# Parabolic equations with singular divergence-free

drift vector
fields

###### Abstract

In this paper, we study an elliptic operator in divergence-form but not necessarily symmetric. In particular, our results can be applied to elliptic operator , where is a time-dependent vector field in , which is divergence-free in distribution sense, i.e. . Suppose . We show the existence of the fundamental solution of the parabolic operator , and show that satisfies the Aronson estimate with a constant depending only on the dimension , the elliptic constant and the norm . Therefore the existence and uniqueness of the parabolic equation are established for initial data in -space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form.

## 1 Introduction

The analysis of the Navier-Stokes equations, which are non-linear partial differential equations describing the motion of incompressible fluids confined in certain spaces, has inspired the large portion of the mathematical analysis of non-linear partial differential equations (see for example [15, 16, 21, 30] and etc.) due to the fundamental work J. Leray [17]. The Navier-Stokes equations are partial differential equations of second-order

(1.1) |

(1.2) |

subject to the no-slip boundary condition if the domain of fluid is finite, where is the velocity vector field of the fluid flow, is the pressure at the instance and location . J. Leray [17] demonstrated the existence of a weak solution which belongs to the space and also to the space . The vorticity exists in space and formally, by differentiating the Navier-Stokes equations, solves the vorticity equation

(1.3) |

where the velocity and the vorticity , which is too a time dependent vector field , are related by the definition that . The resolution of the Navier-Stokes equations remains to be an open mathematical problem (see [16, 36] for example), the current research has thus concentrated on the understanding of the related partial differential equations and on developing numerical approaches.

Observe that both the Navier-Stokes equations and the vorticity equations may be put into the following form

(1.4) |

and

(1.5) |

where the diffusion part is the same and is defined by the parabolic operator

(1.6) |

The elliptic operator is the generator of the so-called Taylor diffusion (see J. T. Taylor [34, 35]) of the flow of fluids. There are two non-linear terms appearing in the Navier-Stokes equations and the vorticity equations, which determine the turbulent nature of flows of fluids (see for example [19, 20]). The parabolic operator has the capability of covering the so-called non-linear convection mechanism – the rate-of-strain (for the Navier-Stokes equations [36, 18]) and the vorticity (in the case of the vorticity equations) can be amplified even more rapidly by an increase of the velocity. It is therefore important to study the parabolic equations associated with the elliptic operator , where is a weak solution of the Navier-Stokes equations. The main feature here is that is a time-dependent vector field with little regularity, which however is solenoidal, that is, for every , in distribution sense, so that the formal adjoint is also a diffusion generator. These special features have significance, and have been explored in several recent articles [13, 28, 29] etc. for example. In this paper we give a through study for a class of such parabolic equations.

Recall that in dimension three, a vector field is divergence-free, i.e. , implies that its corresponding two form (with respect to the Hodge star operation) is closed, that is . In fact the divergence operator coincides with the Hodge dual up to a sign, where is the exterior differentiation. Therefore, according to the Poincaré lemma, is exact, that is, there is a vector field so that . Hence coincides with up to a sign. is a two form with components , where is the usual Kronercker symbols of three elements. is skew-symmetric, and

The elliptic operator can thus be put into a divergence form

where is not necessarily symmetric. The symmetric part is uniformly elliptic, and the skew-symmetric part determines the divergence-free drift vector field .

In the present paper we develop a theory for the linear parabolic equation

under very weak assumptions that is uniformly elliptic and its anti-symmetric part only belongs to the BMO space.

The paper is organized as following. In Section 2, we describe the main result, that is the Aronson estimate which depends only on the elliptic constant and the BMO norm of the anti-symmetric part of , which is the key tool of studying weak solutions. In Section 3, we provide several results which will be used to prove the Aronson estimate in our setting. These results are interesting by their own, including several versions of the compensated compactness, and a density result of the BMO space which seems new. In Section 4, we give the details of the proof of the Aronson estimate, and in the final Section 5, we study the weak solutions to the linear parabolic equations in divergence form (but not necessarily symmetric) under weak assumptions. In particular, we prove the existence and uniqueness of weak solutions to the parabolic equation associated with a non-symmetric diffusion matrix .

## 2 Aronson’s estimate for non-symmetric parabolic equations

Let us begin with the description of our framework. We consider the following type of linear parabolic equations of second order

(2.1) |

where

(2.2) |

is in divergence form but* not necessarily symmetric*, and their
associated diffusion processes in terms of fundamental solutions defined
by (2.1). There is a unique decomposition
such that is symmetric, while
is skew-symmetric. We assume that is uniformly elliptic in the
following sense: there exists a constant such that

(2.3) |

for any , and .

Let us first consider the regular case where are smooth, bounded and possess bounded derivatives of all orders on .

Let be the parabolic linear operator associated with . The formal adjoint of is again a parabolic operator (with vanished zero order term) given by

(2.4) |

where is the transpose of .
It is known that (see A. Friedman [10], Theorem 11
and 12, Chapter 1), under the elliptic condition and smoothness assumptions
on , there is a unique positive *fundamental solution*
of the parabolic operator , and it
is smooth in on and .
Recall that the following properties are satisfied.

