On the Blow-up criterion of Navier-Stokes equation associated with the Weinstein operator

TL;DR

This paper investigates the Navier-Stokes equations associated with the Weinstein operator, establishing existence, uniqueness, and blow-up criteria for solutions in weighted Lebesgue spaces, and characterizing the growth near blow-up time.

Contribution

It introduces a new Navier-Stokes system linked to the Weinstein operator and provides novel blow-up criteria and growth estimates for solutions in weighted function spaces.

Findings

Existence and uniqueness of solutions in $L_{eta}^{p}$ spaces.

Blow-up growth rate characterized near maximal time.

Conditions under which solutions develop singularities.

Abstract

In this paper we give Navier-Stokes system associated with the Weinstein operator $(NSW)$ (see Eq.\eqref{11}), We study the existence and uniqueness of solutions to equations (NSW) in $L_{\alpha}^{p}\left(\mathbb{R}_{+}^{d+1}\right), 2 \alpha+d+2<p \leq \infty$, and we proved some properties of the maximal solution of equation. If the maximum time $T^*$is finite, we establish that the growth of $\left\| u ( {t}) \right\|_{L ^ {p}_{\alpha}} $ is at least of the order of ${\left(T^{*}-t\right)^{{-\frac{2 p}{p-2 \alpha-d-2}}}},$ fo rall $t$ in $\left[0, T^{*}\right]$, also we give some blow-up results.