Decay of solutions to the three-dimensional generalized Navier

Abstract

In this paper, temporal decay estimates for weak solutions to the three dimensional generalized Navier–Stokes equations are established firstly. With these estimates at disposal, algebraic time decay for higher order Sobolev norms of small initial data solutions are obtained. The decay rates are optimal in the sense that they coincide with ones of the corresponding generalized heat equation.

Keywords

generalized Navier–Stokes equations decay Fourier-splitting method

1. Introduction

The incompressible Navier–Stokes equations can be written as $\{\begin{matrix} u_{t} + (u \cdot \nabla) u - ν Δ u = - \nabla p, \\ div u = 0, \\ u (x, 0) = u_{0} (x), \end{matrix}$ (1.1) where $x \in R^{n}$ , $n ⩾ 2$ , $t > 0$ , the vector field $u = u (x, t)$ denotes the velocity of the fluid, $p = p (x, t)$ is the pressure of the fluid and the positive ν is the viscosity coefficient.

Whether or not weak solutions of (1.1) decay to zero in $L^{2}$ as time tends to infinity was posed by Leray in his pioneering paper [11,12]. Kato [8] gave the first affirmative answer for strong solutions with small data to system (1.1). Algebraic decay rates for weak solutions to system (1.1) were first obtained by Schonbek [18], in which she proved that there exists a Leray–Hopf weak solution of (1.1) in three space dimension with arbitrary data in $L^{1} \cap L^{2}$ which satisfies ${∥ u (t) ∥}_{2} ⩽ C {(t + 1)}^{- \frac{1}{4}},$ where the constant C only depends on the $L^{1}$ and $L^{2}$ norms of the initial data. Later the method in [18] was extended by Schonbek [19], Kajikiya and Miyakawa [7] and Wiegner [25]. It was shown that the rate of decay for Leray–Hopf solutions of (1.1) in three space dimension with large data in $L^{p} \cap L^{2}$ , $1 ⩽ p < 2$ is the same as for solutions of the heat equation, that is ${∥ u (t) ∥}_{2} ⩽ C {(t + 1)}^{- \frac{3}{4} (\frac{2}{p} - 1)},$ where the constant C only depends on the $L^{p}$ and $L^{2}$ norms of the initial data. It is referred to [2,3,5,10,15,24] for more studies.

In this paper, we consider the large-time behavior of solutions to the Cauchy problem for the incompressible generalized Navier–Stokes equations in three space dimension $\{\begin{matrix} u_{t} + (u \cdot \nabla) u + ν Λ^{2 α} u = - \nabla p, \\ div u = 0, \\ u (x, 0) = u_{0} (x) . \end{matrix}$ (1.2) Here $Λ^{2 α}$ is defined through Fourier transform (see [21]) $\hat{Λ^{2 α} f} (ξ) = | ξ |^{2 α} \hat{f} (ξ), \hat{f} (ξ) = \int_{R^{n}} f (x) e^{- 2 π i x \cdot ξ} d x .$ It is known that if $(u (x, t), p (x, t))$ is a solution to the three-dimensional generalized Navier–Stokes equations, then for any $λ > 0$ , the scaling solution $(u_{λ} (x, t), p_{λ} (x, t)) = (λ^{2 α - 1} u (λ x, λ^{2 α} t), λ^{4 α - 2} p (λ x, λ^{2 α} t))$ also solves the generalized Navier–Stokes equations. The corresponding energy is $E (u_{λ}) = \frac{1}{2} sup_{t} \int_{R^{3}} | u_{λ} |^{2} d x + \int_{0}^{\infty} \int_{R^{3}} {| Λ^{α} u_{λ} |}^{2} d x d t = λ^{4 α - 5} E (u) .$ It follows that $E (u_{λ}) \to \infty$ as $λ \to 0$ when $α < \frac{5}{4}$ . In this sense, we say that the three-dimensional generalized Navier–Stokes equations is supercritical if $α < \frac{5}{4}$ , critical for $α = \frac{5}{4}$ and subcritical with $α > \frac{5}{4}$ . So far, it has been proved that when $α ⩾ \frac{5}{4}$ , the three-dimensional generalized Navier–Stokes equations has a global and unique regular solution (see [13] or [26]). However, when $α < \frac{5}{4}$ , the three-dimensional fractional Navier–Stokes system is still supercritical and their global well-posedness theories remain open. Meanwhile, the Navier–Stokes equations with fractional dissipation have been studied from a mathematical viewpoint and some interesting results have been obtained. Tao [22] showed the global regularity for a logarithmically supercritical hyper-dissipative Navier–Stokes system. In [9], it was shown that the Hausdorff dimension of the space singular set at time of first blow up for smooth solution to the three-dimensional fractional Navier–Stokes equations is at most $5 - 4 α$ as $1 < α < \frac{5}{4}$ .

In this paper, we will focus on the asymptotic behavior of solution of (1.2) in the supercritical case, that is, when $α < \frac{5}{4}$ . Motivated by [18], [19] and [20], we will show that the weak solutions to (1.2) which are to be constructed in the end of the paper, subject to large initial data, decay in $L^{2}$ at a uniform algebraic rate. The decay estimates for the higher order Sobolev norms whose global in time existence is guaranteed for sufficiently small initial data [27] will also be established. Our main results can be stated as follows.

Theorem 1.1.

Assume $0 < α < \frac{5}{4}$ . Then for divergence-free vector-field $u_{0} \in L^{2} (R^{3}) \cap L^{p} (R^{3})$ with $max (1, \frac{1}{3 - 2 α}) ⩽ p < 2$ , the Cauchy problem (1.2) admits a weak solution such that ${∥ u (t) ∥}_{2}^{2} ⩽ C {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)},$ (1.3) where the constant C depends on α, the $L^{p}$ and $L^{2}$ norms of the initial data.

