Special Properties of Plane Solenoidal Fields

Abstract

Algebraic and integral identities have been obtained for a pair or a triple of plane solenoidal fields. We obtain sufficient potentiality conditions for a plane vector field. The integral identities are also important for exact a priori estimates.

1. Introduction

The well-known classical Helmholtz result for the decomposition of the vector field using the sum of the solenoidal and potential components is generalized. This generalization is known as the Helmholtz–Weyl decomposition (see, for example, [1]). A more exact Lebesgue space $L_2\left(R^n\right)$ of vector fields $u=(u_1,\dots ,u_n)$ is represented by a direct sum

$$L_2\left(R^n\right)=H_2\left(R^n\right)\oplus G_2\left(R^n\right)$$

where $H_2\left(R^n\right)$ is a closure of all smooth solenoidal fields and $G_2\left(R^n\right)$ is a closure of all smooth potential fields with respect to the norm of space $L_2\left(R^n\right)$. For different dimensions (see [2,3,4]), the space denoted by $H_2\left(R^n\right)$ can be considered as a closure of all smooth finite solenoidal fields. We remark that it is impossible to replace space $R^n$ with a domain $\mathsf{\Omega }\ne R^n$ (see [5]). Some conditions for these replacements can be found in [6].

Helmholtz–Weyl decomposition implies an integral identity:

$${\int }_{R^n}u·gdx=0$$

(1)

for any fields $u\in H_2\left(R^n\right)$ and $g\in G_2\left(R^n\right)$, that provide the key to studying the Navier–Stokes equations. In principle, the different results connected with integral identities for solenoidal fields in space were obtained in [7]. The other dimensions are considered in [8,9].

A new portion of integral identities is found in [10], where the rotor is used. Hence, a distinction between the properties of plane and spatial solenoidal fields was emphasized. This idea was first considered in [11]. Various examples of flows in space confirm the poorness of plane–parallel fluids in comparison with spatial fluids.

The main goal of this work is to show new applications of new integral identities for plane solenoidal fields. First, we provide sufficient conditions for a plane field to be a potential field. These identities can be considered as the origin of new exact a priori estimates (see, for example, [10]), as the origin of new conservation laws, from which we can include the initial data from the Cauchy problem. These integral identities can be considered as elements of latent symmetry. Hydrodynamics is very rich in such elements (see, for example, the important review [12] or book [13]). Therefore, the comparison of different symmetry applications is very useful in this way.

In the following, we apply notations. Let $u:R^2\to R^2,u=(u_1,u_2)$, be any vector field. Symbols such as

$$u_{k,i}=\frac{\partial u_k}{\partial x_i},u_{k,ij}=\frac{{\partial }^2{u}_k}{\partial x_i\partial x_j}$$

and so forth mean a partial differentiation or differentiation in distributions. They are also denoted by another symbol

$$D^{\alpha }u,$$

where $\alpha =({\alpha }_1,{\alpha }_2)$ is a multi–index of a partial derivative. Naturally, $\Delta$ is the Laplace operator. Below, if we do not specifically note the repeated lower indices $i,j,k,m$ mean summation within the boundaries of their changes. For example,

$$u_{i,jm}{u}_{k,im}{u}_{k,j}=\sum _{i,j,m,k=1}^2{u}_{i,jm}{u}_{k,im}{u}_{k,j},$$

$$u_i{u}_{k,i}▵u_k=\sum _{i,k=1}^2{u}_i{u}_{k,i}▵u_k$$

etc. Furthermore, the rotor coordinates are denoted by

$$c_{ki}\left(u\right)=u_{k,i}-u_{i,k}$$

and are interpreted as elements of a skew-symmetric matrix C of the second order.

Typically, a finite field vanishes out of a disk.

2. Main Results

For simplicity, we will confine ourselves to the formulation of the main results for the entire plane and smooth fields. The main results are described by Theorems 1 and 2. Their proof relies on Lemmas 1 and 2. These simple lemmas may be interesting as separate statements.

Lemma 1.

