Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

Sparse univariate polynomials with many roots over finite fields

Suppose q is a prime power and $f\in {\mathbb{F}}_q[x]$ is a univariate polynomial with exactly t monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2{(q-1)}^{\frac{t-2}{t-1}}$ on the number of cosets in ${\mathbb{F}}_q^{⁎}$ needed to cover the roots of f in ${\mathbb{F}}_q^{⁎}$. Here, we give explicit f with root structure approaching this bound: When q is a perfect $(t-1)$-st power we give an explicit t-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of ${\mathbb{F}}_q^{⁎}$. Over prime fields ${\mathbb{F}}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $\mathrm{\Omega }\left(\frac{\log p}{\log \log p}\right)$ distinct roots in ${\mathbb{F}}_p$.