New Approach for Vorticity Estimates of Solutions of the Navier-Stokes Equations

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arXiv:2210.04129v1 [math.AP] 9 Oct 2022

New Approach for Vorticity Estimates of Solutions of the

Navier-Stokes Equations

Gui-Qiang G. Chen*and Zhongmin Qian†

Abstract

We develop a new approach for regularity estimates, especially vorticity estimates, of solutions

of the three-dimensional Navier-Stokes equations with periodic initial data, by exploiting carefully

formulated linearized vorticity equations. An appealing feature of the linearized vorticity equations

is the inheritance of the divergence-free property of solutions, so that it can intrinsically be employed

to construct and estimate solutions of the Navier-Stokes equations. New regularity estimates of

strong solutions of the three-dimensional Navier-Stokes equations are obtained by deriving new

explicit a priori estimates for the heat kernel (i.e., the fundamental solution) of the corresponding

heterogeneous drift-diffusion operator. These new a priori estimates are derived by using various

functional integral representations of the heat kernel in terms of the associated diffusion processes

and their conditional laws, including a Bismut-type formula for the gradient of the heat kernel. Then

the a priori estimates of solutions of the linearized vorticity equations are established by employing

a Feynman-Kac-type formula. The existence of strong solutions and their regularity estimates up to

a time proportional to the reciprocal of the square of the maximum initial vorticity are established.

All the estimates established in this paper contain known constants that can be explicitly computed.

Key words: Gradient estimates, vorticity estimates, approach, iteration scheme, a priori esti-

mates, strong solutions, Navier-Stokes equations, vorticity equations, heat kernel, fundamental solu-

tion, stochastic diffusion process, heterogeneous drift-diffusion operator, conditional laws, Bismut-

type formula, Feynman-Kac-type formula.

MSC classiﬁcations: Primary: 35Q30, 35Q35, 35B65, 35B45, 35D35, 76D05; Secondary:

35K45, 35A08, 35B30, 35Q51

1 Introduction

We are concerned with the quantitative regularity estimates of solutions of the Navier-Stokes equations

in R3. In this paper, we consider the Cauchy problem for the Navier-Stokes equations for x∈R3and

t≥0:

(

∂

tu+u·∇u+∇P=

∆u,

∇·u=0,(1.1)

subject to the periodic initial condition:

u|t=0=u0,(1.2)

satisfying that u0(x+Lk) = u0(x)for all k∈Z3and ∇·u0=0, where

>0 is the kinematic constant,

and L>0 is the period of the initial data. Although periodic ﬂows are special in nature, the periodic

solutions can be considered as ideal solutions for turbulent motions away from their physical boundaries,

or as models for homogeneous turbulent ﬂows.

*Mathematical Institute, University of Oxford, Oxford OX2 6GG. Email: chengq@maths.ox.ac.uk

†Mathematical Institute, University of Oxford, Oxford OX2 6GG, and OSCAR, Suzhou, China. Email:

qianz@maths.ox.ac.uk

The Cauchy problem (1.1)–(1.2) for the Navier-Stokes equations (1.1) with periodic initial data (1.2)

seeks for a velocity vector ﬁeld u(x,t)and a scalar pressure P(x,t)that are periodic functions with period

Lsatisfying the initial condition that u(x,0) = u0(x), where u0(x)is the periodic initial velocity in (1.2),

which is divergence-free, i.e.,∇·u0=0.

The mathematical study of global solutions of the Navier-Stokes equations (1.1) was initiated in

the work of Leray [23,24] and Hopf [16,17], in which global weak solutions were constructed and

investigated. Since then, many properties and features of the solutions of the Navier-Stokes equations

(1.1) have been understood; see [14,15,21,36,39,40] and the references cited therein. The Navier-

Stokes equations (1.1) have been studied by using various methods; the mathematical analysis of (1.1)

has been mainly based on several functional analysis methods and on the results on certain functional

spaces such as the Sobolev spaces and the Besov spaces. Great progress has been made in the past

decades, the existences of local and global solutions of the Navier-Stokes equations (1.1) have been

studied (cf. [10,13,20,26,34,35], besides the references cited therein and above). The partial regularity

of the weak solutions of the Navier-Stokes equations (1.1) has also received intensive study; see [5,25,

30,31,32,33] and the references cited therein. However, the quantitative regularity of solutions, the

global existence of strong solutions, and the uniqueness of weak solutions of the three-dimensional (3-D)

Navier-Stokes equations remain to be the major open problems in Mathematics.

