A Some comparison lemmas of the Čaplygin type

On approximate solutions of the incompressible Euler and Navier-Stokes equations

Carlo Morosi, Livio Pizzocchero(1)

Dipartimento di Matematica, Politecnico di Milano,

P.za L. da Vinci 32, I-20133 Milano, Italy

e–mail: carlo.morosi@polimi.it

Dipartimento di Matematica, Università di Milano

Via C. Saldini 50, I-20133 Milano, Italy

and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy

e–mail: livio.pizzocchero@unimi.it

We consider the incompressible Euler or Navier-Stokes (NS) equations on a torus , in the functional setting of the Sobolev spaces of divergence free, zero mean vector fields on , for . We present a general theory of approximate solutions for the Euler/NS Cauchy problem; this allows to infer a lower bound on the time of existence of the exact solution analyzing a posteriori any approximate solution , and also to construct a function such that for all . Both and are determined solving suitable “control inequalities”, depending on the error of ; the fully quantitative implementation of this scheme depends on some previous estimates of ours on the Euler/NS quadratic nonlinearity [15] [16]. To keep in touch with the existing literature on the subject, our results are compared with a setting for approximate Euler/NS solutions proposed in [3]. As a first application of the present framework, we consider the Galerkin approximate solutions of the Euler/NS Cauchy problem, with a specific initial datum considered in [2]: in this case our methods allow, amongst else, to prove global existence for the NS Cauchy problem when the viscosity is above an explicitly given bound.

Keywords: Navier-Stokes equations, existence and regularity theory, theoretical approximation.

AMS 2000 Subject classifications: 35Q30, 76D03, 76D05.

1 Introduction

In recent years, there has been some activity about approximate solutions of the Euler and Navier-Stokes (NS) equations, viewed as tools to infer accurate a posteriori estimates on the exact solutions. We mention, in particular: the works by Chernyshenko et al. [3], Dashti and Robinson [4], Robinson and Sadowski [18], and our papers [12] [13] [14]. The present work seats within the same research area; here we consider the incompressible Euler/NS equations

(1.1)

where: is the divergence free velocity field; the space variables belong to the torus (and yield the derivatives ); is the Laplacian; (); is the Leray projection onto the space of divergence free vector fields; is the viscosity coefficient, so that in the Euler case and in the NS case; is the Leray projected density of external forces. The dimension is arbitrary in the general setting of the paper, but we put in a final application.

The functional setting that we consider for Eq. (1.1) relies on the Sobolev spaces

(1.2)

with indicating the mean over ; for any real , the above space is equipped with the inner product and with the corresponding norm . One of the main issues in this setting is the behavior of the bilinear map

(1.3)

in the above mentioned Sobolev spaces. It is well known that there are positive constants and fulfilling the “basic inequality”

(1.4)

and the so-called “Kato inequality”

(1.5)

fully quantitative upper and lower bounds on and were derived in our previous works [15] [16], for reasons related to the present setting and described more precisely in the sequel.

Independently of the problem to estimate and , the above two inequalities play a major role in the very interesting paper [3] on approximate Euler/NS solutions and a posteriori estimates on exact solutions. To give an idea of the framework of [3] we describe a result therein, using notations closer to our setting.

Consider the Euler/NS equation (1.1) with a specified initial condition ; let be an approximate solution of this Cauchy problem. Given (and assuming suitable regularity for ), let possess the differential error estimator , the datum error estimator and the growth estimators ; this means that, for ,

(1.6)
(1.7)
(1.8)

(with , etc.). According to [3], Eq. (1.1) with datum has an exact (strong, -valued) solution on a time interval , if (with the estimators for ) fulfills the inequality

(1.9)

The present work aims to refine, to some extent, the approach of [3] and to apply it to get fully quantitative estimates on the exact solution of the Euler/NS Cauchy problem on , with some specific initial datum. Our main result can be described as follows: assuming suitable regularity for , and intending as above, suppose there is a function , with , fulfilling the control inequalities

(1.10)

(with the right upper Dini derivative, see Section 2). Then, the solution of the Euler/NS equation (1.1) with initial datum exists (in a classical sense) on the time interval , and its distance from the approximate solution admits the bound

(1.11)

Some features distinguishing our approach from [3] are the following ones.