1) for any and .

2) For every and , as a function of , solves the parabolic equation on :

(2.5) |

3) Chapman-Kolmogorov’s equation holds

(2.6) |

4) For any bounded continuous function and , it holds that

(2.7) |

for every .

For , let denote the corresponding linear operator defined by

(2.8) |

where is Borel measurable, either non-negative, or/and bounded. By (2.6)

(2.9) |

for any .

Define for .
Then is the fundamental solution to
in the sense that for every fixed , as a function of ,
solves the *backward* parabolic equation

(2.10) |

on . It follows that the fundamental solution also solves the backward parabolic equation:

(2.11) |

which holds for any and .

We are now in a position to state the key result of the present paper.

###### Theorem 2.1

There is a constant depending only on the dimension , the elliptic constant , and the norm of the skew-symmetric part such that

(2.12) |

for any and , where the norm of is defined by

The fundamental heat kernel estimate (2.12) for parabolic equations has a long history. Two side estimate (2.12) was first established in D. G. Aronson [2, 1] for uniformly elliptic operators in divergence form where is symmetric (so that ), his constant depends only on the elliptic constant and the dimension . The estimate (2.12) is therefore referred to as the Aronson estimate. A weaker but global estimate similar to (2.12) under the same assumption as in D. G. Aronson [2] already appeared in the Appendix of J. Nash [26]. D. G. Aronson [2, 3] indicated that his estimate can be established for a general elliptic operator, and a written proof is available in E. B. Fabes and D. W. Stroock [9], D. W. Stroock [31] and J. R. Norris and D. W. Stroock [27] too. In these papers, the Aronson estimate (2.12) was established for the following type of uniformly elliptic operator

where is symmetric and uniformly elliptic. For this case, their constant depends on the dimension, the elliptic constant and the -norms of , and .

A related topic to the Aronson estimate is the regularity of solutions
to the parabolic equation (see for a complete survey of classical
results [15]). If the elliptic operator
is symmetric and is in divergence form, it was J. Nash [26]
who proved the Hölder continuity of bounded solutions and also proved
that the Hölder exponent depends only on the dimension and the elliptic
constant . Under the same setting as that of J. Nash [26],
in 1964, J. Moser [23] established the Harnack inequality
for positive solutions of the parabolic equation , based on
which G. Aronson was able to derive his estimate (2.12).
E. B. Fabes and D. W. Stroock [9] showed that
J. Moser’s Harnack inequality can be derived from Aronson estimate
together with J. Nash’s idea, and D. W. Stroock [31]
further demonstrated that both the Hölder continuity of classical
solutions and Moser’s Harnack inequality for positive solutions can
be established by utilizing the two side Aronson estimate (2.12).
J. Nash’s idea in [26] and the techniques in J. Moser
[23, 24, 25]* *have been investigated
intensively during the past decades. Many excellent results have been
obtained in more general settings, but mainly under the symmetric
setting of Dirichlet forms [11]. See for example
A. A. Grigor’yan [12], E. B. Davies [7]
and D. W. Stroock [32] for a small sample of
references, and see also the literature therein.

The case that is non-symmetric has received intensive study only recently, due to the connection with the Navier-Stokes equations and the blow-up behavior of their solutions. In H. Osada [28], the Aronson estimate (2.12) was obtained for an elliptic operator in divergence form as ours, where may not be symmetric, his constant in (2.12) however depends on the dimension , the elliptic constant and the -norm of the skew-symmetric part . In a recent work by G. Seregin, L. Silvestre, V. Šverák, and A. Zlatoš [29], who noticed that a large portion of Nash’s arguments also work for an elliptic operator with divergence-free drifts, i.e. where the elliptic operator has a form such that . In particular, they mentioned that the fundamental solution of the heat operator

satisfies the diagonal decay estimate

for all . They further proved the Harnack inequality in this case, and their constants depend on the dimension and the of the vector field.

Our work was motivated by the observation made by G. Seregin and etc. [29], and the approach put forward by E. Davies [6], E. B. Fabes and D. Stroock [9], and D. W. Stroock [31]. We follow the approach in E. B. Davies and D. W. Stroock to the non-symmetric case, and adopt their arguments to our case by overcoming the difficulties arising from the singularities of the skew-symmetric part . In a sense, the present work is to complete the program initiated by G. Seregin and etc. [29] by bringing in the techniques developed over years by various authors.

As in D. W. Stroock [31], as consequences of the Aronson estimate, we have the following continuity theorem and the Harnack inequality.

###### Theorem 2.2

There exist and depending only on the dimension , the elliptic constant and the -norm of the skew-symmetric part , such that for every

(2.13) |

for all , with .

###### Theorem 2.3

[Harnack Inequality] There exists a constant depending only on and such that given any with and set , we have

(2.14) |

for any , .

