# Symplectic and variational formulations
of compressible and incompressible Navier-Stokes equation

G. de Saxcé
Univ. Lille, CNRS, Centrale Lille, UMR 9013 – LaMcube
Laboratoire de mécanique multiphysique multiéchelle,
F-59000, Lille, France 111 Email address for correspondence: gery.de-
<EMAIL_ADDRESS>
###### Abstract

In a previous paper, we proposed a symplectic version of Brezis-Ekeland-
Nayroles principle based on the concepts of Hamiltonian inclusions and
symplectic polar functions. We illustrated it by application to the standard
plasticity in small deformations. The object of this work is to generalize the
previous formalism to dissipative media in large deformations and Eulerian
description. This aim is reached in three steps. Firstly, we develop a
Lagrangian formalism for the reversible media based on the calculus of
variation by jet theory. Next, we propose a corresponding Hamiltonian
formalism for such media. Finally, we deduce from it a symplectic minimum
principle for dissipative media and we show how to obtain a minimum principle
for unstationary compressible and incompressible Navier-Stokes equation.

Keywords: Dynamical dissipative systems; Hamiltonian methods; Brezis-Ekeland-
Nayroles principle; convex dissipation; Navier-Stokes equation

##  1 Introduction

This paper was written out to mark the occasion of the 200${}^{\,th}$ birthday
of Navier’s works that spearheaded the Navier-Stokes equation.

Among the tools of the differential geometry, one of the most used is the
geometry of the Riemannian spaces, equipped with a symmetric 2-covariant
tensor, the metric

$g(dz,dz^{\prime})=dz^{T}K\,dz^{\prime}$

allowing to measure lengths, but there is another interesting geometry, the
one of the symplectic space, equipped with an antisymmetric 2-covariant tensor

$\omega(dz,dz^{\prime})=dz^{T}J\,dz^{\prime}$

allowing to measure areas ([35], [57]). One of the origins of the symplectic
geometry is the dynamics of reversible systems. We are working with the phase
space of which the elements are of the form

$z=\left[\begin{array}[]{c}\xi\\\ \eta\\\ \end{array}\right]$

where $\xi$ are the degrees of freedom and $\eta$ are the corresponding
momenta. It is equipped with the canonical symplectic form

$\omega(dz,dz^{\prime})=\left[d\xi,d\eta\right]\left[\begin{array}[]{cc}0&I\\\
-I&0\\\ \end{array}\right]\,\left[\begin{array}[]{c}d\xi^{\prime}\\\
d\eta^{\prime}\\\ \end{array}\right]$

where $I$ is the identity operator. Introducing the Hamiltonian vector field
(or symplectic gradient)

$\dot{z}=X_{H}=J\cdot\nabla_{z}H(t,z)$

restitues the canonical equations

$\dot{\xi}=\nabla_{\eta}H,\qquad\dot{\eta}=-\nabla_{\xi}H$

Considering the lagrangian $L$ associated to the Hamiltonian $H$, this
formulation can be recast in integral form thanks to

Hamilton’s variational principle: the natural evolution of the systems
minimizes the action

$\alpha\left[\xi\right]=\int^{T}_{0}L(\xi,\dot{\xi},t)\,\mbox{ d}t$

among the paths $t\mapsto\xi(t)$ satisfying

$\xi(0)=\xi_{0},\quad\xi(T)=\xi_{T}$

It is worth to remark that the application of this principle is strictly
limited to the reversible systems but unfortunately it fails for dissipative
ones. For dissipative systems, several authors proposed unified frameworks
inspired from both overmentioned geometries. We quickly review now these
theoretical frameworks:

  * •

The GENERIC systems (General Equation for Non-Equilibrium Reversible-
Irreversible Coupling) were introduced by Grmla and Öttinger ([25], [46]).
They combine the Hamiltonian formulation and the Onsager one, according to the
evolution law

$\dot{z}=J\cdot\nabla_{z}H(z)+K\cdot\nabla_{z}S(z)$

where Onsager term is built from a symmetric and positive-definite operator
$K$ and an entropy-like function $S$. A variational formulation of GENERIC can
be found in [37].

  * •

The Port-Hamiltonian systems were introduced by Brockett [4] and van der
Schaft [61]

$\dot{z}=(J-R)\cdot\nabla_{z}H(z)$

where the symmetric and positive-definite operator $R$ modelizes the resistive
effects (because of the minus sign).

