Abstract
In this note, we prove a new \(L^4\)-estimate of the velocity by the technique of Hardy space \({\mathcal {H}}^1\) and BMO.
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1 Introduction
In this note, we consider the Cauchy problem of the 2D Navier-Stokes equations:
Here u is the velocity field and \(\pi \) is the pressure.
In [1], Beirão da Veiga showed the well-known estimate:
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
Using the \(L^\infty \)-estimate of the heat equation, we have
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
In the following proof, we will use the Hardy-BMO duality and the following div-curl lemma [4]:
and
with
where we have used the well-known relation
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
Testing (1.1) by \(|u|^2u\) and using (1.2), (1.8), (1.9) and (1.10), we find that
which gives (1.7) by the Gronwall lemma.
This completes the proof.
\(\square \)
References
Beirão da Veiga, H.: Existence and asymptotic behavior for strong solutions of the Navier–Stokes equations in the whole space. Indiana Univ. Math. J. 36(1), 149–166 (1987)
Giga, Y., Matsui, S., Sawada, O.: Global existence of two-dimensional Navier–Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3, 302–315 (2001)
Sawada, O., Taniuchi, Y.: A remark on \(L^\infty \) solutions to the 2-D Navier–Stokes equation. J. Math. Fluid Mech. 9, 533–542 (2007)
Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
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Dedicated to Professor Hideo Kozono on the occasion of his sixtieth birthday.
Mathematical Fluid Mechanics and Related Topics: In Honor of Professor Hideo Kozono’s 60th Birthday. This article is part of the topical collection dedicated to Prof. Hideo Kozono on the occasion of his 60th birthday, edited by Kazuhiro Ishige, Tohru Ozawa, Senjo Shimizu, and Yasushi Taniuchi.
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Fan, J., Ozawa, T. A note on 2D Navier–Stokes equations. Partial Differ. Equ. Appl. 2, 73 (2021). https://doi.org/10.1007/s42985-021-00129-0
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DOI: https://doi.org/10.1007/s42985-021-00129-0