1.1 Main theorem

Assume that the periodic initial data function u0is smooth and divergence-free. Then any strong solution

u(x,t)of the Cauchy problem (1.1)–(1.2) must have a constant mean velocity, so that it is assumed

without loss of generality that

[0,L]3

u(x,t)dx=0,(1.3)

due to the Galilean invariance of system (1.1). Denote the vorticity:

(x,t) = ∇∧u(x,t)(1.4)

which is the curl of the velocity. Then

0=∇∧u0is the initial vorticity.

In this paper, we develop a new approach, i.e., an iteration scheme, to construct and estimate strong

solutions of the Cauchy problem (1.1)–(1.2) more sharply than the existing results, by developing an

array of useful mathematical tools in Analysis. The main results for the quantitative estimates of strong

solutions can be stated in the following theorem.

Theorem 1.1 (Main Theorem).There are two universal constants C1>0and C2>0such that there

exists a unique strong solution u(x,t)of the Cauchy problem (1.1)–(1.2)for all t ∈[0,T0]with

T0=C1

2L−4k

0k−2

∞,(1.5)

so that the following estimates for u(x,t)hold:For all 0<t≤T0and x ∈R3,

|u(x,t)| ≤C2Lk

0k∞,|∇u(x,t)| ≤ C2L

√

0k∞,(1.6)

(x,t)| ≤C2k

0k∞,|∇

(x,t)| ≤ C2

√

0k∞(1.7)

where k

0k∞is the L∞-norm of the initial vorticity

The two positive constants C1and C2in Theorem 1.1 are computable, which can be worked out by

tracing all the universal constants in the proof.

The Cauchy problem (1.1)–(1.2) of the Navier-Stokes equations with periodic initial data u0(x)has

been studied traditionally in the Fourier space, which is particularly the case in turbulence literature;

see, for example, [2,7,11]. Our approach is necessary to departure from the well-known methods.

The quantitative regularity estimates, Theorem 1.1, will be proved by developing a new approach via an

array of mathematical tools from the theory of partial differential equations, stochastic analysis, and the

Hodge theory on the torus.

1.2 New approach – An iteration scheme for the construction and estimates of strong

solutions

We now describe brieﬂy the new approach – an iteration scheme – developed in this paper to prove

Theorem 1.1; see §6–§7 for details.

First of all, by using the dimensionless scaling,

U(x,t) = L

u(Lx,L2

has period 1 and solves the Navier-Stokes equations (1.1) with

2. Thus, without loss of generality,

we will assume that the viscosity constant

2and L=1 in what follows. Furthermore, from now on,

by a (time-dependent) periodic tensor ﬁeld f(x,t)on R3, or equivalently by saying that a tensor ﬁeld

f(x,t)is periodic, we mean that f(x,t)is a tensor ﬁeld on R3depending on the time parameter tand

satisﬁes that f(x+k,t) = f(x,t)for all x∈R3,t≥0, and k∈Z3.

Our new iteration scheme for the construction of strong solutions of the Cauchy problem (1.1)–(1.2)

is based on the vorticity equation for

=∇∧u:

∂

+ (u·∇)

−A(u)

−1

2∆

=0,(1.8)

where A(u)is the total derivative of u, a tensor ﬁeld with components A(u)i

∂

xjui. Although there

are several formulations of the vorticity equations, one of our main observations is that this version of

formulation serves our aims particularly well.

Suppose that b(x,t)is a periodic, smooth, and divergence-free vector ﬁeld such that b(x,0) = u0(x).

Then we deﬁne a vector ﬁeld w(x,t), which should be a candidate of the vorticity (while wis not in

general the vorticity of b), by solving the Cauchy problem of the following linear parabolic equations:

(

∂

tw+ (b·∇)w−A(b)w−1

2∆w=0,

w(·,0) =

0,(1.9)

where A(b) = (A(b)i

j) = (

∂

xjbi)is the total derivative of b. The unique solution w(x,t)has two properties

that are important to our approach:

(i) It can be shown that ∇·w=0 (divergence-free again); this is the key property that makes the

linear parabolic equations (1.9) appealing and workable to our task.