  * •

The rate-independent systems proposed by Mielke and Theil [39], Mielke [40]
and developped with applications in Mielke and Roubíček [41], are based on two
fundamental conditions:

    * –

The stability condition: $\nabla_{\xi}E(\xi)\cdot w+\Phi(w)\geq 0,\;\forall w$

    * –

The power balance: $\nabla_{\xi}E(\xi)\cdot\dot{\xi}+\Phi(\dot{\xi})=0$

where $E$ in the energy functional and $\Phi$ is a 1-homogeneous dissipation
potential depending on the velocity.

  * •

The Hamiltonian inclusions were proposed by Buliga [5],

$\dot{z}=J\cdot\nabla_{z}H(z)+J\cdot\nabla_{\dot{z}}\Phi(\dot{z})$ (1)

where $\Phi$ is a convex dissipation potential. Because of the first term, It
is clearly related to the two former formalisms (GENERIC and Port-Hamiltonian
systems) but it is also inspired from the rate-independent systems because of
the dissipation potential. Nevertheless it is important to remark that $\Phi$
is not necessarily 1-homogeneous or even homogeneous.

This Hamiltonian inclusion framework is compatible with the theory of internal
variables and generalized standard materials applied successfully to the
modelling of the material behaviour in plasticity, viscoplasticity and damage
[28]. As such constitutive laws are non smooth, the non differentiability of
the dissipation potential is no longer required and the second term in the
expression (1) is a non unique generalized gradient belonging to a set called
the symplectic subdifferential [5]. The constitutive law is multivoque and the
suitable mathematical tool is the convex analysis. As such non smooth laws are
not necessary to build a variational principle for Navier-Stokes equation, we
continue in this paper considering the more regular case where the dissipative
law is univoque and the dissipation potential is differentiable but its
convexity property remains essential.

In our opinion, the main advantage of Hamiltonian inclusion framework is that
it leads naturally to a variational formulation with a minimum principle.
Following Brezis and Ekeland [3] and Nayroles [43], Buliga and the author
proposed in [6] a symplectic version of the Brezis-Ekeland-Nayroles principle
(in short SBEN) of which the functional is built from three functions, the
symplectic form (for the dynamics), the dissipation potential (for the
irreversible behaviour) and the Hamiltonian (for the reversible behavior)
through the Hamiltonian vector field.

Our main goal was to propose new numerical methods downstream for plasticity
and viscoplasticity of structures in small deformations using the finite
element method, more precisely to compute the whole loading history in
contrast to now widely used incremental methods but that suffer from drawbacks
with respect to stability and convergence criteria. We verified the
feasability of these new numerical methods in statics for thick tubes [7] and
plates [8] and in dynamics [9]. Also we proposed a non incremental variational
principle for brittle fracture using mainly the concept of driving force and
with an enlightening discussion of crack stability criteria of the literature
[12].

The next step is extending this formalism in large deformation. For the
solids, the suitable approach is working with the material or Lagrangian
representation. In [10], we applied the SBEN principle to the plasticity with
the additive decomposition of the Green-Lagrange tensor and the multiplicative
decomposition of the deformation gradient.

