1. Introduction

In this paper, we consider the energy equality for distributional solutions to the following fractional Navier-Stokes equations

(1.1)\begin{equation} \left\{\begin{array}{@{}ll} \partial_{t} u+u \cdot \nabla u+\mu(-\Delta)^{\alpha} u+\nabla p=0, & x \in \mathbb{R}^{3}, t>0 \\ \nabla \cdot u=0, & x \in \mathbb{R}^{3}, t>0 \\ \left.u \right|_{t=0}=u_{0}(x), & x \in \mathbb{R}^{3} \end{array}\right. \end{equation}

with $\mu >0$ is the kinematic viscosity, for simplicity, we set $\mu = 1$ in the sequel. Here $u=u(x, t)\in \mathbb {R}^3$ and $p=p(x, t)\in \mathbb {R}$ are non-dimensional quantities corresponding to the flow velocity and the total kinetic pressure at the point $(x, t)$, respectively. $u_{0}(x)$ is the initial velocity field satisfying that $\nabla \cdot u_{0}=0$. We denote the Fourier transform of the function $f$ by $\hat {f},$ then fractional Laplacian is defined by

\[ \widehat{\left(-\Delta\right)^{\alpha} f}(\xi)=|\xi|^{2 \alpha} \hat{f}(\xi), \]

more details on $(-\Delta )^{\alpha }$ can be found in [Reference Stein16], as a notation, we take $\Lambda$ as $(-\Delta )^{\frac {1}{2}}$.

The fractional Navier-Stokes equations (1.1) were first considered by Lions [Reference Lions10], they showed that if $\alpha \geq \frac {5}{4}$, then equations (1.1) has a unique global smooth solution for any smooth initial data (also see [Reference Wu19]). In [Reference Wu19], Wu showed that when $\alpha >0$, equations (1.1) with $u_{0} \in L^{2}$ possess a global weak solution and local in time strong solution for given initial value $u_{0} \in H^{1}$. Where, the weak solution means that $(u, p)$ satisfies (1.1) in the distribution sense and satisfies so-called energy inequality. Katz-Pavlovié [Reference Katz and Pavlović7] showed that if $1<\alpha <\frac {5}{4}$, the Hausdorff dimension of the singular set at the time of first possible blow-up is at most $(5-4\alpha )$. Recently, Tang-Yu [Reference Tang and Yu17] studied the partial regularity of fractional Navier-Stokes equations (1.1) with $\frac {3}{4} < \alpha < 1$, more precisely, they proved that the suitable weak solution is regular away from a relatively closed singular set whose $(5-4\alpha )$-dimensional Hausdorff measure is zero. Further partial regularity results of equations (1.1), we refer readers to [Reference Chen and Wei2, Reference Ren, Wang and Wu15]. It is well-know that in case $\alpha <\frac {5}{4}$, it remains unknown whether or not the solution will preserve sufficiently smooth initial regularity. On the other hand, for $\alpha \geq \frac {5}{4}$, it is easy to prove that the Leray-Hopf weak solution of the fractional Navier-Stokes equations is energy-conserved by a standard mollifying procedure, and a taking limits argument. From this point of view, there is a rather subtle relationship between energy equality and regularity for weak solutions. Therefore, it is natural to consider the energy conservation problem of the weak solution to the fractional Navier-Stokes equations and expect to have a better understanding for the regularity of the equations (1.1).

When $\alpha =1$, equations (1.1) reduce to the classical incompressible Navier-Stokes equations:

(1.2)\begin{equation} \left\{\begin{array}{@{}ll} \partial_{t} u+u \cdot \nabla u-\Delta u+\nabla p=0, & x \in \mathbb{R}^{3}, t>0 \\ \nabla \cdot u=0, & x \in \mathbb{R}^{3}, t>0 \\ \left.u \right|_{t=0}=u_{0}(x), & x \in \mathbb{R}^{3}. \end{array}\right. \end{equation}

It is well known since the work of Leray [Reference Leray9] and Hopf [Reference Hopf6], that for any $u_0\in L^2_\sigma (\mathbb {R}^3)$ one can construct a global weak solutions to (1.2), namely, a function $u$ that, for each $T>0$, is in the class

(1.3)\begin{equation} u\in L^\infty(0,T;L^2_\sigma(\mathbb {R}^3))\cap L^2(0,T;H^{1}(\mathbb {R}^3)) \end{equation}

and solves (1.1) in a distributional sense. Here, $L^2_\sigma (\mathbb {R}^3)$ is the subspace of $L^2(\mathbb {R}^3)$ of divergence-free vector functions. In addition, such a $u$ satisfies the so-called energy inequality:

