On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows

Fabian Bleitner; Bartosz Protas

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arXiv · 2607.02739

On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows

Fabian Bleitner·Bartosz Protas

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Abstract

In this paper we consider solutions $\boldsymbol{u}$ of the three-dimensional Navier-Stokes system and investigate sharpness of the a priori bound \begin{align*} \frac{d}{dt}\|\boldsymbol{u}\|_q^q \leq C\|\boldsymbol{u}\|_q^{q\frac{q-1}{q-3}}, \qquad q > 3. \end{align*} This bound is closely related to the Ladyzhenskaya-Prodi-Serrin conditions characterizing classical solutions of the Navier-Stokes system. Velocity fields maximizing the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ under certain constraints are found as solutions of a suitable optimization problem which is solved numerically using a Riemannian conjugate gradient approach. The results obtained for different $q$ and increasing values of $\|\boldsymbol{u}\|_q$ indicate that the bound is indeed sharp, up to a numerical prefactor, and therefore cannot be fundamentally improved. Additionally, the results also suggest that the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ diverges as $q\to 3$.

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Fabian Bleitner, Bartosz Protas. 2026-07-02. On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows. https://arxiv.org/abs/2607.02739

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