A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces

Abstract

We exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to 𝐿^2/(2−𝑟)((0,𝑇);ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³))), where ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³)) is the multipliers between Sobolev spaces whose definition is given later for 0<𝑟<1, then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.

1. Introduction

Consider the Navier-Stokes equations in ℝ³:𝜕_𝑡𝑢+𝑢⋅∇𝑢−Δ𝑢+∇𝑝=0,(𝑥,𝑡)∈ℝ³×(),()0,∞div𝑢=0,𝑥,𝑡∈ℝ³×(),𝑢()0,∞𝑥,0=𝑢₀(𝑥),𝑥∈ℝ³, (1.1) where 𝑢=𝑢(𝑥,𝑡) is the velocity field, 𝑝=𝑝(𝑥,𝑡) is the scalar pressure, and 𝑢₀(𝑥) with div 𝑢₀=0 in the sense of distribution is the initial velocity field. For simplicity, we assume that the external force has a scalar potential and is included into the pressure gradient.

In the famous paper, Leray [1] and Hopf [2] constructed a weak solution 𝑢 of (1.1) for arbitrary 𝑢₀∈𝐿²(ℝ³) with div𝑢₀=0. The solution is called the Leray-Hopf weak solution. Regularity of such Leray-Hopf weak solutions is one of the most significant open problems in mathematical fluid mechanics. We note here that there are partial regularity results from Scheffer and from Caffarelli et al., see [3, 4] and references therein. Besides, more work was pioneered by Serrin [5] and extended and improved by Giga [6], Struwe [7, 8], and Zhou [9]. Further results can be found in [10–16] and references therein.

Introducing the class 𝐿^𝛾((0,𝑇);𝐿^𝛼(ℝ³)), Serrin [5] showed that if we have a Leray-Hopf weak solution 𝑢 belonging to 𝐿^𝛾((0,𝑇);𝐿^𝛼(ℝ³)) with the exponents 𝛼 and 𝛾 satisfying 2/𝛾+3/𝛼<1, 2<𝛾<∞, 3<𝛼<∞, then the solution 𝑢(𝑥,𝑡)∈𝐶^∞((0,𝑇)×ℝ³), while the limit case 2/𝛾+3/𝛼=1 was shown much later by Sohr [17] (see also [18]).

Regularity results including assumptions on the pressure gradient have been given by Zhou [15], and it was extended later by Struwe [8] to any dimension 𝑛≥3. It is shown that if the gradient of pressure ∇𝑝∈𝐿^𝛼((0,𝑇);𝐿^𝑞(ℝ³)) with 2/𝛼+3/𝑞≤3, then the corresponding weak solution is actually strong. For the recent work on the regularity problem containing the pressure, velocity field, and the quotient of pressure-velocity, we refer to [19–21] for details.

The purpose of this short paper is to establish a regularity criterion in terms of the pressure gradient for weak solutions to the Navier-Stokes equations in the class 𝐿^2/(2−𝑟)((0,𝑇);ℳ(𝐻̇^𝑟→𝐻̇^−𝑟)). This work is motivated by the recent results [22, 23] on the Navier-Stokes equations. It is an unusual, larger space considered in the current paper than 𝐿^3/2𝑟 (the following Lemma 2.3) and possesses more information. Obviously, the present result extends some previous ones. For more facts concerning regularity of weak solutions, we refer the readers to the celebrated papers [24–30].

2. Preliminaries

We recall the definition of the multiplier space, which was introduced in [31] (see also [32, 33]). The space ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³)) of pointwise multipliers, which map 𝐻̇^𝑟 into 𝐻̇^−𝑟, is defined in the following way.

Definition 2.1. For 0≤𝑟<3/2, ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³)) is a Banach space of all distributions 𝑓 on ℝ³ such that there exists a constant 𝐶 such that for all 𝑢∈𝒟(ℝ³) we have 𝑓𝑢∈𝐻̇^−𝑟(ℝ³) and 𝑓𝑢_{̇𝐻−𝑟}≤𝐶‖𝑢‖_̇𝐻𝑟, (2.1) where we denote by 𝐻̇^𝑟(ℝ³) the completion of the space 𝐶^∞₀(ℝ³) with respect to the norm ‖𝑢‖_̇𝐻𝑟=‖(−Δ)^𝑟/2𝑢‖_𝐿2 and denote by 𝒟(ℝ³) the Schwarz class.

