1 Introduction

In this note, we consider the Cauchy problem of the 2D Navier-Stokes equations:

$$\begin{aligned}&\partial _tu+(u\cdot \nabla )u+\nabla \pi -\Delta u=0,\end{aligned}$$
(1.1)
$$\begin{aligned}&\mathrm {div}\,u=0,\end{aligned}$$
(1.2)
$$\begin{aligned}&u(\cdot ,0)=u_0,\ \mathrm {div}\,u_0=0\ \ \text{ in }\ \ {\mathbb {R}}^2. \end{aligned}$$
(1.3)

Here u is the velocity field and \(\pi \) is the pressure.

In [1], Beirão da Veiga showed the well-known estimate:

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^p}\le \Vert u_0\Vert _{L^p}\exp \{C\Vert u\Vert _{L^q(0,t;L^p)}^q\} \end{aligned}$$
(1.4)

with the space dimension \(n<p<\infty \) and \(\frac{2}{q}+\frac{n}{p}=1\) and \(C>0\) is an absolute constant by the standard \(L^p\)-energy method.

The Navier–Stokes equation (1.1) can be written in the form of the following nonlinear heat equation

$$\begin{aligned} \partial _tu-\Delta u=-\mathrm {div}\,(u\otimes u)-\nabla \pi . \end{aligned}$$
(1.5)

Using the \(L^\infty \)-estimate of the heat equation, we have

$$\begin{aligned} \Vert u\Vert _{L^\infty ({\mathbb {R}}^n\times (0,T))}\lesssim & {} \Vert u\otimes u\Vert _{L^s({\mathbb {R}}^n\times (0,T))}+\Vert \pi \Vert _{L^s({\mathbb {R}}^n\times (0,T))}+\Vert u_0\Vert _{L^\infty }\nonumber \\\lesssim & {} \Vert u\Vert ^2_{L^{2s}({\mathbb {R}}^n\times (0,T))}+\Vert u_0\Vert _{L^\infty } \end{aligned}$$
(1.6)

with \(s>n+2\) and \(u_0\in L^2\cap L^\infty \).

For other \(L^\infty \)-estimates of the velocity, we refer to [2, 3].

The aim of this note is to prove a different estimate. We will prove

Theorem 1.1

Let \(n=2\) and \(u_0\in L^2\cap L^4\). Then solutions of the Navier–Stokes equations satisfy the \(L^4\)-estimate of the form

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^4}\le \Vert u_0\Vert _{L^4}\exp \left( C\int _0^t\Vert \nabla u\Vert _{L^2}^2\mathrm {d}\tau \right) . \end{aligned}$$
(1.7)

In the following proof, we will use the Hardy-BMO duality and the following div-curl lemma [4]:

$$\begin{aligned} u\cdot \nabla \pi \in {\mathcal {H}}^1 \end{aligned}$$
(1.8)

and

$$\begin{aligned} \Vert u\cdot \nabla \pi \Vert _{{\mathcal {H}}^1}\lesssim \Vert u\Vert _{L^4}\Vert \nabla \pi \Vert _{L^\frac{4}{3}} \end{aligned}$$
(1.9)

with

$$\begin{aligned} \Vert \nabla \pi \Vert _{L^\frac{4}{3}}\lesssim \Vert u\cdot \nabla u\Vert _{L^\frac{4}{3}}\lesssim \Vert u\Vert _{L^4}\Vert \nabla u\Vert _{L^2}, \end{aligned}$$
(1.10)

where we have used the well-known relation

$$\begin{aligned} \pi =\sum ^2_{j,k=1}R_jR_k u_ju_k \end{aligned}$$

with the Riesz operator \(R_j\) and its boundedness in \(L^{\frac{4}{3}}\).

2 Proof of Theorem 1.1

This section is devoted to the proof Theorem 1.1. We assume that the solution is smooth and only need to show the a priori estimates.

Testing (1.1) by u and using (1.2), we see that

$$\begin{aligned} \frac{1}{2}\int _{{\mathbb {R}}^2}|u|^2\mathrm {dx}+\int _0^t\int _{{\mathbb {R}}^2}|\nabla u|^2\mathrm {dx}\mathrm {d}\tau =\frac{1}{2}\int _{{\mathbb {R}}^2}|u_0|^2\mathrm {dx}. \end{aligned}$$
(2.1)

Testing (1.1) by \(|u|^2u\) and using (1.2), (1.8), (1.9) and (1.10), we find that

$$\begin{aligned}&\frac{1}{4}\frac{\mathrm {d}}{\mathrm {dt}}\int _{{\mathbb {R}}^2}|u|^4\mathrm {dx}+\int _{{\mathbb {R}}^2}|u|^2|\nabla u|^2\mathrm {dx}+\frac{1}{2}\int _{{\mathbb {R}}^2}|\nabla |u|^2|^2\mathrm {dx}\\&\quad =-\int _{{\mathbb {R}}^2}(u\cdot \nabla \pi )|u|^2\mathrm {dx}\\&\quad \lesssim \Vert u\cdot \nabla \pi \Vert _{{\mathcal {H}}^1}\Vert |u|^2\Vert _{BMO}\\&\quad \lesssim \Vert u\Vert _{L^4}\Vert \nabla \pi \Vert _{L^\frac{4}{3}}\Vert \nabla |u|^2\Vert _{L^2}\\&\quad \lesssim \Vert u\Vert _{L^4}^2\Vert \nabla u\Vert _{L^2}\Vert \nabla |u|^2\Vert _{L^2}\\&\quad \le \frac{1}{4}\Vert \nabla |u|^2\Vert _{L^2}+C\Vert u\Vert _{L^4}^4\Vert \nabla u\Vert _{L^2}^2, \end{aligned}$$

which gives (1.7) by the Gronwall lemma.

This completes the proof.

\(\square \)