On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $ L^{p} $-space$ ^\dagger $

<abstract><p>We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $ L^{\frac{n}{2}}(\mathbb{R}^{n}) $ for the parabolic-elliptic Keller-Segel system, $ n\geq4 $. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-$ L^{\frac{n}{2}} $ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $ L^{q}((0, T);L^{p}) $-norms with $ q\neq\infty $), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $ C([0, T);L^{\frac{n}{2}}) $, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $ \gamma $ of the chemoattractant is also analyzed.</p></abstract>

Optimality of Serrin type extension criteria to the Navier-Stokes equations

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ L θ (0, T; U ˙ ∞ , 1 / θ , ∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$ ) for 2/θ + α = 1, 0 < α < 1 or u ∈ L 2(0, T; V ˙ ∞ , ∞ , 2 0 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$ ), where U ˙ p , β , σ s $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V ˙ p , q , θ s $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space B ˙ p , q s $\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$ . Our method is based on a bilinear estimate and a logarithmic interpolation inequality.