(i) Differently from (1.9), our control inequalities (1.10) depend explicitly on and thus could allow a more accurate analysis of the influence of viscosity on the regularity of the Euler/NS solutions.

(ii) Our approach promises better lower bounds on the time of existence of . For example, for and under specific assumptions illustrated in the paper, the inequalities (1.10) have solutions with , implying the global nature of the NS solution ; on the contrary, if or are nonzero the inequality (1.9) cannot have a solution with very large , since the right hand side is bounded by and thus vanishes for .

(iii) In [3] there is not an explicit bound on the distance between and , such as (1.11) (however, our analysis yielding (1.11) is greatly indebted to [3] and, in a sense, it mainly refines and completes a chain of inequalities for appearing therein).

(iv) The constants and in the inequalities (1.4) (1.5) are not evaluated in [3]. On the contrary, here we have at hand our previous results [15] [16] on these constants; thus, in specific applications, we can implement the control inequalities (1.10) and their outcome (1.11) in a fully quantitative way.

As an example of our approach, in the final part of the paper we consider the Euler/NS equations on with a specific initial datum . Independently of the approach developed here, this datum has been already considered in an interesting paper by Behr, Neas and Wu [2], where it is indicated as the origin of a possible blow-up for the Euler equations. However, in the cited work the blow-up is conjectured on the grounds of a merely “experimental” analysis of a finite number of terms in the power series solving formally the Euler Cauchy problem.

In the present work, dealing with the initial datum of [2] both for and for , a different approach to the Cauchy problem is developed using the familiar Galerkin approximation (with a convenient set of Fourier modes), combined with our general setting for approximate solutions based on the control inequalities (1.10); in this case, the Sobolev order is , is the Galerkin solution and we use for it the required estimators, to be substituted in the control inequalities (1.10) (with the values for the constants and obtained in [15] [16]). We search for a solution fulfilling Eqs. (1.10) as equalities (i.e., with replaced by ); this gives rise to an ordinary Cauchy problem for , which is solved very easily and reliably by numerical means. Admittedly, our computations are preliminary: they were performed using MATHEMATICA on a PC, with a fairly small set of Fourier modes for the Galerkin approximation; we plan to develop the same approach with more powerful computational tools in a subsequent work.

In a few words, our results are as follows: in the case , the solution of the control equations (1.10) exists on a finite time interval (after which it blows up); so, we can grant existence for the Euler Cauchy problem on the interval , where we also have the estimate (1.11) on the distance between the exact solution and the Galerkin approximate solution . (Unfortunately, is less than the blow-up time suggested in [2] for the Euler Cauchy problem, so we cannot disprove the conjecture of the cited paper; the situation could change using many more Galerkin modes, which is our aim for the future). For , the situation is similar: blows up in a finite time , and we can grant existence for the NS Cauchy problem only up to . On the contrary, for , our approach grants global existence for the NS Cauchy problem (and a bound of the type (1.11) on the full interval ).

To conclude this Introduction, let us describe the organization of the paper. In Section 2 we present some preliminaries: these concern mainly the Sobolev spaces on , in view of their applications to the Euler/NS equations (1.1). In Section 3 we define formally the Euler/NS Cauchy problem, the general notion of approximate solution for this problem and the related error estimators. In Section 4, that contains the main theoretical results of the paper, we develop the general framework yielding the control inequalities (1.10), and prove the estimate (1.11) on the distance between the exact solution of the Euler/NS Cauchy problem and an approximate solution (here we also give more details on the connections of the present work with [3]). In Section 5 we present some analytical solutions of the control inequalities (1.10), under specific assumptions for their estimators and supposing, for simplicity, that the external forcing in (1.1) is zero; as anticipated, in certain cases our analytical solutions for the control inequalities are global, thus ensuring global existence for the Euler/NS Cauchy problem. In Section 6 we describe the general Galerkin method for (1.1); in particular, we give error and growth estimators for the Galerkin approximate solutions, to be used with our control inequalities (1.10). In Section 7 we consider the Galerkin method with the initial datum of [2], both for an for . In Appendix A we review some comparison lemmas of the Čaplygin type about differential inequalities; these are employed in Section 4 in relation to the control inequalities. In Appendix B, for completeness we report the proof of an essentially known statement on the Galerkin approximants for the Euler/NS Cauchy problem.

