"Restricted Euler Equations" Model Is Not Very Suitable for Revealing Properties Inherent to Euler Equations
Vladimir A. Dobrynskii*
Institute for Metal Physics of N.A.S.U., Kiev, Ukraine
For the restricted Euler equations, we prove the following: 1) If there is a time instant such that the perfect vector alignment between the vorticity vector and the strain matrix eigenvector happens at some point inside the 3D incompressible restricted Euler flow, it continues then permanently and keeps forever in the sense that one happens successively at all points which belong to the given point trajectory generated by the flow. 2) If the aforementioned trajectory exists, then, depending on initial data, one can either blow up for a finite time or not. What is interesting in doing so is that the aforesaid finite-time blow ups can be various and very different. In particular, we found the blow up such that the strain matrix eigenvalues all go at infinity (which can be both positive and negative) whereas the vorticity remains bounded at the same time. Such kind the vorticity behavior should be considered, generally speaking, as slightly unexpected if to take into account the well-known Beale-Kato-Majda (abbr. BKM) criterion for solutions of the genuine Euler equations to blow up at a finite-time. On the other hand, there are solutions to the restricted Euler equations which blow up at a finite time by a scenario similar to that is described in the BKM criterion. Summarizing all what is the aforesaid we see that the restricted Euler equations are not suitable enough for revealing properties inherent to the Euler equations.
Finite-Time Blow up, the Euler Equations, the Restricted Euler Equations, Alignment, Vorticity, Strain, Vortex Stretching
Received: July 2, 2015
Accepted: August 3, 2015
Published online: August 13, 2015
@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/
The problem of global unambiguous solvability for the 3D Euler equations of ideal incompressible fluid as well as that of whether the 3D incompressible Euler equations solution can develop a finite time singularity from smooth initial data remain the most challenging open problems in the ideal incompressible fluid mathematical theory. Their main difficulty consists in that a dependence of pressure in the fluid flow on its velocity is non-local. In order to overcome this difficulty, a so-called model of the "restricted Euler equations" is proposed. (For the model definition, see the following paper text.) In this model, the original pressure term of the Euler equations is changed by a certain local expression. The model idea goes back to Viellefosse [1, 2], Novikov  and Cantwell . Later, the given model is studied in [5, 6, 7] by Liu, Tadmor and Wei. This paper prolongs exploration of the restricted Euler equations properties which was begun in .
We start of the Cauchy problem for the 3D incompressible Euler equations in although results presented below all remain valid for such of the kind flows inside open both bounded and unbounded subsets of having smooth boundary as well as for the periodic flows in a torus. So, we consider the following problem:
where is the material derivative, is the gradient operator with respect to , is the given initial flow velocity satisfying Here and in what follows 3D vectors and scalars are designated by bold letters and ordinary ones respectively. That is ,and. As a rule, we shall omit the designation on the space-time co-ordinates if those are designated by and such the omission will not lead to misunderstanding, and, on the contrary, we shall point out such the dependence otherwise. For the given velocity and pressure let us introduce the matrices
where and is the Kronecker delta symbol, i.e. when
and otherwise. We recall that is related to the vorticity
by means of the formulas
where is the Eddington alternating skew-symmetric tensor with the normalization . Other designations we use are following. Operations of a vector (external) product and scalar (inner) one are designated by the signs and respectively. Thus, for any and ,
. We shall denote the vector by .
In what follows we shall use notions of the fluid particle trajectory, the Lagrangian particle marker, the Lagrangian variable also. Let us set
Definition. The function of is said to be the fluid particle trajectory of the flow if it is a unique solution of the following differential equation:
where is the classical solution of Eq.(1) such that.
The pair is called the Lagrangian variables. Here is the Lagrangian particle marker and is time. A totality of all the solutions of Eq.(2) generates a fluid particle trajectory mapping Since transition from the Euler fluid flow discription to the Lagrangian one results in changing of the variable on the variable , it is suitable to call the Lagrangian variable also. In what follows we shall assume always that and are considered for the maximal time interval of existence of the classic (strong) solution of Eq.(1) and shall not indicate a function dependence on the Lagrangian variables if it will not generate ambiguity and misunderstanding.
