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

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.