Cauchy problem for time-space fractional incompressible Navier-Stokes equations in $$\mathbb {R}^n$$

Abstract

In this paper, Cauchy problem for incompressible Navier-Stokes equations with time fractional differential operator and fractional Laplacian in \(\mathbb {R}^n\) (\(n\ge 2\)) is investigated. The global and local existence and uniqueness of mild solutions are obtained with the help of Banach fixed point theorem when the initial data belongs to \(L^{p_{c}}(\mathbb {R}^n)\) \((p_c=\frac{n}{\alpha -1})\). In addition, the decay properties of mild solutions to the considered time-space fractional equations are constructed. Moreover, it is shown that when the initial data belongs to \(L^{p_{c}}(\mathbb {R}^n)\cap L^{p}(\mathbb {R}^n)\) with \(1<p<p_c\), the existence and uniqueness of global and local mild solutions can also be established. At the end of this paper, the integrability of mild solutions is discussed.