## Abstract

A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time *t*=0 and at a later time *t*=*T*. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media.

## 1. Introduction

Green & Naghdi (1991, 1992, 1993, 1995*a–d*, 1996) developed an analysis in a rational way to produce fully consistent theories of thermoelasticity which incorporates thermal pulse transmission in a very logical manner, and nonlinear fluid behaviour. We believe that only extensive mathematical and physical analyses of the developments of Green & Naghdi will reveal the usefulness of their theories and it is to this goal that the present paper is addressed.

In the papers cited above, Green & Naghdi developed a theory for describing the behaviour of a continuous body which relies on an entropy balance law rather than an entropy inequality. Their thermodynamics introduces a quantity *T* which is the ‘empirical’ temperature and a term(1.1)which they refer to as the thermal displacement variable. Observe that * x* is the spatial coordinate in the reference configuration of the body. Green & Naghdi (1992, 1993) showed how thermoelasticity may be incorporated within their theory, and several aspects of this branch of thermoelasticity have been explained in Quintanilla & Straughan (2000, 2002, 2004, 2005) and Puri & Jordan (2004). Green & Naghdi (1995

*d*) developed a theory for thermoviscous fluids, and of particular relevance here Green & Naghdi (1996) produced a novel theory of viscous fluids which involves vorticity and spin of vorticity. They allow two temperatures and are motivated by turbulence and the work of Marshall & Naghdi (1989

*a*,

*b*). However, Green & Naghdi (1996) pointed out that their theory is of relevance in its own right. We here restrict attention to the Green & Naghdi (1996) model for isothermal, incompressible viscous flow.

Recently, a new class of non-standard problems have been shown to be relevant to many applied mathematical situations. This is where the data are not given at time *t*=0, but instead as a linear combination at times *t*=0 and *t*=*T*, see Payne & Schaefer (2002), Ames *et al*. (2004*a*,*b*), Payne *et al*. (2004, in press) and references therein. Such problems were originally introduced as a means of stabilizing solutions to the improperly posed problem when the data is given at *t*=*T* and one wishes to compute the solution backward in time, cf. Ames *et al*. (1998), Ames & Payne (1999) and references therein. Accurately solving such problems is essential to many real-world situations, such as image reconstruction from noisy data, see e.g. Ames & Straughan (1997, ch. 8). The work most relevant to the present study is that of Payne *et al*. (2004) who develop bounds for a solution to a non-standard problem for the Navier–Stokes and Stokes equations. The object of this paper is to obtain solution estimates in appropriate measures within the context of the Green & Naghdi (1996) theory of viscous flow given data as a linear combination at *t*=0 and *t*=*T*. We also derive a solution bound for an equivalent problem for the nonlinear Brinkman–Forchheimer equations for flow in porous media. While Payne and his co-workers have successfully studied such problems for wave-like equations and the Navier–Stokes and Stokes equations, we believe this is the first such study in the important area of higher order fluids (as the Green–Naghdi theory is) and in porous media. We obtain solution estimates for the velocity in both the Green & Naghdi (1996) nonlinear theory and for the Brinkman–Forchheimer equations.

## 2. Equations for viscous flow involving higher derivatives

There has recently been tremendous interest in equations which contain higher derivatives than the Navier–Stokes equations with a view to obtaining better regularity than for the Navier–Stokes theory, and concerning turbulence, see e.g. Chen *et al*. (1999), Foias *et al*. (2001, 2002) and the many references therein. These papers compare the higher order equations to the theory of second grade fluids of Dunn & Fosdick (1974). The theory of Green & Naghdi (1996) also contains higher order derivatives than are present in the Navier–Stokes equations, but before discussing this theory we deem it relevant to very briefly review other related theories.

In Bleustein & Green (1967), the theory for a dipolar fluid is developed. This theory is believed capable of describing flow where the fluid may contain long chain molecules; it is shown in Green *et al*. (1965), Bleustein & Green (1967) and Green & Rivlin (1967) how a dipolar theory is equivalent to a director theory and one interpretation of a director is a long molecule. For constant temperature and incompressible flow, the dipolar fluid equations of Bleustein & Green (1967), with the dipolar inertia tensor given by Green & Naghdi (1970) are(2.1)In these equations standard indicial notation is assumed with a superposed dot denoting the material derivative, Δ is the Laplacian, *v*_{i}, *p*, *ρ* and *f*_{i} are the velocity, pressure, (constant) density and body force (*f*_{i} is actually composed of a classical body force and a dipolar one). The constants *d*^{2}, *ℓ*^{2} and *ν* are the micro-inertia coefficient, the micro-length coefficient and viscosity, respectively.