Remark 1.1.

Here and in the following the $L^{p}$ -norm of a function f is denoted by ${∥ f ∥}_{p}$ and the $H^{s}$ - norm by ${∥ f ∥}_{H^{s}}$ . We will also set $ν = 1$ for simplicity.

Remark 1.2.

The rate of decay in (1.3) is the same as that of the solutions to the generalized heat equation $v_{t} + ν Λ^{2 α} v = 0$ with the same initial data $u_{0}$ (see Lemma 3.1 in [17]).

Theorem 1.2.

Assume $0 < α < \frac{5}{4}$ and divergence-free vector-field $u_{0} \in L^{2} (R^{3}) \cap L^{p} (R^{3})$ with p satisfying one of the conditions: $\begin{array}{rcl} (1) 0 < α ⩽ \frac{1}{2}, 1 ⩽ p ⩽ \frac{6}{4 α + 3}; \\ (2) \frac{1}{2} < α ⩽ 1, 1 ⩽ p ⩽ \frac{6}{4 α + 1}; \\ (3) 1 < α < \frac{5}{4}, \frac{1}{3 - 2 α} ⩽ p < 2 . \end{array}$ Then, for $m \in N$ (the set of positive integers) there exists a $T_{0} > 0$ such that the small global-in-time solution satisfies ${∥ D^{m} u (t) ∥}_{2}^{2} ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1) - \frac{m}{α}}$ for all $t > T_{0}$ , where the constant C depends on m, α and ${∥ u}_{0} ∥_{L^{2} \cap L^{p}}$ .

Remark 1.3.

In Theorem 1.2, the restrictions on p are based on Theorem 1.1 which requires that the lower bound of p is $max (1, \frac{1}{3 - 2 α})$ . However, in Theorem 1.2, for technical reason, we have a little bit more restrictions on the upper bound of p. Precisely, in the case of $0 < α ⩽ \frac{1}{2}$ , p is chosen to satisfy $1 ⩽ p ⩽ \frac{6}{4 α + 3}$ . The point in the proof is to show ${(t + 1)}^{- \frac{3}{4 α} (\frac{2}{p} - 1)} ⩽ {(t + 1)}^{- 1}$ in (3.14). In the case of $\frac{1}{2} < α ⩽ 1$ , p is chosen to satisfy $1 ⩽ p ⩽ \frac{6}{4 α + 1}$ . The point in the proof is to show ${(t + 1)}^{- \frac{3}{4 α - 2} (\frac{2}{p} - 1)} ⩽ {(t + 1)}^{- 1}$ in (3.11). While in the case $1 < α < \frac{5}{4}$ , the restriction on p is the same as in Theorem 1.1.

Remark 1.4.

Similar results have been established respectively in [4] for the classical Navier–Stokes equations and in [20] for the Hall-magnetohydrodynamic equations.

Our results extend the corresponding ones in [18,19] and [20] to the supercritical cases and the Fourier splitting method due to M. Schonbek with appropriate modification will be applied. In comparison with [18–20], new ingredients in our proof are as follows. Firstly, in the proof of Theorems 1.1 and 1.2, we choose $g (t) = {(\frac{γ}{t + 1})}^{\frac{1}{2 α}}$ as a test function with γ being a constant to be determined. In fact, γ has to be adjusted according to the change of α in order to get the optimal decay (see (2.13)). This is different from the one in [18,19] in which $α = 1$ and $γ = n$ (the dimension of the space) are fixed. Secondly, in the proof of Theorem 1.2, we firstly establish decay estimates for s-order Sobolev norms in Proposition 3.3 ${∥ u (t) ∥}_{H^{s}}^{2} ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)}, s > \frac{5}{2} - α, 0 < α < \frac{5}{4} .$ However, the decay rates are not optimal. Then, we establish a unified inequality for $m = 1, 2, \dots$ $\frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} ⩽ {C (t + 1)}^{- 1} {∥ Λ^{m} u ∥}_{2}^{2},$ (1.4) which is different from Lemma 3.2. of [19], in which $\frac{d}{d t} {∥ D^{m} u ∥}_{2}^{2} + {∥ D^{m + 1} u ∥}_{2}^{2} ⩽ {C (t + 1)}^{- 1} {∥ D^{m} u ∥}_{2}^{2} + R_{m}, m = 1, 2, \dots,$ with $R_{m} = \{\begin{matrix} 0 & for m = 1, 2, \\ \sum_{1 ⩽ j ⩽ \frac{m}{2}} {∥ D^{j} u ∥}_{\infty}^{2} ∥ {D^{m - j} u ∥}_{2}^{2} & for m ⩾ 3 . \end{matrix}$ Thirdly, based on (1.4), we will use an iterative scheme. By induction for m, we get the optimal decay estimates for high order Sobolev norms.

The paper unfolds as follows: Section 2 is devoted to the proof of Theorem 1.1 whereas Section 3 deals with the proof of Theorem 1.2. The existence of weak solutions is postponed in the final section of the paper.

2. Proof of Theorem 1.1

In this section, we will prove Theorem 1.1. The following are two key lemmas.