(1.4)\begin{equation} \|u(t)\|^2_{L^2}+2\int_0^t\|\nabla u(\tau)\|^2_{L^2}\,{\rm d}\tau\le\|u_0\|^2_{L^2},\ \ \forall\, t\ge 0. \end{equation}

Much about the solutions of the Navier-Stokes equation is unknown, including uniqueness and regularity. The main barrier is the fact that the energy equality, which states that for any smooth solution $u$, it obeys the following basic energy law:

(1.5)\begin{equation} \|u(t)\|^2_{L^2}+2\int_0^t\|\nabla u(\tau)\|^2_{L^2}\,{\rm d}\tau=\|u_0\|^2_{L^2},\ \ \forall\, t\ge 0. \end{equation}

In the context of weak solutions even in the class $u \in L^2H^1,$ such a manipulation is not feasible due to lack of sufficient regularity to integrate by parts. This leaves room for additional mechanisms of energy dissipation due to the work of the nonlinear term. A natural question that remains open is whether energy equality, which should be expected from a physical point of view, is valid for weak solutions. Thus, an interesting question is: how badly behaved $u$ can keep the energy conservation. Lions [Reference Lions11] and Ladyzhenskaya [Reference Ladyzhenskaia, Solonnikov and Ural'tseva8] proved independently that such solutions satisfy the (global) energy equality (1.5) under the additional assumption $u \in L^{4} L^{4}$. Shinbrot [Reference Shinbrot18] generalized the Lions-Ladyzhenskaya condition to

(1.6)\begin{equation} u\in L^r(0,T;L^s(\mathbb{R}^d)) \ \text{with}\ \frac{2}{r}+\frac{2}{s} \leq 1, s \geq 4. \end{equation}

Yu in [Reference Yu20] given a new proof to the Shinbrot energy conservation criterion.

When considering distributional solutions (see definition 1.1) of 3D incompressible Navier-Stokes equations (1.2), in this case there is not any available regularity on velocity field $u$, apart the solution being in $L_{\mathrm {loc}}^{2}(\mathbb {R}^{3} \times [0, T))$. The interest for distributional solutions dates back to Foias [Reference Foias4], who proved their uniqueness under the solution in Serrin class (i.e., $u\in L^r(0,T;L^s(\Omega ))$ with $\frac {2}{r}+\frac {3}{s} = 1$, $s >3$). Later, Fabes, Jones and Riviere [Reference Fabes, Jones and Riviere3] proved the existence of distributional solutions for the Cauchy problem, while the case of the initial-boundary value problem has been studied mainly starting from the work of Amann [Reference Amann1]. Recently, The possible connection between distributional solutions and the energy equality has been considered by Galdi [Reference Galdi5], who proved that if distributional solution in $L^4 (0, T; L^4(\mathbb {R}^3))$, and with initial data $u_0$ in $L^2(\mathbb {R}^3)$, then energy equality (1.5) holds true. The key observation is the use of the duality argument and the above conditions to improve the regularity of the solution (i.e., $L^{\infty }(0, T; L^{2}(\mathbb {R}^3)) \cap L^{2}(0, T; H^{1}(\mathbb {R}^3))$).

Inspired by the above mentioned works on energy conservation of classical Navier-Stokes equations (1.2), the purpose of this note is to prove that, actually, for generalize and extend Galdi's energy conservation result of classical Navier-Stokes equations to fractional Navier-Stokes equations (1.1). More precisely, setting

\[ \mathcal{D}_T:=\{\varphi\in C_0^\infty(\mathbb{R}^3\times [0,T)):\ \operatorname{div}\varphi=0\}. \]

The main result of this paper is

Theorem 1.2 Suppose that $1\leq \alpha <\infty$ and $u\in L_{\mathrm {loc}}^{2}(\mathbb {R}^{3} \times [0, T))$ be a distributional solution in the sense of definition 1.1 to system (1.1). If

\[ u \in L^{\frac{4\alpha}{2\alpha-1}}\left(0, T ; L^{4}(\mathbb{R}^{3})\right), \]

then

\[ \int_{\mathbb{R}^{3}}|u(t, x)|^{2}\,{\rm d}x+2 \int_{0}^{t} \int_{\mathbb{R}^{3}}|\Lambda^\alpha u(x,\tau)|^{2}\,{\rm d}x\,{\rm d}\tau=\int_{\mathbb{R}^{3}}\left|u_{0}\right|^{2}\,{\rm d}x \]

for any $t \in [0, T]$.