The norm of ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³)) is given by the operator norm of pointwise multiplication 𝑓_{̇𝐻ℳ(𝑟(ℝ3̇𝐻)→−𝑟(ℝ3))}=sup𝑓𝑢_{̇𝐻−𝑟}∶‖𝑢‖_̇𝐻𝑟≤1,𝑢∈𝒟ℝ³. (2.2)

Remark 2.2. Equivalently, we will say that 𝑓∈ℳ(𝐻̇^𝑟(ℝ³)→𝐻̇^−𝑟(ℝ³)) if and only if the inequality 𝑓𝑢,𝑣≤𝐶‖𝑢‖_̇𝐻𝑟‖𝑣‖_̇𝐻𝑟 (2.3) holds for all 𝑢,𝑣∈𝒟(ℝ³).

Lemma 2.3. Let 0≤𝑟<3/2. Then the following embedding: 𝐿^3/2𝑟ℝ³⊂ℳ𝐻̇^𝑟ℝ³⟶𝐻̇^−𝑟ℝ³ (2.4) holds.

Proof. Indeed, let 𝑓∈𝐿^3/2𝑟(ℝ³). By using the following well-known Sobolev embedding: 𝐿^𝑞ℝ³⊂𝐻̇^−𝑟ℝ³ (2.5) with 3/𝑞=3/2+𝑟, we have by Hölder’s inequality 𝑓𝑔_{̇𝐻−𝑟}≤𝐶𝑓𝑔_𝐿𝑞≤𝐶𝑓_𝐿3/2𝑟𝑔_𝐿𝜎≤𝐶𝑓_{𝐿(3/2𝑟)}𝑔_̇𝐻𝑟𝐻̇^𝑟ℝ³⊂𝐿^𝜎ℝ³, (2.6) where 1/𝜎=1/2−𝑟/3. Then, it follows that 𝑓_{̇𝐻ℳ(𝑟→̇𝐻−𝑟)}=sup_{𝑔̇𝐻𝑟≤1}𝑓𝑔_{̇𝐻−𝑟}≤𝐶𝑓_𝐿3/2𝑟. (2.7) This completes the proof.

Example 2.4. Due to the well-known inequality 𝑢|𝑥|_𝐿²≤2‖∇𝑢‖_𝐿2, (2.8) we see that |𝑥|⁻²∈ℳ(𝐻̇¹(ℝ³)→𝐻̇⁻¹(ℝ³)).

Indeed, since the functions of class 𝐶^∞₀(ℝ³) are dense in 𝐻̇¹(ℝ³) in the norm ‖⋅‖_{̇𝐻𝑟(ℝ3)}, suppose 𝑢,𝑣∈𝐶^∞₀(ℝ³). Then by virtue of the Cauchy-Schwarz inequality we obtain|𝑥|⁻²≤𝑢𝑢,𝑣|𝑥|_𝐿²𝑣|𝑥|_𝐿²≤4‖∇𝑢‖_𝐿2‖∇𝑣‖_𝐿2, (2.9) and thus for 𝑢,𝑣∈𝒟,‖𝑢‖_̇𝐻𝑟≤1, and ‖𝑣‖_̇𝐻𝑟≤1|𝑥|⁻²_{̇𝐻ℳ(𝑟→̇𝐻−𝑟)}=sup_{‖𝑢‖𝑢∈𝒟,̇𝐻𝑟≤1}|𝑥|⁻²𝑢_{̇𝐻−𝑟}=sup_{𝑢,𝑣∈𝒟}|𝑥|⁻²𝑢,𝑣≤4<∞. (2.10)

3. Regularity Theorem

Now we state our result as following.

Theorem 3.1. Let 𝑢₀∈𝐿²(ℝ³)∩𝐿^𝑞(ℝ³) for some 𝑞≥3 and ∇⋅𝑢₀=0 in the sense of distributions. Suppose that 𝑢(𝑡,𝑥) is a Leray-Hopf solution of (1.1) in [0,𝑇). If the pressure gradient satisfies ∇𝑝∈𝐿^2/(2−𝑟)(0,𝑇);ℳ𝐻̇^𝑟⟶𝐻̇^−𝑟with0<𝑟<1, (3.1) then 𝑢(𝑡,𝑥) is a regular solution in the sense that 𝑢∈𝐶^∞[0,𝑇]×ℝ³. (3.2)