Lemma 2.1.
Let u be a smooth solution to the Cauchy problem (1.2) with initial data $u_{0} \in L^{p} (R^{3}) \cap L^{2} (R^{3})$ , $1 ⩽ p < 2$ . Then there exists a constant $C > 0$ depending only on ${∥ u}_{0} ∥_{2}$ such that $| \hat{u} (ξ, t) | ⩽ C (| \hat{u_{0}} (ξ) | + \frac{1}{| ξ |^{2 α - 1}}) .$ (2.1)
Proof.
Taking the Fourier transform of the first equation of (1.2) yields ${\hat{u}}_{t} (ξ, t) + | ξ |^{2 α} \hat{u} (ξ, t) = H (ξ, t),$ (2.2) where $H (ξ, t) = - \hat{u \cdot \nabla u} (ξ, t) - \hat{\nabla p} (ξ, t) .$ Multiplying (2.2) by the integrating factor $e^{| ξ |^{2 α} t}$ gives $\frac{d}{d t} (e^{| ξ |^{2 α} t} \hat{u} (ξ, t)) = e^{| ξ |^{2 α} t} H (ξ, t) .$ Integrating in time from 0 to t, we obtain $\hat{u} (ξ, t) = e^{- | ξ |^{2 α} t} \hat{u_{0}} (ξ) + \int_{0}^{t} e^{- | ξ |^{2 α} (t - s)} H (ξ, s) d s .$ (2.3) Hence $| \hat{u} (ξ, t) | ⩽ | \hat{u_{0}} (ξ) | + \int_{0}^{t} e^{- | ξ |^{2 α} (t - s)} | H (ξ, s) | d s .$ (2.4)

To complete the proof we need to establish an estimate for $H (ξ, s)$ . Taking the divergence operator on the first equation of (1.2) yields $- Δ p = \sum_{i, j = 1}^{3} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} (u^{i} u^{j}) .$ Note that the Fourier transform is a bounded map from $L^{1}$ into $L^{\infty}$ . It follows that $\begin{array}{rcl} | \hat{\nabla p} (ξ, t) | & ⩽ & | ξ | | \hat{p} (ξ, t) | \\ ⩽ & \sum_{i, j = 1}^{3} \frac{| ξ_{i} ξ_{j} |}{| ξ |} | \hat{u^{i} u^{j}} (ξ, t) | \\ ⩽ & C | ξ | {∥ u (t) u (t) ∥}_{1} \\ ⩽ & C | ξ | {∥ u (t) ∥}_{2}^{2} . \end{array}$ Similarly, for the convective term, we have by using the divergence free condition $\begin{array}{rcl} | \hat{u \cdot \nabla u} (ξ, t) | & ⩽ & \sum_{i}^{3} | ξ | | \hat{u^{i} u} (ξ, t) | \\ ⩽ & C | ξ | {∥ u (t) u (t) ∥}_{1} \\ ⩽ & C | ξ | {∥ u (t) ∥}_{2}^{2} . \end{array}$ Combining the above two estimates, we have $| H (ξ, t) | ⩽ C | ξ | {∥ u (t) ∥}_{2}^{2} .$ (2.5) By inserting (2.5) into (2.4) and using the boundedness of the $L^{2}$ norm of the solution, we obtain $\begin{array}{rcl} | \hat{u} (ξ, t) | & ⩽ & | \hat{u_{0}} (ξ) | + \frac{C}{| ξ |^{2 α - 1}} {∥ u_{0} ∥}_{2}^{2} (1 - e^{- | ξ |^{2 α} t}) \\ ⩽ & C (| \hat{u_{0}} (ξ) | + \frac{1}{| ξ |^{2 α - 1}}) . \end{array}$ □
Lemma 2.2.
Let $u_{0} \in L^{p} (R^{3})$ with $1 ⩽ p ⩽ 2$ , then $\int_{S (t)} {| \hat{u_{0}} (ξ) |}^{2} d ξ ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)},$ (2.6) where $S (t) = {ξ \in R^{3} : | ξ | ⩽ g (t)}, g (t) = {(\frac{γ}{t + 1})}^{\frac{1}{2 α}}$ (2.7) and the constant C depends on γ, α and the $L^{p}$ norm of $u_{0}$ .
Proof.
Let $F$ be the Fourier transform. By Riesz theorem, if $1 ⩽ p ⩽ 2$ , we have $F : L^{p} \to L^{q}$ and ${{∥ F u}_{0} ∥}_{q} ⩽ {C {∥ u}_{0} ∥}_{p},$ (2.8) where $\frac{1}{p} + \frac{1}{q} = 1$ . Therefore, we get by the Hölder inequality $\int_{S (t)} | \hat{u_{0}} |^{2} d ξ ⩽ {(\int_{S (t)} | \hat{u_{0}} |^{q} d ξ)}^{\frac{2}{q}} {{(\int}_{S (t)} d ξ)}^{1 - \frac{2}{q}} .$ (2.9) Combining (2.8) and (2.9) yields $\int_{S (t)} | \hat{u_{0}} |^{2} d ξ ⩽ C {(\int_{S (t)} d ξ)}^{1 - \frac{2}{q}} .$ Since $\frac{1}{p} + \frac{1}{q} = 1$ and the volume of $S (t)$ is $| S (t) | = C g^{3} (t)$ , we thus get $\int_{S (t)} {| \hat{u_{0}} (ξ) |}^{2} d ξ ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ □

In the rest of this section, we first present a formal argument by the Fourier-splitting method (see [18]). Proof of Theorem 1.1.
By taking $L^{2}$ -inner product on bothsides of the first equation of (1.2) with u, we get $\frac{d}{d t} {∥ u (t) ∥}_{2}^{2} = - 2 {∥ Λ^{α} u (t) ∥}_{2}^{2} .$ Applying the Plancherel theorem to the above equation yields $\frac{d}{d t} \int_{R^{3}} {| \hat{u} (ξ) |}^{2} d ξ = - 2 \int_{R^{3}} {| ξ |}^{2 α} {| \hat{u} (ξ) |}^{2} d ξ .$ Let $S (t) = {ξ \in R^{3} : | ξ | ⩽ g (t)}, g (t) = {(\frac{γ}{t + 1})}^{\frac{1}{2 α}},$ (2.10) where γ is a constant to be determined. Then $\begin{array}{rclr} \frac{d}{d t} \int_{R^{3}} {| \hat{u} (ξ) |}^{2} d ξ & ⩽ & - g^{2 α} (t) \int_{| ξ | ⩾ g (t)} {| \hat{u} (ξ) |}^{2} d ξ - \int_{| ξ | ⩽ g (t)} {| ξ |^{2 α} | \hat{u} (ξ) |}^{2} d ξ \\ ⩽ & - g^{2 α} (t) \int_{R^{3}} {| \hat{u} (ξ) |}^{2} d ξ + g^{2 α} (t) \int_{| ξ | ⩽ g (t)} {| \hat{u} (ξ) |}^{2} d ξ . & (2.11) \end{array}$ Multiplying (2.11) by the integrating factor $G (t) = e^{\int_{0}^{t} g^{2 α} (τ) d τ}$ yields $\frac{d}{d t} (G (t) {∥ u (t) ∥}_{2}^{2}) ⩽ g^{2 α} (t) G (t) \int_{| ξ | ⩽ g (t)} {| \hat{u} (ξ) |}^{2} d ξ .$ Using (2.10), we get $G (t) = {(t + 1)}^{γ}$ . Hence $\frac{d}{d t} ({(t + 1)}^{γ} {{∥ u (t) ∥}_{2}^{2}) ⩽ γ (t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} {| \hat{u} (ξ) |}^{2} d ξ .$ (2.12)

To complete the proof we will use Lemmas 2.1 and 2.2 to estimate the right-hand side of (2.12). Indeed, by plugging (2.1) into the right-hand side of (2.12) and using (2.6) we have $\begin{array}{rcl} \frac{d}{d t} ({(t + 1)}^{γ} {∥ u (t) ∥}_{2}^{2}) & ⩽ & C {(t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} {| \hat{u_{0}} (ξ) |}^{2} d ξ + C {(t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} \frac{1}{| ξ |^{2 (2 α - 1)}} d ξ \\ ⩽ & C {(t + 1)}^{γ - 1 - \frac{3}{2 α} (\frac{2}{p} - 1)} + C {(t + 1)}^{γ - 1 - \frac{5 - 4 α}{2 α}} . \end{array}$ Integrating the last inequality in time from 0 to t yields ${{∥ u (t) ∥}_{2}^{2} ⩽ C ((t + 1)}^{- γ} + {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} + {(t + 1)}^{- \frac{5 - 4 α}{2 α}}) .$ (2.13)

Case 1: Assume $0 < α ⩽ \frac{1}{2}$ . Since $p ⩾ 1 ⩾ \frac{3}{4 - 2 α}$ , we have $\frac{3}{2 α} (\frac{2}{p} - 1) ⩽ \frac{5 - 4 α}{2 α}$ . Hence, by choosing $γ = \frac{5 - 2 α}{2 α}$ , we have ${∥ u (t) ∥}_{2}^{2} ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$

Case 2: Assume $\frac{1}{2} < α ⩽ 1$ . If $1 ⩽ p < \frac{3}{4 - 2 α}$ , one has $\frac{3}{2 α} (\frac{2}{p} - 1) > \frac{5 - 4 α}{2 α}$ . Hence, by choosing $γ = 3$ we have ${∥ u (t) ∥}_{2}^{2} ⩽ {C (t + 1)}^{- \frac{5 - 4 α}{2 α}} .$ (2.14) If $\frac{3}{4 - 2 α} ⩽ p < 2$ , one has $\frac{3}{2 α} (\frac{2}{p} - 1) ⩽ \frac{5 - 4 α}{2 α}$ . Hence, by choosing $γ = \frac{3}{2 α}$ we have ${∥ u (t) ∥}_{2}^{2} ⩽ {C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ (2.15)

Now we will improve the rate in (2.14). To this end, motivated by [19], we will use the preliminary decay rate of (2.14) to show that $| \hat{u} (ξ, t) | ⩽ | \hat{u_{0}} (ξ) | + C for ξ \in S (t),$ and then a bootstrap argument will lead to the higher decay rate $\frac{3}{2 α} (\frac{2}{p} - 1)$ in (2.14). Replacing H by its upper bound $C | ξ | {∥ u (t) ∥}_{2}^{2}$ in (2.5) and using the decay rate in (2.14), on one hand, for $ξ \in S (t)$ , for $\frac{1}{2} < α ⩽ 1$ and $α \neq \frac{5}{6}$ we have $\begin{array}{rclr} \int_{0}^{t} e^{- | ξ |^{2 α} (t - s)} | H (ξ, s) | d s & ⩽ & C | ξ | \int_{0}^{t} {(s + 1)}^{- \frac{5 - 4 α}{2 α}} d s \\ ⩽ & C \frac{2 α}{6 α - 5} | ξ | ({(t + 1)}^{\frac{6 α - 5}{2 α}} - 1) \\ ⩽ & C \frac{2 α}{6 α - 5} {(t + 1)}^{- \frac{1}{2 α}} {((t + 1)}^{\frac{6 α - 5}{2 α}} - 1) \\ ⩽ & C . & (2.16) \end{array}$ On the other hand, for $α = \frac{5}{6}$ we have $\begin{array}{rclr} \int_{0}^{t} e^{- | ξ |^{2 α} (t - s)} | H (ξ, s) | d s & ⩽ & C | ξ | \int_{0}^{t} {(s + 1)}^{- 1} d s \\ ⩽ & C {(t + 1)}^{- \frac{3}{5}} ln (t + 1) \\ ⩽ & C . & (2.17) \end{array}$ Hence by (2.3), (2.16) and (2.17) $| \hat{u} (ξ, t) | ⩽ | \hat{u_{0}} (ξ) | + C for ξ \in S (t) .$ This combined with (2.12) yields $\begin{array}{rcl} \frac{d}{d t} ({(t + 1)}^{γ} {∥ u (t) ∥}_{2}^{2}) & ⩽ & C {(t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} {| \hat{u_{0}} (ξ) |}^{2} d ξ + {C (t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} d ξ \\ ⩽ & C {(t + 1)}^{γ - 1 - \frac{3}{2 α} (\frac{2}{p} - 1)} + {C (t + 1)}^{γ - 1 - \frac{3}{2 α}} . \end{array}$ Integrating the last inequality in time yields $\begin{array}{rcl} {∥ u (t) ∥}_{2}^{2} & ⩽ & C ({{(t + 1)}^{- γ} + (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} + {(t + 1)}^{- \frac{3}{2 α}}) \\ ⩽ & C ({{(t + 1)}^{- γ} + (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)}) . \end{array}$ By choosing γ suitably large, we have ${{∥ u (t) ∥}_{2}^{2} ⩽ C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$