3. Main Results
Given ,we define a vorticity direction field by the formula
where .Let For , we define then the following scalar fields
where by is designated the angle between and . Notice that is well-defined provided that
because Since we see
Proof. This formula is a corollary of that for the two-fold vector product. Indeed, according to the two-fold vector product formula
. Taking a scalar product of both the given equality sides with we find that
Let us derive differential equations for and . To do this let us compute the material derivatives of and .Keeping in mind that and , we obtain, on one hand, that
and, on the other hand, that
Similarly, we find, on one hand, that
and, on the other hand, that
Substituting instead of in Eq. (4) and subtracting Eq.(3) from the obtained equality term-wise we get finally the following differential equation
Definition. Two non-zero vectors of and are said to be aligned perfectly if ,i.e. when their vector (external) product is equal to zero.
Considering Eq.(5) one can easily see that can be a solution to Eq.(5). Keeping in mind one can easily find the differential equation for :
Considering Eq.(6) one can easily see that can be solutions to Eq.(6). In particular, this is the case when and are aligned perfectly. Indeed, by definition, the non-zero vectors and are aligned perfectly provided that , i.e. when .
To define a notion of the restricted Euler equations, let us compute all
partial derivatives of the first equation of Eq.(1). In doing so we obtain the following equality
Taking the symmetrical and antisymmetrical parts of Eq.(3) we find that
In general the Hessian is the non-local non-isotropic matrix. Computing
of Eqs. (7, 8) we find that , where is a trace of the matrix .
Definition. If to make violence and to assume to be local and isotropic of the kind , we obtain then a model which is now called the "restricted Euler equations" .
Here, the elliptic pressure constraint,,is concerned solely with the diagonal elements of .
Let us study the restricted Euler equations. Since, for any and , , is the eigenvector of .
The latter implies in turn that identically, for all and .1
This leads to considerable simplification of Eqs.(5,6) which take the following form:
If to use the Lagrangian variable, we can then rewrite Eq. (9) legitimately as
Integrating Eq. (10) we obtain two families of solutions of the kind
where It is evident from Eq.(11)that for provided as well as
for provided too. Thus we proved the following.
Proposition 1. Given the restricted Euler equations. If there is such that are aligned perfectly, then are aligned perfectly forever, for all admissible .
Let be the eigenvectors of associated with its eigenvalues There is a generally accepted convention respecting ordering of It is not assumed in what follows. It is not difficult to show that if is an eigenvector for i.e., for example, then, in the frame of the operators of and take the following looks:
In order to prove what is just aforesaid let us find now what is a matrix form of in the orthonormal frame of eigenvectors of provided that is one of the eigenvectors of .
Lemma 2. Given be an eigenvector for associated with its eigenvalue . Then , are the very same for .
Lemma 3. Let be an eigenvector for or .Then is the same for .
Proof of the lemmas. First of all notice that Therefore, due to the definition of , if is the eigenvector for associated with its eigenvalue , the one is the eigenvector for associated with the very same eigenvalue and vice versa.
Remark. Keeping in mind that the eigenvalues of all are real we can see that if is the eigenvector for or, then one is associated to the mutual real eigenvalue of and.
It can be easily to check that there is a rotation symmetry of solutions to the restricted Euler equations in the sense of if is a solution pair to the restricted Euler equations, then is the same pair for any real orthogonal matrix (i.e. such that ). It is well-known also (see, e.g., , p. 258) that any real symmetric matrix is orthogonally similar to the diagonal one. Therefore, without loss of generality, we can assume that the aforementioned coordinate transformation is made from the very beginning and the Cartesian coordinates are chosen in such a manner that they coincides accurate to translation on some constant vector with the eigenvector frame of and in doing so the vorticity direction coincides, for instance, with that of the -axis, i.e. that It is evident that, in the given frame,is the diagonal matrix whose non-zero entries are Keeping in mind this and that one can specify better entries of in such a manner: Since is the eigenvector for and, we find, as a corollary, that and also that Therefore, in the frame under consideration, the matrices should have in general the following look:
Remark. We notice attention that although the matrices and have a canonical look in the eigenvector frame of , the ones are not normal. It can be easily checked if to compute products and to compare them then.