2 Preliminaries

Dini derivatives. Consider a function

(2.1)

(with ). The right, lower and upper Dini derivatives of at any point are, respectively,

(2.2)
(2.3)

Of course,

(2.4)

furthermore, the opposite function is such that

(2.5)

The left, lower and upper Dini derivatives , are defined similarly, with ; however, left derivatives are not used in this paper. Of course, all Dini derivatives coincide with the usual derivative if this exists. Sobolev spaces of vector fields on the torus; Laplacian, Leray projection, and so on. We work in any space dimension (using as indices in ). For in we put and write for the complex conjugate -tuple ; carries the inner product and the norm . We often restrict the previous operations to .

We consider the torus , i.e., the product of copies of ; a point of is generically written as . In the sequel we often refer to the space

(2.6)

of the real distributions on , and to the space

(2.7)

Elements of can be interpreted as “generalized functions ”; in the sequel, we call them (distributional) vector fields on . and will be equipped with their weak topologies. For more details on distributions (and on the function spaces mentioned in the sequel) we refer, e.g., to [14].

Using distributional derivatives, we can give a meaning to several differential operators acting on vector fields, e.g. the Laplacian and the divergence . Any real distribution has a mean (the right hand side in this definition indicates the action of on the constant test function ). Passing to a vector field , we can define componentwise the mean .

Each vector field has a unique (weakly convergent) Fourier series expansion

(2.8)

of course, the -th component of is (i.e., it equals the action of on the test function ). Due to the reality of , the Fourier coefficients have the property for all ; one has .

In the sequel we often refer to the space of zero mean vector fields, of the divergence free (or solenoidal) vector fields and to their intersection; these are, respectively,

(2.9)

Elements of and are characterized, respectively, by the conditions and for all ; handling with the Fourier components of vector fields in , it is convenient to put

(2.10)

Of course, for each one has ; this suggests to define, for any ,

(2.11)

We denote with the space of square integrable vector fields ; this is a real Hilbert space with the inner product and the norm . For any , let us consider the Sobolev space

(2.12)

this is a real Hilbert space with the inner product and the norm

(2.13)
(2.14)

Clearly, implies and . In this paper, we mainly fix the attention on the divergence free Sobolev space

(2.15)

(); this is a closed subspace of , and thus a real Hilbert space with the restriction of .

For let us consider the space of vector fields of class in the ordinary sense; if , equip this with the norm

(2.16)

There is a well known Sobolev imbedding

(2.17)

involving suitable positive constants .

For arbitrary , we have

(2.18)

and is continuous between the above spaces; furthermore,

(2.19)

(all the above statements are made evident by the Fourier representations).

By definition, the Leray projection is the map

(2.20)

here is the orthogonal projection of onto (given explicitly by and , for and ). For each real , we have a continuous linear map

(2.21)

(which, in fact, is the orthogonal projection of onto the subspace ). The fundamental bilinear map in the Euler/NS equations. In this subsection we assume . With this condition, we have a continuous bilinear map , , where is the vector field on with components . By composition with the Leray projection , we obtain a continuous bilinear map

(2.22)

This appears in the Euler/NS equations, and is referred in the sequel as the fundamental bilinear map for such equations. As is known, for all and of Fourier components and , has Fourier components and

(2.23)

for all , where is the already mentioned projection of onto .