Case 3: Assume $1 ⩽ α < \frac{5}{4}$ and $1 ⩽ \frac{1}{3 - 2 α} ⩽ p < \frac{3}{4 - 2 α}$ . Similar to the proof of (2.12), (2.13) and (2.14), we have ${{∥ u (t) ∥}_{2}^{2} ⩽ C (t + 1)}^{- \frac{5 - 4 α}{2 α}} .$ (2.18) By using (2.18), we have $\begin{array}{rclr} \int_{0}^{t} e^{- | ξ |^{2 α} (t - s)} | H (ξ, s) | d s & ⩽ & C | ξ | \int_{0}^{t} {(s + 1)}^{- \frac{5 - 4 α}{2 α}} d s \\ ⩽ & C \frac{2 α}{6 α - 5} | ξ | ({(t + 1)}^{\frac{6 α - 5}{2 α}} - 1) \\ ⩽ & C \frac{2 α}{6 α - 5} {(t + 1)}^{- \frac{1}{2 α}} {((t + 1)}^{\frac{6 α - 5}{2 α}} - 1) \\ ⩽ & C {(t + 1)}^{\frac{3 α - 3}{α}} . & (2.19) \end{array}$ Thanks to (2.3), we have $| \hat{u} (ξ, t) | ⩽ C (| \hat{u_{0}} (ξ) | + {(t + 1)}^{\frac{3 α - 3}{α}})$ . By plugging the last inequality into (2.12) again, we obtain $\frac{d}{d t} ({(t + 1)}^{γ} {∥ u (t) ∥}_{2}^{2}) ⩽ {C (t + 1)}^{γ - 1 - \frac{3}{2 α} (\frac{2}{p} - 1)} + C {(t + 1)}^{γ - 1 - \frac{3}{2 α} + \frac{6 α - 6}{α}} .$ Integrating the last inequality in time and choosing γ suitably yield ${{∥ u (t) ∥}_{2}^{2} ⩽ C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ Similar to the proof of (2.12), (2.13) and (2.15), we know that in the case of $1 ⩽ α < \frac{5}{4}$ and $\frac{3}{4 - 2 α} ⩽ p < 2$ ${{∥ u (t) ∥}_{2}^{2} ⩽ C (t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ Therefore, combining the three cases yields the formal proof of Theorem 1.1.

We apply the a prior estimates to the approximate solutions constructed in the Appendix part to make the above proof rigorous. Let us recall that $u_{N}$ is a solution of the approximate equation $\{\begin{matrix} \partial_{t} u_{N} + P J_{N} (u_{N} \cdot \nabla u_{N}) + Λ^{2 α} u_{N} = 0, \\ div u_{N} = 0, \\ u_{N} (x, 0) = J_{N} u_{0}, \end{matrix}$ where $J_{N}$ is the spectral cutoff defined by $\hat{J_{N} f} (ξ) = 1_{[0, N]} (| ξ |) \hat{f} (ξ)$ and P is the Leray projector over divergence-free vector-fields.

It is shown that the $u_{N}$ converge strongly in $L^{2} (0, T; L_{loc}^{2} (R^{3}))$ to a weak solution of the generalized three-dimensional Navier–Stokes equation (1.2) in the Appendix part. Hence the $L^{2}$ decay of $u_{N}$ will imply the $L^{2}$ decay of the weak solution of (1.2). □

3. Proof of Theorem 1.2

In this section, we start with the following small data global regularity result (Theorem 4.3 in [27]).

Theorem 3.1.

If $u_{0} \in B_{2, q}^{r} (R^{d})$ with $1 < q < \infty, r > 1, r > 1 + \frac{d}{2} - \frac{2 α}{q}$ and ${{∥ u}_{0} ∥}_{B_{2, q}^{r} (R^{d})} < C_{3} ν,$ for some suitable constant $C_{3}$ , then the IVP (1.2) has a global and unique solution $(u, p)$ satisfying $\begin{array}{rcl} u \in L^{\infty} ([0, \infty); B_{2, q}^{r}) \cap L^{q} ([0, \infty); B_{2, q}^{r + \frac{2 α}{q}}) \cap C ([0, \infty); B_{2, \infty}^{r - 1}), \\ p \in L^{\infty} ([0, \infty); B_{2, q}^{r}) \end{array}$ and ${∥ u (\cdot, t) ∥}_{B_{2, q}^{r} (R^{d})} < C_{3} ν for any t > 0 .$

Here $B_{p, q}^{s}$ with $1 ⩽ p$ , $q ⩽ \infty$ and $s ⩾ 0$ is a Besov space, which can be seen in [16] for more details.

Remark 3.1.