Really, keeping in mind that , we can rewrite the first equation of Eq.(8)in such a manner:
Remark. It is worth to give attention that Eq.(12) is distinctly
distinguished from that presented in [5,7] (see, for example, Eq.(6.2) or Eq.(1.6)). This difference occurs because of different objects under study: we consider the strain matrix while the authors of [5-7] examine the velocity gradient matrix .
Proposition 2. Given the restricted Euler equations. If i.e. is the eigenvector for associated with its eigenvalue, then the dynamics of is governed by Eq.(12).
Proof. by definition. Differentiating this relation with respect to $t$ we find Computing a scalar product of the given equality terms with we obtain . Differentiation of the same relation with respect to leads to
Multiplying this equality by first and taking then a scalar product with we have Therefore,
Observing that and that as well as and combining these facts with ones stated above one can easily see due to Eq.(13) that the proposition statement is valid in fact.
Subtracting the last equation of Eq.(12) from the previous one termwise we find after easy transformations that
Integrating this equation we obtain
On the other hand, recalling that and taking into account that is the eigenvector for associated with its eigenvalue we see that . Using the Lagrangian variables converts it to the following . Integrating of the latter gives
Comparing Eqs.(14-15) one can see that
Thus, in the case under consideration, dynamics of is the very same as that of difference of the matrix eigenvalues which are not associated with Substituting instead of in Eq.(16) we find that
If to substitute the expressions of and involving and in the first two equations of Eq.(12) only we obtain the next ones:
When we can easily integrate the first equation of Eq.(17).
We see that does not blow up for a finite time provided and blows up at for any . In doing so as
. But what is interesting is as .
Unfortunately, we can say nothing respecting whether such the behavior of contradicts to the celebrated BKM criterion  or not because the latter involves norms of in the Sobolev space and integrals with respect to time of those of in the space. Let us pay attention also to that enlarges at infinity as although the difference remains bounded up to time . It is a corollary of and the behavior of stated above.
Let us change a variable setting In doing so Eq.(17) takes the look
Multiplying the former of Eq.(19) on the latter crosswise we obtain or, otherwise,
Therefore, there is a first integral
where We take into account here that and Solving Eq.(20) we find
Substituting the given expression in Eq.(19) we obtain
When we can easily integrate the first equation of Eq.(21).
Namely: We see that has no blow up for a finite time provided and blows up at for any . The same takes place for and too.
What is the aforesaid can be summarized in view of the following.
Proposition 3. Given the restricted Euler equations. If there is
such that are aligned perfectly, then, for certain initial conditions, the eigenvalues of lose their initial smoothness for a finite time by means of the so-called "finite-time blow up". As for the vorticity, it can as undergo the similar blow up at the same time as cannot.
Thus, an approach based on studying of the perfect alignment between the vorticity vector and the strain matrix eigenvectors permits us to deepen some results in  and to specify scenarios of the finite time blow ups of the restricted Euler equations solutions. To be more precise, Lemmas 6.1 - 6.2  inform that the aforesaid solutions blow up at a finite time for almost all initial gradient matrix But they say nothing how it occurs and what happens at the blow up time. The principal results presented above describe two concrete very different scenarios of the aforesaid blow ups.
Remark. We would like to attract a reader attention once more to that a
situation when the eigenvalues of blow up for a finite time and in doing so the vorticity does not is, generally speaking, unexpected if to take into account the celebrated BKM criterion for solutions to the (genuine!) Euler equations to blow up at a finite time.
Summarizing all what is the aforesaid respecting the blow ups of solutions to the restricted Euler equations we can infer that their properties differ from those of solutions to the genuine Euler equations very essentially. Really, as is just marked above there are parameter data and initial conditions such that the eigenvalues of the strain matrix for the restricted Euler equations blow up for a finite time and in doing so their vorticity does not and vice versa. However, such kind a behavior is completely not possible for the Euler equations because, as it was shown by G.Ponce in , a finite-time blow up of the Euler equations vorticity accompanies the very same one of the strain matrix and, as a corollary, of its eigenvalues and vice versa. Therefore, the model under study cannot be considered as suitable one for recognizing and study of properties inherent to the Euler equations solutions. Nevertheless, in itself it is of considerable interest because exhibits unexpected features.
1 Indeed, taking into account that according to the two-fold vector product formula, what is the aforesaid can be reformulated with aid of the Lagrangian variables as follows: for all and ,