Proof. In order to prove this result, we have to do a priori estimates for the Navier-Stokes equations and then show that the solution satisfies the well-known Serrin regularity condition. Multiply both sides of the first equation of (1.1) by 4𝑢|𝑢|² and integrate by parts to obtain (see, e.g., [30]) 𝑑‖‖𝑑𝑡𝑢(⋅,𝑡)⁴_𝐿4−4_ℝ3(Δ𝑢)⋅𝑢|𝑢|²𝑑𝑥=−4_ℝ3∇𝑝⋅𝑢|𝑢|²𝑑𝑥, (3.3) for 𝑡∈(0,𝑇). Then we have 𝑑‖‖𝑑𝑡𝑢(⋅,𝑡)⁴_𝐿4+2∇|𝑢|²²_𝐿2≤−4_ℝ3∇𝑝⋅𝑢|𝑢|²𝑑𝑥, (3.4) where we have used 4_ℝ3(Δ𝑢)⋅𝑢|𝑢|²𝑑𝑥≤−2_ℝ3∇|𝑢|²²𝑑𝑥. (3.5)
Let us estimate the integral𝐼=_ℝ3∇𝑝⋅𝑢|𝑢|²𝑑𝑥 (3.6) on the right-hand side of (3.4). By the Hölder inequality and the Young inequality, we have 𝐼≤∇𝑝⋅𝑢_{̇𝐻−𝑟}|𝑢|²_̇𝐻𝑟≤∇𝑝_{̇𝐻ℳ(𝑟→̇𝐻−𝑟)}‖𝑢‖_̇𝐻𝑟|𝑢|²_̇𝐻𝑟≤𝐶∇𝑝_{̇𝐻ℳ(𝑟→̇𝐻−𝑟)}‖𝑢‖_𝐿^1−𝑟₂‖∇𝑢‖^𝑟_𝐿2|𝑢|²_𝐿^1−𝑟₂∇|𝑢|^2𝑟_𝐿2≤𝐶∇𝑝_{̇𝐻ℳ(𝑟→̇𝐻−𝑟)}|𝑢|²_𝐿^1−𝑟₂∇|𝑢|^2𝑟_𝐿2≤𝐶(𝜖)∇𝑝_̇𝐻^2/(2−𝑟)_{ℳ(𝑟→̇𝐻−𝑟)}|𝑢|²_𝐿^{2(1−𝑟)/(2−𝑟)}₂+𝜖∇|𝑢|²²_𝐿2=𝐶(𝜖)∇𝑝_̇𝐻^2/(2−𝑟)_{ℳ(𝑟→̇𝐻−𝑟)}‖𝑢‖_𝐿^4𝜉₄+𝜖∇|𝑢|²²_𝐿2, (3.7) where 𝜉=(1−𝑟)/(2−𝑟)<1; we have used the inequality ‖𝑤‖_̇𝐻𝑟≤𝐶‖𝑤‖_𝐿^1−𝑟₂‖∇𝑤‖^𝑟_𝐿2 (3.8) and the Young inequality with 𝜖: 𝑎𝑏≤𝜖𝑎^𝑝+𝐶(𝜖)𝑏^𝑞(),1𝑎,𝑏>0,𝜖>0𝑝+1𝑞=1, (3.9) for 𝐶(𝜖)=(𝜖𝑝)^−𝑞/𝑝𝑞⁻¹. Hence by (3.4) and the above inequality, we derive 𝑑‖‖𝑑𝑡𝑢(⋅,𝑡)⁴_𝐿4+(2−𝜖)∇|𝑢|²²_𝐿2≤𝐶(𝜖)∇𝑝_̇𝐻^2/(2−𝑟)_{ℳ(𝑟→̇𝐻−𝑟)}‖𝑢‖_𝐿^4𝜉₄. (3.10) Now by Gronwall’s lemma (see for instance in [28, Lemma 2]), we have ‖𝑢(⋅,𝑡)‖⁴_𝐿4≤𝐶‖𝑢(0)‖⁴_𝐿4+^𝑡₀∇𝑝_̇𝐻^2/(2−𝑟)_{ℳ(𝑟→̇𝐻−𝑟)}𝑑𝜏^1/(1−𝜉)=𝐶‖𝑢(0)‖⁴_𝐿4+∇𝑝_𝐿^2/(2−𝑟)_{2/(2−𝑟)()̇𝐻0,𝑇;ℳ(𝑟→̇𝐻−𝑟)}^1/(1−𝜉). (3.11) Due to the integrability of the pressure gradient, it follows that 𝑢∈𝐿^∞0,𝑇;𝐿⁴ℝ³. (3.12) Consequently 𝑢 falls into the well-known Serrin’s regularity framework. Therefore, the smoothness of 𝑢 follows immediately. This completes the proof of Theorem 3.1.

Remark 3.2. By a strong solution we mean a weak solution of the Navier-Stokes equation such that 𝑢∈𝐿^∞(0,𝑇);𝐻¹∩𝐿²(0,𝑇);𝐻². (3.13) It is wellknown that strong solutions are regular (we say classical) and unique in the class of weak solutions.

Acknowledgment

The author thanks the anonymous referee for his/her comments on this paper.