By the fact that $H^{s}$ coincides with $B_{2, 2}^{s}$ , it follows from Theorem 3.1 that the Cauchy problem (1.2) has a global and unique solution in the usual Sobolev space $H^{s}$ with $s > \frac{5}{2} - α$ if we choose $r = s, q = 2, d = 3 .$ We can also obtain that $\frac{d}{d t} {∥ u ∥}_{H^{s}}^{2} ⩽ - C {∥ u ∥}_{H^{s + α}}^{2} .$ (3.1)

In fact, by Theorem 4.3 in [27], one has $\frac{d}{d t} {∥ u ∥}_{B_{2, q}^{r}}^{q} ⩽ - c (C_{3} ν - {∥ u ∥}_{B_{2, q}^{r}}) {∥ u ∥}_{B_{2, q}^{r + \frac{2 α}{q}}}^{q} .$ (3.2) By the hypothesis on initial condition ${{∥ u}_{0} ∥}_{B_{2, q}^{r}} < C_{3} ν$ , we can suppose that $T_{*} < \infty$ is the first time such that ${∥ u (T_{*}) ∥}_{B_{2, q}^{r}} = C_{3} ν .$ Then for any $0 ⩽ t ⩽ T_{*}$ , we have ${∥ u (t) ∥}_{B_{2, q}^{r}} ⩽ C_{3} ν .$ This combined with (3.2) implies that ${∥ u (t) ∥}_{B_{2, q}^{r}}^{q}$ is a non-increasing function of t for $0 ⩽ t ⩽ T_{*}$ . Then $C_{3} ν - {∥ u (T_{*}) ∥}_{B_{2, q}^{r}} ⩾ C_{3} ν - {∥ u_{0} ∥}_{B_{2, q}^{r}} > 0$ which contradicts with ${∥ u (T_{*}) ∥}_{B_{2, q}^{r}} = C_{3} ν$ . Therefore, ${∥ u (t) ∥}_{B_{2, q}^{r}} < C_{3} ν for any t > 0 .$ Using (3.2) again yields $\frac{d}{d t} {∥ u ∥}_{B_{2, q}^{r}}^{q} ⩽ - C {∥ u ∥}_{B_{2, q}^{r + \frac{2 α}{q}}}^{q}$ for some $C > 0$ . In particular, choosing $q = 2$ , $r = s$ , we can get (3.1).

Lemma 3.2.

Let $u_{0} \in L^{p} (R^{3}) \cap L^{2} (R^{3})$ , $1 ⩽ p < 2$ with $div u_{0} = 0$ . Then, for any $| ξ | ⩽ 1$ and for any $j ⩾ 0$ we have $| \hat{Λ^{j} u} (ξ, t) | ⩽ C (| \hat{u_{0}} (ξ) | + \frac{1}{| ξ |^{2 α - 1}}),$ (3.3) where C depends only on $∥ u_{0} ∥_{L^{p} \cap L^{2}}$ .

Proof.

Since $| ξ | ⩽ 1$ , we have $\begin{array}{rcl} | \hat{Λ^{j} u} (ξ, t) | & ⩽ & | ξ |^{j} | \hat{u} (ξ, t) | \\ ⩽ & | \hat{u} (ξ, t) | . \end{array}$ The result of Lemma 2.1 leads to the desired (3.3). □

The following are decay estimates for high order Sobolev norms.

Proposition 3.3.

Assume $0 < α < \frac{5}{4}$ . Let $s > \frac{5}{2} - α$ , $max (1, \frac{1}{3 - 2 α}) ⩽ p < 2$ and $u_{0} \in L^{p} (R^{3}) \cap H^{s} (R^{3})$ with $div u_{0} = 0$ . Then, there exists a $T_{0} > 0$ such that for any $t > T_{0}$ the global-in-time solution established in Theorem 3.1 satisfying ${∥ u (t) ∥}_{H^{s}}^{2} ⩽ C {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)},$ (3.4) where C depends on α and $∥ u_{0} ∥_{H^{s} \cap L^{p}}$ .

Proof.

We adopt to the Fourier-splitting method again. The Fourier transform of (3.1) can be written as $\frac{d}{d t} \int_{R^{3}} {| \hat{u} (ξ, t) |}^{2} + {| \hat{Λ^{s} u} (ξ, t) |}^{2} d ξ ⩽ - \int_{R^{3}} | ξ |^{2 α} ({| \hat{u} (ξ, t) |}^{2} + {| \hat{Λ^{s} u} (ξ, t) |}^{2}) d ξ .$ In a similar fashion as the proof of Theorem 1.1, we have $\begin{array}{rcl} \frac{d}{d t} ({(t + 1)}^{γ} \int_{R^{3}} {| \hat{u} (ξ, t) |}^{2} + {| \hat{Λ^{s} u} (ξ, t) |}^{2} d ξ) \\ ⩽ γ {(t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} {| \hat{u} (ξ, t) |}^{2} + {| \hat{Λ^{s} u} (ξ, t) |}^{2} d ξ . \end{array}$ As a consequence, by using the result of Lemma 3.2, there exists a $T_{0} > 0$ such that for any $t > T_{0}$ , we have $\frac{d}{d t} ({(t + 1)}^{γ} \int_{R^{3}} {| \hat{u} (ξ, t) |}^{2} + {| \hat{Λ^{s} u} (ξ, t) |}^{2} d ξ) ⩽ γ {(t + 1)}^{γ - 1} \int_{| ξ | ⩽ g (t)} {| \hat{u} (ξ, t) |}^{2} d ξ .$ Arguing as for proving (1.3), we get that for any $t > T_{0}$ ${∥ u (t) ∥}_{H^{s}}^{2} ⩽ C {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ Thus, the proof is finished. □

To prove Theorem 1.2, we need the following product and commutator estimates.

Lemma 3.4.