For every three smooth solenoidal plane fields $u,v,w$, the following algebraic identities are true:

$$w_{i,j}{u}_{k,i}{v}_{k,j}+u_{i,j}{w}_{k,i}{v}_{k,j}\equiv 0,$$

(2)

$$w_{i,jm}{u}_{k,im}{v}_{k,j}+u_{i,jm}{w}_{k,im}{v}_{k,j}\equiv 0,$$

(3)

$$u_{i,jm}{u}_{k,im}{w}_{k,j}\equiv 0,$$

(4)

$$D^{\alpha }{u}_{i,jm}{D}^{\alpha }{u}_{k,im}{D}^{\beta }{w}_{k,j}\equiv 0,$$

(5)

$$D^{\alpha }{w}_{i,jm}{D}^{\beta }{u}_{k,im}{D}^{\gamma }{v}_{k,j}+D^{\beta }{u}_{i,jm}{D}^{\alpha }{u}_{k,im}{D}^{\gamma }{v}_{k,j}\equiv 0.$$

(6)

In particular,

$$\Delta u_{i,j}{u}_{k,i}\Delta u_{k,j}+u_{i,j}\Delta u_{k,i}\Delta u_{k,j}\equiv 0,$$

(7)

$$u_{i,jl}{u}_{k,im}{u}_{k,jml}\equiv 0.$$

(8)

Proof of Lemma 1.

The grouping of terms in (2) implies equality

$$w_{i,j}{u}_{k,i}{v}_{k,j}+u_{i,j}{w}_{k,i}{v}_{k,j}$$

$$=2(u_{1,1}^{}{v}_{1,1}^{}{w}_{1,1}^{}+u_{2,2}^{}{v}_{2,2}^{}{w}_{2,2}^{})+(u_{1,2}^{}{w}_{2,1}^{}+u_{2,1}^{}{w}_{1,2}^{})divv+$$

$$(v_{1,2}^{}{w}_{1,2}^{}+v_{2,1}^{}{w}_{2,1}^{})divu+(u_{1,2}^{}{v}_{1,2}^{}+u_{2,1}^{}{v}_{2,1}^{})divw.$$

Therefore, we have the first equality of lemma because $u_{1,1}^{}=-u_{2,2}^{}$, etc.

Fix m. In (2) replacements u by $u_{,m}$, w by $w_{,m}$ imply (3) after summation with respect to m. Choosing in (3) $w=u$ and replacing v with w, we have (4).

Now, in (4), let us replace u by $D^{\alpha }u$, w with $D^{\beta }w$. Then, we have (5). Formula (6) follows from (2) since fields $w_{,m},u_{,m},D^{\alpha }w,D^{\beta }u,D^{\gamma }v$ are solenoidal. Identity (7) as a corollary of (2). Equality (8) is verified in the same way. It is sufficient to group

$$u_{i,jl}{u}_{k,im}{u}_{k,jml}=(u_{1,1l}{u}_{1,1m}{u}_{1,1ml}+u_{2,2l}{u}_{2,2m}{u}_{2,2ml})+(u_{2,1l}{u}_{1,2m}{u}_{1,1ml}+u_{1,2l}{u}_{2,1m}{u}_{2,2ml})+$$

$$(u_{1,2l}{u}_{1,1m}{u}_{1,2ml}+u_{2,2l}{u}_{1,2m}{u}_{1,2ml})+(u_{1,1l}{u}_{2,1m}{u}_{2,1ml}+u_{2,1l}{u}_{2,2m}{u}_{2,1ml}).$$

Here, every sum in brackets vanishes.

Lemma is proved. □

Lemma 2.

For every pair of smooth finite solenoidal plane fields $u,w$, the following integral identities are true:

$${\int }_{R^2}{w}_{i,jm}{u}_{k,im}{u}_{k,j}dx=0$$

(9)

$${\int }_{R^2}{u}_{i,jm}{w}_{k,im}{u}_{k,j}dx=0,$$

(10)

$${\int }_{R^2}\Delta u_{i,j}{u}_{k,im}{u}_{k,jm}dx=0,$$

(11)

$${\int }_{R^2}{D}^{\beta }{w}_{i,jm}{D}^{\alpha }{u}_{k,im}{D}^{\alpha }{u}_{k,j}dx=0,$$

(12)

$${\int }_{R^2}{D}^{\alpha }{u}_{i,jm}{D}^{\beta }{w}_{k,im}{D}^{\alpha }{u}_{k,j}dx=0,$$

(13)

$${\int }_{R^2}{u}_{i,jl}{u}_{k,iml}{u}_{k,jm}dx=0.$$

(14)

Proof of Lemma 2.