Boundary conditions for this theory are discussed in some detail by Bleustein & Green (1967) and by Green & Naghdi (1968).

In order to facilitate comparison with other models we rewrite the *d*^{2} terms not involving a partial time derivative. These terms are(2.2)where we have expanded the left-hand side and cancelled terms out. The last term in equation (2.2) is a gradient and so we may (and do) absorb it in the pressure. The second term *v*_{i,jk}(*v*_{j,k}−*v*_{k,j}) is composed of a contraction in *jk*, the first of which is symmetric while the second is skew-symmetric and so the whole term is zero. Thus, the *d*^{2} terms not involving a partial time derivative reduce to(2.3)The dipolar equations (2.1) may thus be rewritten, for a modified pressure *p*,(2.4)The properties of a dipolar fluid have recently been the subject of extensive analytical investigation by Jordan & Puri (1999, 2002), Puri & Jordan (1999*a*,*b*) and Akyildiz & Bellout (2004).

The Navier–Stokes-alpha equations, or viscous Camassa–Holm equations are, cf. Foias *et al*. (2001, 2002), dividing by their and without body force,(2.5)The properties of these equations are studied in much detail in Chen *et al*. (1999), Foias *et al*. (2001, 2002) and references therein. Foias *et al*. (2002) is an in-depth study of existence, regularity and attractor properties.

The sets of equations (2.4) and (2.5) are, evidently, essentially the same. By comparing the coefficients *d*^{2}, *ν*, *νℓ*^{2} in (2.4) to , *ν*, in equation (2.5), equation (2.4) is more general in that *d*^{2} and *ℓ*^{2} are independent.

A very related set of equations is derived by Green & Naghdi (1996) for isothermal viscous flow. Their Eq. (57), divided by the constant density *ρ*, are(2.6)where *ρ*, *μ*, *μ*_{1} are positive constants. By comparing the coefficients of equation (2.4) with (2.6), since *ρ*, *μ*, *μ*_{1} are independent, the coefficients in equation (2.6) appear to have the same level of independence as those for the dipolar fluid. Of course, the Green–Naghdi equations (2.6) is different from the dipolar fluid and Camassa–Holm equations because they do not contain the term −*d*^{2}*v*_{j,i}Δ*v*_{j}. (It has been pointed out to us by a referee that in the case of parallel (1-D) flow the dipolar fluid equations, see Jordan & Puri 2002, eq. (2.15), and the Green–Naghdi model yield the same equation of motion.)

In §3, we study equation (2.6) on a bounded domain *Ω* in or . The boundary of *Ω* is denoted by *Γ* and we assume that the body force is zero or conservative, and hence may be absorbed in *p*.

## 3. A non-standard problem for the Green–Naghdi equations

We consider equations (2.6) on the domain *Ω*×(0, *T*) for a fixed time *T*>0. The boundary conditions are(3.1)In equation (3.1) is the couple on the boundary, , where * n* is the unit outward normal to

*Γ*and . The vector

*w*

_{i}is the vorticity, so

*=curl*

**w***, i.e.*

**v***w*

_{i}=

*ϵ*

_{irs}

*v*

_{s,r}. Also, are tangent vectors on the boundary. This second boundary condition is suggested by Green & Naghdi (1996, p. 243). The second boundary condition may be rewritten as(3.2)In addition to equation (3.1) we consider the non-standard conditions (in lieu of initial or final data)(3.3)for

*α*a constant and

*g*

_{i}given. Note that if

*α*=0 we have the backward in time problem for equation (2.6). We consider condition (3.3) rather than the more complicated condition of Payne

*et al*. (2004)

*v*

_{i}(

*T*)+

*ω*

_{i}(0)=

*g*

_{i}, where we have suppressed the

*values and*

**x***=(*

**ω***α*

_{1}

*v*

_{1},

*α*

_{2}

*v*

_{2},

*α*

_{3}

*v*

_{3}) for possibly different

*α*

_{i}. We could consider different

*α*

_{i}at the expense of more technicality.