To avoid any confusion, it must be pointed out that, although the SBEN
approach uses tools of symplectic geometry, its application to Navier-Stokes
equation has nothing to do with the formalism of Hamiltonian fluid dynamics
[52] which applies only to non dissipative fluids. The problem of identifying
a variational principle for the motion of a viscous fluid does not admit a
unique solution. The pionnering works are due to Helmholtz [29] (based on a
minimum dissipation principle) and Rayleigh [48], both considering restrictive
hypotheses on the boundary conditions and flow features. A more general result
was proposed by Millikan [42] and Bateman [2]. Onsager discussed the validity
of this principle and introduced a description of the production of entropy in
terms of thermodynamic forces and fluxes [45]. Glansdorff and Prigogine [24]
derived the approach of the fluid flow variational that was re-elaborated by
Gyamati [26]. Lebon and Lambermont proposed in [33] a general variational
principle for unstationary dissipative thermodynamics processes. Lebon et al.
[34] applied it to problems with unsteady laminar viscous flows. Sciubba
proposed a variational principle for Navier-Stokes equation based on the
concept of exergy in [54] and [55]. Besides, the last reference presents a
comprehensive survey of previous works on the topics. To deduce Navier-Stokes
equation in a thermodynamical framework, Hamilton’s variational principle was
modified by Fukagawa and Fujitani [16] by introducing a nonholonomic
constraint and Lagrange multipliers. Applying the optimal control theory, they
discuss their approach in connection with the Hamiltonian formulation.
Likewise, Gay-Balmaz and Yoshimura [18] proposed a Lagrangian variational
description of the Navier-Stokes-Fourier system by introducing new dual
variables of which the time rates are the production of entropy and the
temperature. The variational formulation in spatial representation is obtained
using the Euler–Poincaré reduction with advected parameters proposed by Holm
et al. [30]. In [49], Razafindralandy and Hamdouni proposed for incompressible
Navier-Stokes equation to use a bi-Lagrangian but this approach leads, by
adding adjoint variables, to double the number of unknown fields and
equations. The nearest approach to the present one is Ghoussoub and Moameni
work on the application of anti-selfdual Lagrangian to the resolution of the
Navier-Stokes evolutions ([21], [22], [23]). Using convex analysis and
functional analysis tools and under suitable assumptions, they prove the
existence of solutions as minimizers of a given functional for particular
boundary conditons but their approach is quite general and may apply to many
other situations. In [47], Parker and Schneider use a variational method for
finding periodic orbits in the incompressible Navier–Stokes equations but
without using their dynamical or symplectic structure.

Our variational approach to Navier-Stokes is different because developped from
a geometric point of view as in Gay-Balmaz and Yoshimura work [18]. It leads
to construct a problem of minimizing a functional equivalent to compressible
Navier-Stokes equation. This equation is covariant in the sense that it
includes the gravitation and satisfies Galileo’s principle of relativity. The
limit case of the incompressible flow is deduced by well-known variational
techniques. The SBEN principle is a geometric method to built variational
principles for dissipative systems, not only for solids but also for fluids.
The main technical difficulties is that, unlike the applications to solid
mechanics obtained customarily working with the material representation, the
modelization of fluids requires to work in the spatial or Eulerian
representation.

Our goal is reached in three steps. In Section 2, we develop for the non
dissipative systems a Lagrangian formalism based on the calculus of variation
performed on the jet space of order one. In Section 3, we propose a
corresponding Hamiltonian formalism. In Section 4, we deduce a symplectic
variational principle of minimum for the dissipative systems in Eulerian
representation. We apply it to the compressible Navier-Stokes equation in
Section 5 and to the limit case of incompressible Navier-Stokes equation in
Section 6.

##  2 Lagrangian formalism for a reversible continuum

We are working in the space-time, set of events $X=(t,x)$ where $x$ is the
position at time $t$. In contrast to the usual representation of the motion of
a continuum by a map $x=f(t,x_{0})$ where $x_{0}$ are the material coordinates
of the particle, our point of view, inspired from Hilbert-Einstein action in
General Relativity [56], is to represent it by
$x_{0}=\kappa(t,x)=\kappa_{t}(x)=\kappa(X)$. As usual, the deformation
gradient is denoted

$F=\nabla_{x_{0}}x$ (2)

Let $\Omega$ be a bounded open subset of the space-time corresponding to the
motion of the continuum. The equations of balance of the linear momentum and
the energy of a reversible continuum are deduced from a space-time action

$\alpha\left[x_{0}\right]=\int_{\Omega}\mathcal{L}\left(X,x_{0},\nabla_{X}x_{0}\right)\mbox{d}^{4}X$

of Hamilton’s principle, using a special form of the calculus of variation by
replacing the original field $x_{0}$ by its first jet prolongation
$j^{1}x_{0}$, a map from $\Omega$ into the jet bundle
$J^{1}(\Omega,\mathbb{R}^{3})$ such that
$j^{1}x_{0}(X)=(X,x_{0}(X),\nabla_{X}x_{0}(X))$, that leads to perform
variations not only on the field and its derivatives but also on the variable
$X$. To explicit them, we consider a new parameterization given by a regular
map $X=\psi\ (Y)$ of class $C^{1}$ and we perform the variation of the
function $\psi$, the new variable being $Y$. After calculating the variation
of the action, we will consider the particular case where the function $\psi$
is the identity of $\Omega$. Hence we start with