For the second integral, we exchange summation indices j and m. Then

$$2{\int }_{R^2}{w}_{i,jm}{u}_{k,im}{u}_{k,j}dx={\int }_{R^2}{w}_{i,jm}{u}_{k,im}{u}_{k,j}dx+{\int }_{R^2}{w}_{i,jm}{u}_{k,ij}{u}_{k,m}dx$$

$$={\int }_{R^2}{w}_{i,jm}{\left(u_{k,m}{u}_{k,j}\right)}_{,i}dx=-{\int }_{R^2}div{w}_{,jm}\left(u_{k,m}{u}_{k,j}\right)dx=0.$$

Equality (9) is proved.

Replace (9) w with u. After that, we replace u by $u+tw$, where t is any number. Then,

$${\int }_{R^2}(u_{i,jm}+t{w}_{i,jm})(u_{k,im}+t{w}_{k,im})(u_{k,j}+t{w}_{k,j})dx=0$$

for all t. Therefore, at the first power t, the coefficient vanishes

$${\int }_{R^2}(w_{i,jm}{u}_{k,im}{u}_{k,j}+u_{i,jm}{w}_{k,im}{u}_{k,j}+u_{i,jm}{u}_{k,im}{w}_{k,j})dx=0.$$

From (9) and (3), it follows that (10). Now, in (9), we take $w=\Delta u$. Integrating by parts, we obtain (11) from (2), where we choose $w=\Delta u,v=u$ and field u in (2) is replaced by $\Delta u$.

If in (9) and (10), we replace w with $D^{\beta }w$, u with $D^{\alpha }u$, then we get (12) and (13). Formula (14) follows immediately from (11) and (8) since

$${\int }_{R^2}\Delta u_{i,j}{u}_{k,im}{u}_{k,jm}dx=-{\int }_{R^2}(u_{i,jl}{u}_{k,iml}{u}_{k,jm}+u_{i,jl}{u}_{k,im}{u}_{k,jml})dx.$$

Lemma is proved. □

Remark 1.

The integral identity (9) holds for spatial solenoidal fields where the integral over plane must be replaced by the integral over whole space.

Theorem 1.

Let $u,v$ be a pair of smooth solenoidal plane fields, and one of them is finite. Then,

(1) a vector field $g^1=(g_1^1,g_2^1)$ where

$$g_k^1=u_{i,k}▵v_i+u_{i,kj}{v}_{i,j}+u_{i,j}{v}_{k,ij},k=1,2,$$

(15)

is potential;

(2) a vector field $g^2=(g_1^2,g_2^2)$ where

$$g_k^2=u_{k,i}▵v_i+v_{k,i}▵u_i+u_{k,ij}{v}_{i,j}+u_{i,j}{v}_{k,ij},k=1,2,$$

(16)

is potential;

(3) a vector field $g^3=(g_1^3,g_2^3)$ where

$$g_k^3=u_{i,k}▵v_i+u_{i,j}{c}_{ki,j}\left(v\right),k=1,2,$$

(17)

is potential;

(4) a vector field $g^4=(g_1^4,g_2^4)$ where

$$g_k^4=c_{ki}\left(u\right)▵v_i+c_{ki}\left(v\right)▵u_i,k=1,2,$$

(18)

is potential.

Proof of Theorem 1.

Equality (2) we integrate over plane and apply integration formula by parts. Removing derivatives of the field w, we obtain

$${\int }_{R^2}(w_i(u_{k,i}▵v_k+u_{k,ij}{v}_{k,j})+w_k{u}_{i,j}{v}_{k,ij})dx=0.$$

(19)

Now, in the first sum, we exchange summation indices $i,k$. Hence,

$${\int }_{R^2}{w}_k(u_{i,k}▵v_i+u_{i,kj}{v}_{i,j}+u_{i,j}{v}_{k,ij})dx=0$$

for every smooth solenoidal field w. We obtain an identity similar to (1). Therefore, from Helmholtz–Weyl decomposition, it follows that vector field $g^1=(g_1^1,g_2^1)$ defining by (15) is potential.