Let $s > 0$ and $1 < p < \infty$ . Then $\begin{array}{rclr} {∥ Λ^{s} (f g) ∥}_{p} ⩽ C ({∥ Λ^{s} f ∥}_{p_{1}} {∥ g ∥}_{p_{2}} + {∥ Λ^{s} g ∥}_{q_{1}} {∥ f ∥}_{q_{2}}), & (3.5) \\ {∥ Λ^{s} (f g) - f Λ^{s} g ∥}_{p} ⩽ C ({∥ Λ^{s} f ∥}_{p_{1}} {∥ g ∥}_{p_{2}} + {∥ \nabla f ∥}_{q_{2}} {∥ Λ^{s - 1} g ∥}_{q_{1}}), & (3.6) \end{array}$ where $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{q_{1}} + \frac{1}{q_{2}}$ and $p_{1}, q_{1} \in (1, \infty)$ , $p_{2}, q_{2} \in [1, \infty]$ .

The proof is referred to Lemma 3.1 in [6] and the details are omitted here.

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2.

For any $m \in N$ , applying $Λ^{m}$ on both sides of the first equation of (1.2), multiplying the resulting equation by $Λ^{m} u$ and integrating by part, we obtain $\frac{1}{2} \frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} = - \int_{R^{3}} Λ^{m} u \cdot Λ^{m} (u \cdot \nabla u) d x .$ (3.7) By the Hölder inequality and (3.5), $\begin{array}{rclr} \frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} & ⩽ & C {∥ Λ^{m + α} u ∥}_{2} {∥ Λ^{m - α + 1} (u \otimes u) ∥}_{2} \\ ⩽ & C {∥ Λ^{m + α} u ∥}_{2} {∥ Λ^{m - α + 1} u ∥}_{2} {∥ u ∥}_{\infty} . & (3.8) \end{array}$ Inserting the Gagliardo–Nirenberg inequality ${∥ Λ^{m - α + 1} u ∥}_{2} ⩽ C {∥ Λ^{m + α} u ∥}_{2}^{\frac{1}{α} - 1} {∥ Λ^{m} u ∥}_{2}^{2 - \frac{1}{α}}, \frac{1}{2} < α ⩽ 1$ into (3.8) and using the Young inequality, we get $\begin{array}{rclr} \frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} & ⩽ & C {∥ Λ^{m + α} u ∥}_{2}^{\frac{1}{α}} {∥ Λ^{m} u ∥}_{2}^{2 - \frac{1}{α}} {∥ u ∥}_{\infty} \\ ⩽ & \frac{1}{4} {∥ Λ^{m + α} u ∥}_{2}^{2} + C {∥ u ∥}_{\infty}^{\frac{2 α}{2 α - 1}} {∥ Λ^{m} u ∥}_{2}^{2} . & (3.9) \end{array}$ The Gagliardo–Nirenberg inequality yields ${∥ u ∥}_{\infty} ⩽ C {∥ u ∥}_{2}^{\frac{1}{2}} {∥ Λ^{3} u ∥}_{2}^{\frac{1}{2}} .$ In view of Theorem 1.1 and Proposition 3.3, we have for any $t > T_{0}$ ${∥ u ∥}_{\infty} ⩽ C {(t + 1)}^{- \frac{3}{4 α} (\frac{2}{p} - 1)}, \frac{1}{2} < α ⩽ 1 .$ (3.10) Plugging (3.10) into (3.9), one has for $\frac{1}{2} < α ⩽ 1$ $\begin{array}{rclr} \frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} & ⩽ & C {(t + 1)}^{- \frac{3}{4 α - 2} (\frac{2}{p} - 1)} {∥ Λ^{m} u ∥}_{2}^{2} \\ ⩽ & C {(t + 1)}^{- 1} {∥ Λ^{m} u ∥}_{2}^{2} . & (3.11) \end{array}$

When $0 < α ⩽ \frac{1}{2}$ , we can also establish the same estimate as (3.11). Indeed, by divergence free condition, (3.7) can be written as $\frac{1}{2} \frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} = - \int_{R^{3}} Λ^{m} u \cdot (Λ^{m} (u \cdot \nabla u) - u \cdot \nabla Λ^{m} u) d x .$ (3.12) As a result, using the Hölder inequality and commutator estimate (3.5)–(3.6), we have $\frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {∥ Λ^{m + α} u ∥}_{2}^{2} ⩽ C {∥ \nabla u ∥}_{\infty} {∥ Λ^{m} u ∥}_{2}^{2} .$ (3.13) Combining the estimates of Theorem 1.1 and Proposition 3.3 and using the Gagliardo–Nirenberg inequality yield for any $t > T_{0}$ and $0 < α ⩽ \frac{1}{2}$ $\begin{array}{rclr} {∥ \nabla u ∥}_{\infty} & ⩽ & C {∥ u ∥}_{2}^{\frac{1}{6}} {∥ Λ^{3} u ∥}_{2}^{\frac{5}{6}} \\ ⩽ & C {(t + 1)}^{- \frac{3}{4 α} (\frac{2}{p} - 1)} \\ ⩽ & C {(t + 1)}^{- 1} . & (3.14) \end{array}$ Hence, it is now clear that for $0 < α ⩽ 1$ , $\frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + {{∥ Λ^{m + α} u ∥}_{2}^{2} ⩽ C (t + 1)}^{- 1} {∥ Λ^{m} u ∥}_{2}^{2} .$