$\alpha\left[X,x_{0}\right]=\int_{\Omega^{\prime}}\mathcal{L}\left(\psi(Y),x_{0},\nabla_{Y}x_{0}\cdot\nabla_{X}Y\right)\
\det\left(\nabla_{Y}X\right)d^{4}Y$ (3)

where $\Omega^{\prime}=\psi^{-1}(\Omega)$ and the variables of the functional
are now both $X$ and $x_{0}$. For more details on the jet spaces and the
calculus of variation on such spaces, the reader is referred to ([53], [1],
[14], [36], [11]). For sake of easiness, we introduce

$f=-\nabla_{x_{0}}\mathcal{L},\quad h=\nabla_{X}\mathcal{L},\quad
P=\nabla_{\nabla_{X}x_{0}}\mathcal{L},\quad
T=P\cdot\nabla_{X}x_{0}-\mathcal{L}\,I$ (4)

The technical details of the variation are given in Appendix A. Taking the
variation with respect to $x_{0}$ and $X$, we obtain Euler-Lagrange equations
of variation:

$\nabla_{X}\cdot P+f=0,\qquad\nabla_{X}\cdot T+h=0\ .$ (5)

The first equation leads to a non linear partial derivative system which can
be used to determine the unknown field $x_{0}$. The last one gives extra
conservation conditions obtained in the spirit of Noether’s theorem.

In order to satisfy Galileo’s principle of relativity, we consider the
Lagrangian

$\mathcal{L}=\rho\,\left(\frac{1}{2}\,\parallel v\parallel^{2}+A\cdot v-\phi-
e_{int}(x_{0},C)\right)$ (6)

where $v=-F\cdot(\partial x_{0}/\partial t)$ is the spatial velocity, $A,\phi$
are the vector and scalar potentials of the Galilean gravitation [11],
$e_{int}$ is the specific internal energy depending on the right Cauchy
strains

$C=F^{T}\cdot F$ (7)

and

$\rho=\frac{\rho_{0}(x_{0})}{\det(F)}$ (8)

is the mass density satisfying the equation of balance of mass

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\,v)=0$ (9)

Introducing the linear momentum

$\pi=\nabla_{v}\mathcal{L}=\rho\,(v+A)\ ,$ (10)

the Hamiltonian density

$\mathcal{H}=\pi\cdot
v-\mathcal{L}=\frac{1}{2\,\rho}\,\parallel\pi-\rho\,A\parallel^{2}+\rho\,(\phi+e_{int})\
.$ (11)

and the reversible stresses

$\sigma_{R}=2\rho\,F\cdot\nabla_{C}\,e_{int}\cdot F^{T}$ (12)

In the sequel, the gradient $\nabla_{x}$ with respect to the position will be
denoted $\nabla$ as usual. As shown in [11], it holds

$T=\left[\begin{array}[]{cc}\mathcal{H}&-\pi\\\
\mathcal{H}v-\sigma_{R}v&\sigma_{R}-v\otimes\pi\\\ \end{array}\right],\qquad
h=\left[\begin{array}[]{c}\rho\left(\frac{\partial A}{\partial t}\cdot
v-\frac{\partial\phi}{\partial t}\right)\\\ \rho\,((\nabla A)\cdot
v-\nabla\phi)\\\ \end{array}\right]$ (13)

$T$ is the analogous in Galilean mechanics of the energy-momentum tensor in
relativistic one. Then the equation (5) of variation of the action with
respect to $X=(t,x)$ allows we recover the balance equations of the energy

$\frac{\partial\mathcal{H}}{\partial
t}+\nabla\cdot\left(\mathcal{H}\,v-\sigma_{R}\cdot
v\right)=\rho\left(\frac{\partial\phi}{\partial t}-\frac{\partial A}{\partial
t}\cdot v\right)$

and the balance of linear momentum

$-\frac{\partial\pi}{\partial
t}+\nabla\cdot(\sigma_{R}-v\otimes\pi)+\rho\,((\nabla A)\cdot v-\nabla\phi)=0$
(14)

After classical simplifications owing to the balance of mass and given in
Appendix B, it is reduced to