Now denote *γ*=*μ*_{1}/*ρμ*>0 and *β*=2*μ*_{1}/*ρ*>0. We multiply equation (2.6) by *v*_{i} and integrate over *Ω* using the boundary conditions to find(3.4)where ‖.‖ denotes the norm on *L*^{2}(*Ω*). The boundary condition (3.2) is needed in deriving the *β* term to remove a boundary term involving Δ*v*_{i}∂*v*_{i}/∂*n*. To see this we note that equation (3.2) yield *v*_{3,23}−*v*_{2,33}=0 and *v*_{3,13}−*v*_{1,33}=0 on the boundary whence since * v* is divergence free,

*v*

_{1,33}=0 and

*v*

_{2,33}=0 there. These together with the solenoidal behaviour of

*allows us to show the boundary integral of Δ*

**v***v*

_{i}∂

*v*

_{i}/∂

*n*is zero.

By integration by parts the cubic term on the left of equation (3.4) is seen to be(3.5)For this term is zero. When it is not necessarily zero, although if we employ the dipolar or Camassa–Holm equations the extra term cancels the term in (3.5) out and the cubic term disappears.

We now proceed with for the Green & Naghdi (1996) equations so that the term in expression (3.5) is zero. However, we note that our analysis would work for for the dipolar or Camassa–Holm equations. For one can obtain decay from equation (3.4) for the standard boundary-initial value problem (i.e. without the *u*_{i}(*T*) term in equation (3.3)). To do this we, for example, write(3.6)where a value for *C* may be found in, for example, Straughan (2004, p. 388). An energy analysis then leads to exponential decay of ‖* v*‖

^{2}+

*γ*‖∇

*‖*

**v**^{2}but only for , where

*δ*is a constant defined below. This is conditional decay, i.e. only for a restricted class of initial data.

Proceeding with the condition (3.3) and we put *δ*=2*β*/*γ* when *ν*>*β*/*γ* and *δ*=2*ν* when *ν*<*β*/*γ*. Then, from equation (3.4) we may show(3.7)where *F*=‖* v*‖

^{2}+

*γ*‖∇

*‖*

**v**^{2}and

*D*=‖∇

*‖*

**v**^{2}+

*γ*‖Δ

*‖*

**v**^{2}. Since

*λ*

_{1}‖

*‖*

**v**^{2}≤‖∇

*‖*

**v**^{2}and

*λ*

_{1}‖∇

*‖*

**v**^{2}≤‖Δ

*‖*

**v**^{2}, where

*λ*

_{1}>0 is the first eigenvalue in the fixed membrane problem for

*Ω*, we derive from inequality (3.7)(3.8)Put

*λ*

_{1}

*δ*=2

*ω*and then equation (3.8) leads to(3.9)We do not know the right-hand side of inequality (3.9) and must bound it in terms of

*g*

_{i}. We evaluate inequality (3.9) at

*t*=

*T*and employ equation (3.3) to see that(3.10)With (.,.) denoting the inner product on

*L*

^{2}(

*Ω*), we now use the Cauchy–Schwarz inequality on the (

*,*

**g***(0)) and (∇*

**v***, ∇*

**g***(0)) terms to derive from inequality (3.10)(3.11)To simplify notation put*

**v***=*

**a***, ,*

**g***=*

**c***(0), and . We assume |*

**v***α*|>e

^{−ωT}so that

^{2}>0. Completing the square in inequality (3.11) we may showThe right-hand side of this is positive and so we in turn drop a term from the left and take the square root to findThus,(3.12)Upon using inequality (3.12) in (3.9) we arrive at the estimate(3.13)for 0≤

*t*≤

*T*and . Inequality (3.13) is the required estimate for our non-standard problem.

*We can establish uniqueness for a solution to equations* *(2.6)* *with v*_{i} *and* *given on Γ provided we have bounds on* |Δ* v*|

*and*|

**|*

**v***, where*

***

**v***is a second solution*.

*The proof follows*

*Serrin's*(

*1959*)

*arguments*.