Let $D (t) = {ξ \in R^{3} : | ξ | ⩽ f (t)}, f (t) = {(\frac{l}{t + 1})}^{\frac{1}{2 α}}, l is to be determined later .$ Then, $\begin{array}{rclr} {∥ Λ^{m + α} u ∥}_{2}^{2} & = & \int_{R^{3}} | ξ |^{2 α} {| \hat{Λ^{m} u} (ξ, t) |}^{2} d ξ \\ ⩾ & \int_{| ξ | ⩾ f (t)} | ξ |^{2 α} {| \hat{Λ^{m} u} (ξ, t) |}^{2} d ξ \\ ⩾ & f^{2 α} (t) {∥ Λ^{m} u ∥}_{2}^{2} - f^{2 α + 2} (t) \int_{| ξ | ⩽ f (t)} {| \hat{Λ^{m - 1} u} (ξ, t) |}^{2} d ξ \\ ⩾ & f^{2 α} (t) {∥ Λ^{m} u ∥}_{2}^{2} - f^{2 α + 2} (t) \int_{R^{3}} {| \hat{Λ^{m - 1} u} (ξ, t) |}^{2} d ξ . & (3.15) \end{array}$ Inserting (3.15) with suitable l into (3.11), we get $\frac{d}{d t} {∥ Λ^{m} u ∥}_{2}^{2} + \frac{l}{t + 1} {{∥ Λ^{m} u ∥}_{2}^{2} ⩽ C (\frac{l + 1}{t + 1})}^{\frac{2 α + 2}{2 α}} {∥ Λ^{m - 1} u ∥}_{2}^{2} .$ (3.16) To complete the proof of Theorem 1.2 in the case of $0 < α ⩽ 1$ , we use the induction for m. Assume ${{∥ Λ^{m - 1} u ∥}_{2}^{2} ⩽ C_{m - 1} (t + 1)}^{- ρ_{m - 1}}, ρ_{m - 1} = \frac{3}{2 α} (\frac{2}{p} - 1) + \frac{m - 1}{α},$ then we have after inserting the above assumption into (3.16) $\frac{d}{d t} ({(t + 1)}^{l}) {{∥ Λ^{m} u ∥}_{2}^{2}) ⩽ C_{m - 1} (t + 1)}^{l - ρ_{m - 1} - \frac{α + 1}{α}} .$ (3.17) Integrating (3.17) in time from $T_{0}$ to t yields ${(t + 1)}^{l} {{∥ Λ^{m} u (t) ∥}_{2}^{2} ⩽ (T_{0} + 1)}^{l} {{∥ Λ^{m} u (T_{0}) ∥}_{2}^{2} + C_{m - 1} (t + 1)}^{l - ρ_{m - 1} - \frac{1}{α}} .$ Hence, by choosing suitable l we can get $\begin{array}{rclr} {∥ Λ^{m} u (t) ∥}_{2}^{2} & ⩽ & C_{m} {(t + 1)}^{- ρ_{m - 1} - \frac{1}{α}} \\ ⩽ & C_{m} {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1) - \frac{m}{α}} . & (3.18) \end{array}$

We now consider the case when $1 < α < \frac{5}{4}$ . Since $1 < α < \frac{5}{4}$ , we have $m > \frac{5}{2} - α$ for any $m = 2, 3, 4, \dots .$ Therefore, we get by Remark 3.1 $\frac{d}{d t} {∥ u ∥}_{H^{m}}^{2} ⩽ - C {∥ u ∥}_{H^{m + α}}^{2} .$ (3.19) By choosing $m = 1$ in (3.8) and using the Young inequality, we get $\frac{d}{d t} {∥ D u ∥}_{2}^{2} + {∥ Λ^{1 + α} u ∥}_{2}^{2} ⩽ C {∥ u ∥}_{\infty}^{2} {∥ Λ^{2 - α} u ∥}_{2}^{2} .$ This combined with $\frac{d}{d t} {∥ u ∥}_{2}^{2} = - 2 {∥ Λ^{α} u ∥}_{2}^{2}$ yields $\frac{d}{d t} {∥ u ∥}_{H^{1}}^{2} + {∥ u ∥}_{H^{α + 1}}^{2} ⩽ C {∥ u ∥}_{\infty}^{2} {∥ Λ^{2 - α} u ∥}_{2}^{2} .$ (3.20) In the same way as proving (3.10), we can obtain ${∥ u ∥}_{\infty}^{2} ⩽ C {(t + 1)}^{- \frac{3}{2 α} (\frac{2}{p} - 1)} .$ Since $α > 1$ , ${∥ Λ^{2 - α} u ∥}^{2} ⩽ C {∥ u ∥}_{H^{α + 1}} .$ Therefore, there exists some T such that for all $t > T$ , the right-hand side of (3.20) can be bounded by $\frac{1}{4} {∥ u ∥}_{H^{α + 1}}$ . We thus get $\frac{d}{d t} {∥ u ∥}_{H^{1}}^{2} + C {∥ u ∥}_{H^{α + 1}}^{2} ⩽ 0 .$ This together with (3.19) yield for any $m = 1, 2, 3, 4, \dots$ $\frac{d}{d t} {∥ u ∥}_{H^{m}}^{2} ⩽ - C {∥ u ∥}_{H^{m + α}}^{2} .$ Inserting (3.15) with suitable l into the last inequality yields that $\frac{d}{d t} {∥ u ∥}_{H^{m}}^{2} + \frac{l}{t + 1} {∥ u ∥}_{H^{m}}^{2} ⩽ C {(\frac{l}{t + 1})}^{\frac{2 α + 2}{2 α}} {∥ u ∥}_{H^{m - 1}}^{2} .$ (3.21) Adopting to similar procedure as above, we can also obtain (3.18). Theorem 1.2 is proved. □

Footnotes

Acknowledgement

The authors express their gratitude to the anonymous referee for his/her valuable suggestions to improve the presentations of the paper.

Existence of weak solutions

In this section we show that the generalized Navier–Stokes equations with $α > 0$ have a global weak solution corresponding to any prescribed $L^{2}$ initial data.

We start with a definition of weak solutions for (1.2) with $L^{2}$ initial data $u_{0}$ . Let $T > 0$ be arbitrarily fixed.

The following theorem states that there exists global-in-time weak solutions of (1.2).

We will use the Friedrichs method to prove Theorem A.1. Before proof, let us recall the following Picard theorem [14] and Bernstein inequality [1].

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