Equality (2) we integrate again, and apply the integration formula by parts, removing derivatives of the field v. Then, we have the following equality:

$${\int }_{R^2}{v}_k(u_{k,i}▵w_i+u_{k,ij}{w}_{i,j}+u_{i,j}{w}_{k,ij}+w_{k,i}▵u_i)dx=0.$$

Exchanging fields v and w, we get

$${\int }_{R^2}{w}_k(u_{k,i}▵v_i+u_{k,ij}{v}_{i,j}+u_{i,j}{v}_{k,ij}+v_{k,i}▵u_i)dx=0$$

(20)

for every smooth solenoidal field w. Hence, Helmholtz–Weyl decomposition vector field $g^2=(g_1^2,g_2^2)$ defined by (16) is potential.

The field from (17) is also potential. This follows from equality

$$g_k^1=g_k^3+{\left(u_{i,j}{v}_{i,j}\right)}_{,k},k=1,2,$$

and the fact that the field $g^1$ is potential.

Now, we assume that a solenoidal field w is smooth and finite. Then, (see Theorem 2 from [10]) we have

$${\int }_{R^2}{w}_i{c}_{ki}\left(u\right)▵u_{k}dx=0,$$

(21)

In (21) let us replace the field u by the field $u+tv$, where v is any smooth solenoidal field and t is any number. Then,

$${\int }_{R^2}{w}_i{c}_{ki}(u+tv)▵(u_k+t{v}_k)dx\equiv 0$$

for every t. Therefore, at the first power, t the coefficient must equal zero. That is

$${\int }_{R^2}{w}_i(c_{ki}\left(u\right)▵v_k+c_{ki}\left(v\right)▵u_k)dx=0$$

for any smooth solenoidal field w. Then, from Helmholtz–Weyl decomposition, the vector field $g^4=(g_1^4,g_2^4)$ defined by formula (18) is potential. The theorem is proved. □

Theorem 2.

Let u be a smooth solenoidal plane field. Then,

(1) a vector field $g^5=(g_1^5,g_2^5)$ where

$$g_k^5=u_{k,ij}▵u_{i,j}+u_{i,jm}{u}_{k,ijm},k=1,2,$$

is potential;

(2) a vector field $g^6=(g_1^6,g_2^6)$, where

$$g_k^6=D^{\beta }(D^{\alpha }{u}_{k,im}▵D^{\alpha }{u}_{i,j}+D^{\alpha }{u}_{i,jm}{D}^{\alpha }{u}_{k,ijm}),k=1,2,$$

is potential;

(3) a vector field $g^7=(g_1^7,g_2^7)$, where

$$g_k^7=D^{\beta }(D^{\alpha }{u}_{k,i}{D}^{\gamma }(▵v_i)+D^{\alpha }{u}_{k,ij}{D}^{\gamma }{v}_{i,j}+D^{\alpha }{u}_{i,j}{D}^{\gamma }{v}_{k,ij}+D^{\alpha }{v}_{k,i}{D}^{\gamma }(▵u_i)),k=1,2,$$

is potential.

Proof of Theorem 2.

Let w be a smooth solenoidal finite field. In equality (10), we fulfill reiterated integration by parts, removing derivatives of the vector field w. Then, repeating arguments from proof of Theorem 1, we have the necessary statement. The second part we obtain from (13). The potentiality of the field $g^7$ follows immediately from (20) if we replace w with $D^{\beta }w$, u with $D^{\alpha }u$, v with $D^{\gamma }v$. The theorem is proved. □

Remark 2.

Integral identities with two solenoidal fields may be very useful for conservation laws, because we can take an initial velocity in the Cauchy problem for Navier–Stokes equations into account.

Remark 3.

Choosing multi–indices α and β and applying item (3) from Theorem 2, we can construct diverse potential fields summing over α and β.

3. Conclusions

For plane solenoidal vector fields, new algebraic and integral identities are proposed. We have new constructions for a potential field as corollary, where two plane solenoidal fields are used. The importance of integral identities for exact a priori estimates is shown in [10].