## 4. The Brinkman–Forchheimer equations

We have considered a non-standard problem for a viscous fluid theory of Green & Naghdi (1996). However, it is increasingly being recognized in the partial differential equations literature that studies involving the flow of viscous fluids in porous media may be every bit as useful, cf. Doering & Constantin (1998) and van Duijn & Pop (2004). We now, therefore, study an analogous problem for the Brinkman–Forchheimer equations(4.1)Here *u*_{i}, *p* are the velocity and pressure, and *A*, *λ*, *β* are positive constants.

The Brinkman–Forchheimer equations describe flow in a saturated porous material and are much studied in this context, see e.g. Qin & Kaloni (1998), Nield & Bejan (1999), Straughan (2004) and references therein.

Equations (4.1) are defined on a bounded domain on the time-interval (0, *T*) for some *T*<∞, and the boundary conditions are(4.2)We again consider the non-standard condition(4.3)for *α* a constant.

It is worth observing that for *α*=0, a global solution to equations (4.1)–(4.3) does not exist, i.e. for all time. By transforming *t*→*T*−*t* one may show from equations (4.1) (cf. for example, the arguments in Straughan 1998)(4.4)where *γ*=(*λλ*_{1}+1)/*A*, *k*_{2}=2*β*/*Am*^{1/2}, with *λ*_{1} being the first eigenvalue in the membrane problem for *Ω* and with *m* being the volume of *Ω*. Clearly the right-hand side of inequality (4.4) blows-up at timeand so *u*_{i} cannot exist classically beyond this time. This demonstrates care must be taken with the problem (4.1)–(4.3).

Next, multiply equation (4.1)_{1} by *u*_{i} and integrate over *Ω* using the boundary conditions to find(4.5)We employ the Poincaré inequality and the Cauchy–Schwarz inequality to find . Then from equation (4.5) with *E*(*t*)=‖* u*(

*t*)‖

^{2}we may find(4.6)whereWe integrate inequality (4.6), and with

*δ*=

*c*

_{1}/2 find(4.7)0≤

*t*≤

*T*.

Again, we do not know *u*_{i}(0) and so must estimate ‖* u*(0)‖. We wish to retain the

*c*

_{2}term, since this contains the Forchheimer effect (the

*β*term).

To find a lower bound for ‖* u*(0)‖ we put

*f*(

*t*)=

*c*

_{2}(1−e

^{−δt})/

*c*

_{1}, then from inequality (4.7)Use equation (4.3), and then(4.8)Note thatandUpon use of these expressions in inequality (4.8) we may show(4.9)To derive an upper bound for ‖

*(0)‖ observe that from inequality (4.7),Squaring and writing*

**u***u*

_{i}(

*T*)=

*g*

_{i}−

*αu*

_{i}(0) together with the Cauchy–Schwarz inequality now leads to, cf. Payne

*et al*. (2004),From this we now find for |

*α*|>e

^{−δT},(4.10)We finally employ inequalities (4.9) and (4.10) in inequality (4.7) to derive the estimate(4.11)|

*α*|>e

^{−δT}, 0≤

*t*≤

*T*.

*While the bound in inequality* *(4.11)* *is not optimal*, *the system of equations* *(4.1)* *is nonlinear*. *Thus,* *any bound which yields useful information is likely to be important*. *If we instead consider the equivalent problem for the Brinkman equations,* *i.e. take β*=0 *in equations* *(4.1)**, we may find an optimal estimate*. *We do not give details since they follow very closely the arguments of* *Payne et al*. *(2004)* *for the Stokes problem*. *The only difference is the addition of the* −*u*_{i} *term in equations* *(4.1)*. *The Lagrange identity argument of* *Payne et al*. *(2004)* *carries over as does their non-uniqueness proof,* *mutatis mutandis,* *cf*. *Payne et al*. *(2004, pp. 2049–2053)*.

## Acknowledgments

This work was supported by the project ‘Aspects of stability in thermomechanics’ BFM 2003-00309 of the Spanish Ministry of Science and Technology.

## Footnotes

- Received January 14, 2005.
- Accepted April 26, 2005.

- © 2005 The Royal Society