(ii) r[0,1]3w(x,t)dx=0 for all t>0, which is satisﬁed when t=0.

With these, we deﬁne the candidate v(x,t)for the velocity by solving the Poisson equation:

(∆v=−∇∧w,

r[0,1]3v(x,t)dx=0 for any t>0.

Then w=∇∧v, according to the Hodge theory on the torus.

In this way, we construct a mapping Vthat sends b(x,t)to v(x,t). That is, the iteration for obtaining

a strong solution is deﬁned as

u(n)=V(u(n−1))for n=1,2,..., (1.10)

with the initial iteration deﬁned by u(0)(x,t) = u0(x)for all xand t≥0.

Let us point out that the computational schemes for simulations of turbulent ﬂows based on different

formulations of the vorticity equations have been very fruitful in the past, cf. [6,27] for an overview. Our

analysis below shows that the iteration scheme developed in this paper does converge to the strong solu-

tion with inherent gradient estimates of the iteration solutions uniformly. This shows that the iteration

scheme should also be useful for developing numerical algorithms to compute turbulent solutions.

1.3 Assumptions and notations

In order to describe the technical aspects in our study, we introduce a few notations and assumptions

that will be used throughout the paper. By a function or a tensor ﬁeld we mean a function deﬁned on Rd

or a tensor ﬁeld on the torus Td=Rd/Zd; in the latter case, it is identiﬁed with a periodic ﬁeld on Rd

with period 1 in each coordinate variable.

Suppose that f(x,t)for x∈Rdis a tensor ﬁeld depending on a time parameter t≥0. Then we assume

that fis Borel measurable on Rd×[0,∞). For I⊂[0,∞), the L∞-norm of fover Rd×Iis deﬁned to be

kfkL∞(I)=sup

(x,t)∈Rd×I|f(x,t)|,

and for the case when I= [0,∞), the previous norm is simply denoted by kfk∞.

If 0 ≤

<T, the parabolic L∞-norm of fover an interval [

,T]will play an important role, which

is deﬁned by

kfk

→T=sup

(x,t)∈Rd×[

,T]√t−

f(x,t)=

√·−

f

L∞([

,T]) .(1.11)

In the case when T=∞, the interval [

,T]is replaced by [

,∞). From the deﬁnition, it is clear that

kfk

→T≤√T−

kfkL∞([

,T]) ≤√T−

kfk∞.(1.12)

Throughout the paper, the probability density function (PDF) of a normal random variable with mean

zero and variance t>0 is denoted by

Gt(x) = 1

t)d/2exp(−|x|2

2t)for x∈Rd.(1.13)

Notice that Gt−

(y−x)is the fundamental solution of the heat operator

∂

t−1

2∆in the Euclidean space

Rd.

As a convention, the Laplacian ∆and the gradient ∇(in particular, the divergence operation ∇·and

the curl ∇∧), when operating on time-dependent tensor ﬁelds on Rd, apply to space variable xonly.

If b(x,t)is a time-dependent vector ﬁeld on Rdfor t≥0, the heterogeneous drift-diffusion differen-

tial operator of second order:

Lb(x,t)=1

2∆+b(x,t)·∇(1.14)

will play an important role in our study. When no confusion arises, Lb(x,t)is denoted simply by Lb.

Among the technical assumptions on b(x,t), the most essential one is the assumption that b(x,t)is

solenoidal (i.e.,divergence-free). That is, for every t, the divergence ∇·b(·,t)vanishes identically in the

distributional sense. Under this assumption, the formal adjoint operator:

L⋆

b=L−b,

which is again an elliptic operator of the same type. The second assumption is technical for the construc-

tion of probabilistic structures. For simplicity, we assume that b(x,t)is Borel measurable and bounded

over any ﬁnite interval, i.e.,kbkL∞(Rd×[0,T]) <∞for every T>0. The probability density function of the

Lb-diffusion (i.e., the fundamental solution or heat kernel to the parabolic operator L⋆

b(x,t)+

∂

t; see §2) is

denoted by

pb(

,x,t,y)for t>

≥0 and x,y∈Rd.