The Harnack inequality is also established by G. Seregin and etc. in [29] under a bit additional technical conditions than those stated in the theorem above.

## 3 Several technique facts

In this and next several sections, we are going to prove the main result, Aronson estimate. In this section, we prove several technique facts which will be needed in the proof of the main result.

The first result we need is a variation of Coifman-Meyer’s compensated compactness Theorem [5, 4, 16] which highlights the importance of the Hardy spaces in the study of partial differential equations.

We first recall some facts on BMO functions [30, 14]. A function is in if

(3.1) |

where and the supremum is taken over all open balls (in what follows, or denotes the ball centered at with radius ). If define another norm

(3.2) |

for any , John-Nirenberg inequality [14] (see also for example, Appexdix in D. W. Stroock and S. R. S. Varadhan [33]) implies that are equivalent for different .

The original version of the compensated compactness Theorem, which will be used in our proof of the lower bound of Aronson estimate, can be stated as following

###### Proposition 3.1

Let vector fields satisfy , with , ) and , . Then where is the Hardy space, and (

(3.3) |

In particular, there is a constant depending on the dimension and such that

(3.4) |

for any , where . Hence

(3.5) |

for any and for any which is skew-symmetric, .

To prove the upper bound of Aronson estimate, we need the following estimate, in the same spirit of compensated compactness.

###### Proposition 3.2

There is a universal constant depending only on the dimension , such that

(3.6) |

for any and , where denotes the Hardy norm.

Proof. Let be any smooth non-negative function on , with its support in the unit ball such that , and for , . Notice that in , so

where denotes the average integral over the ball , that is, . For the first term on the right-hand side, we have

(3.7) |

where , such that and . Choose

(3.8) |

For the second term on the right-hand side, we integrate by part again to obtain

(3.9) |

By using these estimates we thus conclude that

where is the maximal function. Since , we have , and similarly , . So and

(3.10) |

Given a function , we want to approximate it by a mollified sequence, which is not trivial as it looks. A simple example is a vector field which depends only on , not on the space variables. Then it may not be in and there is no approximations by mollifying sequences. However, the problem considered here allow us to add a constant to it, i.e. consider , where is skew-symmetric so that it will not alter the weak solution formulation of the corresponding parabolic equations. So by subtracting the mean value of on a unit ball, we may assume that

(3.11) |

Then for any

(3.12) | |||||

(3.13) |

where . By the definition of BMO functions, we have

(3.14) |

which implies that for any .

###### Proposition 3.3

Take and with and

Let and satisfies (3.11). Define . Suppose and

(3.15) |

Then locally in for any , and

(3.16) |

Proof. Let , and be fixed. Let denote the average integral over , that is,

For we have

so that

Hence we have proved .

The lattice property in the proposition below of the BMO space should be well known, for completeness, a proof is attached.

###### Proposition 3.4

Suppose , then and . Moreover, we have

(3.17) |

where only depends on .

Proof. Here we only prove it for and follows similar proof. Observe that for any , we have

(3.18) |

Hence for any ball ,

and the proof is done.

## 4 Proof of Aronson’s estimate

The proof follows the main lines as in D. W. Stroock [31] and in particular E. B. Davies [6] from which a clever use of the -transforms from harmonic analysis is borrowed, while we need to overcome several difficulties since is non-symmetric and the skew-symmetry part is singular. These ideas are mainly due to J. Nash [26], J. Moser [22, 23, 24, 25].

Let us begin with the proof of the upper bound.

### 4.1 Proof of the upper bound

In this part we show the upper bound:

(4.1) |

for any and , where depends only on , and .

The main idea of E. B. Davies [6] is to consider the -transform of the fundamental solution and apply Nash and Moser’s iteration to the -transforms of the fundamental solution . J. Nash’s idea is to iterate the -norms of solutions to parabolic equations, and to control the growth of the -norms. The main ingredient in J. Nash’s argument is the clever use of the Nash inequality

(4.2) |

where is a constant depending only on the dimension .

The Nash iteration is neatly described as the following (D. Stroock [31], Lemma I.1.14, page 322).

###### Lemma 4.1

Given positive numbers , , and . Let be positive and non-decreasing, continuous on , and be positive and has continuous derivatives on . Suppose the following differential inequality holds:

(4.3) |

Then there exists a such that

(4.4) |

for every .

The above iteration procedure works in a very general setting, and has been explored since the publication of J. Nash’ paper [26], and it is still the major ingredient in our proof. It is surprising that they work well even in our setting where the diffusion is very singular.

Fortunately as well, E. B. Davies’ idea [6, 7] also works well for our parabolic equations. Following E. B. Davies [6] and D. Stroock [31], given a smooth function on , consider

(4.5) |

and the linear operator

which is defined for non-negative Borel measurable , and for which is smooth with a compact support. It is easy to see that the adjoint operator of can be identified as the following integral operator

That is

for any smooth functions and with compact supports.

###### Lemma 4.2

Let . Let be non-negative, and where . Define

for , and

for .

There is a constant depending only on , such that for any ,

(4.6) |

, and