Solution of the Navier-Stokes problem
The author claims a proof of global existence and uniqueness of solutions of the Navier-Stokes equations. Unfortunately, the proof proposed by the author is incorrect. It is based on a Gronwall-like inequality, viz. \[ \psi(t)\leq \psi(0)+c \int_0^t (t-s)^{-\frac 5 4}\, \psi(s)\, ds. \] As $t_+^{-\frac 5 4}$ is not integrable in a neighbourhood of $0$, one can only conclude (if $\psi$ does not vanish) that $\psi\leq +\infty$. In order to overcome the difficulty of $t_+^{-\frac 5 4}$ not being integrable, the author proposes to deal with the family of distributions \[ \Phi_\lambda= \frac{t_+^{\lambda-1}}{\Gamma(\lambda)}. \] However, for $\lambda <0$ (as for instance $\lambda=-\frac 1 4$) $\Phi_\lambda$ is not a distribution. As a matter of fact, the distributions $\Phi_\lambda$ are well-defined for $\Re \lambda >0$ and have a holomorphic dependence on $\lambda$; moreover, they satisfy \[ \frac{d\Phi_{\lambda+1}}{dx}=\Phi_\lambda. \] This allows one to extend the family by analytic continuation as a holomorphic family of distributions defined for all $\lambda\in\mathbb{C}$. But, in that case, the distribution $\Phi_{-\frac 1 4}$ is not equal to $\frac{t_+^{-\frac 5 4}}{\Gamma(-\frac 1 4)}$, but to its finite part: \[ \int_{\vert t\vert >\epsilon} \frac {t_+^{-\frac 5 4}}{\Gamma(-\frac 1 4)} \varphi(t)\, dt= - \frac{\epsilon^{-\frac 1 4}}{\Gamma(\frac 3 4)}\varphi(\epsilon) +\langle \Phi_{-\frac 1 4} \vert \varphi\rangle +o(1). \] All the formulas the author intends to apply are valid for the finite part of $\frac {t_+^{-\frac 5 4}}{\Gamma(-\frac 1 4)}$ (by analytic continuation of the case $\Re \lambda>0$), but are definitely false for this non-integrable function.
scientific article; zbMATH DE number 7026037
scientific article