Throughout the paper, universal constants (the constants depending only on the dimension d, or

some parameters

,etc. introduced in proofs) are denoted by C1,C2,etc. which may be different at

each occurrence.

1.4 A priori estimates of solutions of the parabolic equations

In order to prove Theorem 1.1, the main effort is to derive precise a priori estimates of solutions of the

Cauchy problem (1.9) for the parabolic equations. To achieve this, there are two tasks to be carried out.

1. The main task is to derive explicit a priori estimates for the fundamental solution (or called

the heat kernel)of the parabolic operator

∂

t−Lband its gradient in terms of the bound of b and the

parabolic norm of ∇b, when b(x,t)is a bounded, divergence-free, and smooth vector ﬁeld on Rd.

Although the regularity theory for linear parabolic equations has been well established (cf. [12,22,

38]), our a priori estimates contain the universal constants depending only on the dimension dand a

parameter

>1 ﬁxed in our estimates. In addition to its explicit form, for a divergence-free vector ﬁeld

b(x,t), the gradient estimate for the heat kernel associated with the parabolic operator

∂

t−Lb, depends

only on the bound of band its ﬁrst-order derivative ∇b.

Theorem 1.2. Let b(x,t)be a smooth, divergence-free, and bounded time-dependent vector ﬁeld on Rd.

Then, for every

>1, there are constants C1and C2depending only on

and dimension d such that

pb(

,x,t,y)≤C1eC2(t−

)kbk2

∞G

(t−

)(y−x),(1.15)

|∇ypb(

,x,t,y)| ≤ C1

√t−

eC2(t−

)kbk2

∞+1

2√t−

k∇bk

→tG

(t−

)(y−x),(1.16)

for all t >

and x,y∈Rd.

These estimates are quite delicate to derive: They are obtained by introducing substantial tools

from stochastic analysis, mainly various functional integral representations for the fundamental solutions

(cf. [29,28]) and a new kind of Bismut’s formulas (cf. [3,4]), together with careful and explicit

computations. Estimates (1.15)–(1.16) will be proved in §4 and §5, respectively.

2. The second task in our study is to prove the explicit a priori estimates to the iteration u(n)=

V(u(n−1))in (1.10), or equivalently, to derive the a priori estimates of the solutions of the Cauchy prob-

lem (1.9)for the linear parabolic equations that deﬁne the nonlinear mapping V , namely the solution of

the Cauchy problem for the parabolic equations:

((

∂

t−L−b)w=A(b)w,

w(·,0) =

0,(1.17)

where, as before, L−bdenotes the time-dependent elliptic operator 1

2∆−b·∇.

The crucial observation is that the term on the right-hand side, A(b)w, is a linear zero-order term,

which differs fundamentally from the linearized Navier-Stokes equations. This crucial difference allows

us to apply the Feynman-Kac-type formula, obtained in this context in [28], to derive the necessary

explicit a priori estimates.

The a priori estimates and technical tools are worked out for a general dimension dand a general

vector ﬁeld b(x,t)that is divergence-free on Rd. Therefore, they have independent interests and are

likely useful for both treating the Navier-Stokes equations with other boundary conditions and dealing

with other linear/nonlinear PDEs.

1.5 Organisation of the paper

In §2, several probabilistic structures associated with a time-dependent vector ﬁeld b(x,t)are reviewed,

and then a functional integration representation formula for the heat kernel of 1

2∆−b·∇and a Bismut-

type formula for the gradient of the heat kernel are established, which are the tools for deriving the a

priori estimates in Theorem 1.2 we need. In §3, several technical potential estimates are established,

which will be used in the proof of Theorem 1.2 that will be carried out in §4–§5. In §6, the linearized

vorticity equations are carefully analyzed, and the main regularity results for the strong solutions of the

Cauchy problem (1.1)–(1.2) with periodic initial data (1.2) for the Navier-Stokes equations (1.1) in R3

will be proved in §7.

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arXiv:2210.04129v1[math.AP]9Oct2022NewApproachforVorticityEstimatesofSolutionsoftheNavier-StokesEquationsGui-QiangG.Chen*andZhongminQian†AbstractWedevelopanewapproachforregularityestimates,especiallyvorticityestimates,ofsolutionsofthethree-dimensionalNavier-Stokesequationswithperiodicinitialdata,bye...

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