The continuity of is equivalent to the existence of a constant such that

(2.24)

we refer to this as the basic inequality about . With the stronger assumption , it is known that there is a constant such that

(2.25)

we call this the Kato inequality, since it originates from Kato’s seminal paper [6] (for completeness, we mention that for as above). In our previous works [15] [16], we derived upper and lower bounds for the sharp constants in the above inequalities; throughout this paper, and are any two constants fulfilling Eqs. (2.24) (2.25).

3 The Cauchy problem for the Euler/NS equations: exact and approximate solutions

From here to the end of Section 6 we fix any space dimension , we consider the Sobolev spaces of vector fields on , and we choose a real number such that

(3.1)

Euler and NS equations: the Cauchy problem. Let us choose a “viscosity coefficient”

(3.2)

a “forcing”

(3.3)

and an initial datum

(3.4)
3.1

Definition. The Cauchy problem for the (incompressible) fluid with viscosity , initial datum and forcing is the following:

(3.5)

(with , depending on ). As usually, we speak of the “Euler Cauchy problem” if , and of the “NS Cauchy problem” if .

3.2

Remark. (i) The map sends continuously into , while sends continuously into ; this explains the appearing of in the previous definition, at least in the NS case. In the Euler case , is absent from (3.5); so, in the previous definition and in the subsequent theoretical developments one could systematically replace with ; this is not done just to avoid tedious distinctions between the Euler and the NS case.

(ii) Consider any function . By the Sobolev imbedding (2.17), one has

(3.6)

The following results are well known, and reported for completeness.

3.3

Proposition. For any and as above, (i)-(iii) hold.

(i) The Cauchy problem (3.5) has a unique maximal (i.e., nonextendable) solution, hereafter denote with , with a suitable domain (). All the other solutions of (3.5) are restrictions of .

(ii) has the property (3.6). Furthermore, if and the forcing is from to , the function is as well from to .

(iii) For any consider the vorticity matrix , of elements (); let . If , one has

(3.7)

this implies

(3.8)

Proof. (i) See [7].

(ii) The property (3.6) holds because . For and of class on , the same smoothness property is granted for by Theorem 6.1 of [5] (with the domain considered therein replaced by ).

(iii) Let . Eq. (3.7) is the celebrated Beale-Kato-Majda blow-up criterion [1] (see also [8]). To prove (3.8) we note that, for each ,

(3.9)

here the first inequality follows from the definition of in terms of the derivatives , and the second one from the Sobolev imbedding (2.17) (with ). The relations (3.7) and (3.9) immediately give Eq. (3.8). Approximate solutions. Our treatment uses systematically the present terminology.

3.4

Definition. An approximate solution of the problem (3.5) is any map (with ). Given such a function, we stipulate (i) (ii).

(i) The differential error of is

(3.10)

the datum error is

(3.11)

(ii) Let . A differential error estimator of order for is a function

(3.12)

Let , . A datum error estimator of order for is a real number

(3.13)

a growth estimator of order for is a function

(3.14)

In particular the function , the number and the function will be called the tautological estimators of order for the differential error, the datum error and the growth of .

4 Main theorems about approximate solutions

Assumptions and notations. Throughout this section we fix a viscosity coefficient, a forcing and an initial datum as in Eqs. (3.2)–(3.4) (recalling the condition (3.1) ). We consider for the Cauchy problem (3.5) an approximate solution

(4.1)

in the sequel and are differential error and datum error estimators of order for , while are growth estimators of orders (see Definition 3.4). Finally, we denote with

(4.2)

the maximal solution of (3.5) (typically unknown, as well as ). Some lemmas. For the sake of brevity, we put

(4.3)

Furthermore, we introduce the function

(4.4)

this is clearly continuous, and in a neighborhood of any instant such that . In the sequel we often consider the right, upper Dini derivative (see Eq. (2.3)), that is just the ordinary derivative at any with .

The forthcoming two lemmas review and partially refine some results of [3] (after adaptation to our slightly different setting; for example, Dini derivatives are not even mentioned in [3]).

4.1

Lemma. One has

(4.5)
(4.6)

Proof. By definition of the differential error , we have

(4.7)

of course, whence, writing ,