{aligned}\) $$\begin{array}{c}\hfill {\int }_0^T\{{\int }_{\mathrm{\Omega }}|f(x,t)|·L(|f(x,t)|)dx{\}}^{2}dt<\infty .\end{array}$$ This resolves a fine structure in the borderline case \(p=1\) $p=1$ and \(q=2\) $q=2$ appearing in results on existence of weak solutions for sources in \(L^q((0,T);L^p(\Omega ;{\mathbb {R}}^2))\) $L^q((0,T);L^p(\mathrm{\Omega };{\mathbb{R}}^2))$ when \(p\in (1,\infty ]\) $p\in (1,\infty ]$ and \(q\in [1,\infty ]\) $q\in [1,\infty ]$ satisfy \(\frac{1}{p}+\frac{1}{q}\le \frac{3}{2}\) $\frac{1}{p}+\frac{1}{q}\le \frac{3}{2}$ , and on nonexistence if here \(p\in [1,\infty )\) $p\in [1,\infty )$ and \(q\in [1,\infty )\) $q\in [1,\infty )$ are such that \(\frac{1}{p}+\frac{1}{q}>\frac{3}{2}\) $\frac{1}{p}+\frac{1}{q}>\frac{3}{2}$ .