Nonlocal Navier–Stokes Equations: Existence and Asymptotic Behavior

Abstract

In this article, we are devoted to studying the following nonlocal Navier–Stokes equations involving the time–space fractional operators $$\begin{array}{r}\{\begin{array}{ll}{\displaystyle {\mathrm{\partial }}_t^{\beta }\mathbf{u}+(-\mathrm{\Delta }{)}^{\alpha }\mathbf{u}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}=\mathrm{\nabla }p+\lambda |\mathbf{u}{|}^{q-2}\mathbf{u}+\mathbf{g}(x,t)}& \text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},\\ \mathrm{div}\mathbf{u}=0& \text{in }\mathrm{\Omega }\times {\mathbb{R}}^{+},\\ \mathbf{u}(x,t)=0& \text{in }({\mathbb{R}}^N\setminus \mathrm{\Omega })\times {\mathbb{R}}^{+},\\ \mathbf{u}(x,0)={\mathbf{u}}_0(x)& \text{in }\mathrm{\Omega },\end{array}\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, $(-\mathrm{\Delta }{)}^{\alpha }$ is the fractional Laplacian with $0<\alpha <1$ , ${\mathrm{\partial }}_t^{\beta }$ is the Riemann–Liouville time fractional derivative with $0<\beta <1$ , $\lambda$ is a parameter, and $\mathbf{g}\in (L^{\mathrm{\infty }}(0,\mathrm{\infty };L^2(\mathrm{\Omega })){)}^N$ . Due to the nonlocal nature of the fractional Laplacian, the classical weak solution frameworks proposed by J. L. Lions and E. Hopf are inapplicable to this problem. To address this challenge, we first construct a regularized problem and establish the existence of its solutions via the Galerkin method combined with fractional calculus techniques. Subsequently, we prove the uniqueness of the global weak solution to the regularized problem. Furthermore, under appropriate assumptions, a decay estimate characterizing the long-time behavior of solutions is derived. Finally, the existence of solutions to the original problem is obtained by analyzing the limiting behavior of the regularized solutions as the regularization parameter vanishes. The main contributions of this work are twofold:

1. (1)

Our problem simultaneously incorporates the Riemann–Liouville time fractional derivative and the fractional Laplacian, extending classical Navier–Stokes theory to a nonlocal time–space framework.

(2)

We develop a new analytical approach to estimate the nonlinear convection term $\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}$ in the presence of fractional operators, overcoming challenges posed by the loss of localization.