$-\rho\,\frac{Dv}{Dt}+\nabla\cdot\sigma_{R}+\rho\,\left(g-2\,\Omega\times
v\right)=0$ (15)

where occurs the material derivative $D/Dt=\partial/\partial t+v\cdot\nabla$,
the gravity $g$ and Coriolis’ vector $\Omega$ defined by

$g=-\nabla\phi-\frac{\partial A}{\partial
t},\qquad\Omega=\frac{1}{2}\,\nabla\times A$ (16)

For a barotropic fluid, $\sigma_{R}=-p\,I$ where $p$ is the pressure then the
balance of linear momentum takes the form of Euler’s equations

$-\rho\,\frac{Dv}{Dt}-\nabla p+\rho\,\left(g-2\,\Omega\times v\right)=0$ (17)

It is worth to remark these two equations of balance are covariant with
respect to Galileo’s principle of relativity. The last term is Coriolis’ force
useful for the applications to environmental fluids (atmosphere, oceans,
Earth’s interior).

##  3 Hamiltonian formalism and canonical equations for a reversible continuum

Let $\Omega_{t}$ the set of positions occupied by the material particles of
the continuum at time $t$. As $v$ are the components of a 1-contravariant
tensor and $\mathcal{H}=\pi\cdot v-\mathcal{L}$ is a density, $\pi$ are the
components of a 1-covariant and antisymmetric 3-contravariant tensor. Then the
field $x\mapsto\pi(t,x)$ defined on $\Omega_{t}$ is a section of the fiber
bundle $\bigwedge^{3}(T\,\Omega_{t})\otimes T^{*}\Omega_{t}$ of which the
coordinates $(x,\pi)$ in local charts are taken as canonical variables. The
total energy at time $t$ is

$H\left[x_{0},\pi\right]=\int_{\Omega_{t}}\mathcal{H}\left(x,x_{0},\nabla
x_{0},\pi\right)d^{3}x$

where, for sake of easiness, the dependence with respect to time $t$ is no
longer explicitly expressed in the sequel.

We claim that the motion of the continuum is described by the canonical
equations

$\zeta=\left(\frac{dx}{dt},\frac{\partial\pi}{\partial
t}\right)=\left(v,\frac{\partial\pi}{\partial t}\right)=X_{H}$

obtained by the calculus of the Hamiltonian vector field $X_{H}$ for the
canonical symplectic form

$\omega(\zeta,\zeta^{\prime})=\int_{\Omega_{t}}\left(\frac{dx}{dt}\cdot\frac{\partial\pi^{\prime}}{\partial
t}-\frac{\partial\pi}{\partial
t}\cdot\frac{dx^{\prime}}{dt}\right)\mbox{d}^{3}x$ (18)

As $H$ is a functional, $X_{H}$ is a variational derivative that can be
calculated by the jet theory as in the previous section. Hence we consider a
new parameterization $x=\psi\ (y)$ of class $C^{1}$ and we perform the
variation of the function $\psi$, the new variable being $y$. After
calculating the variation of the functional, we will consider the particular
case where the function $\psi$ is the identity of $\Omega_{t}$. Hence we start
with

$H\left[x,x_{0},\pi^{\prime}\right]=\int_{\Omega^{\prime}_{t}}\mathcal{H}\left(\psi(y),x_{0},\nabla_{y}x_{0}\cdot\nabla
y,\det\left(\nabla y\right)\,(\nabla y)^{T}\cdot\pi^{\prime}\right)\
\det\left(\nabla_{y}x\right)d^{3}y\ ,$ (19)

where $\Omega^{\prime}_{t}=\psi^{-1}(\Omega_{t})$ and $\pi^{\prime}$ are the
components of the linear momentum in the coordinates $y$. A calculus similar
to the one of the previous section and detailled in Appendix C leads to the
following expression of the canonical equations

$\frac{dx}{dt}=\nabla_{\pi}\mathcal{H}$ $\frac{\partial\pi}{\partial
t}=-\nabla\mathcal{H}$ $-\nabla\cdot\left[\nabla_{\nabla
x_{0}}\,\mathcal{H}\cdot\nabla
x_{0}-\left(\mathcal{H}-\nabla_{\pi}\mathcal{H}\cdot\pi\right)\,I+\nabla_{\pi}\mathcal{H}\otimes\pi\right]$

where the extra terms of the jet theory are given by the last line.

Let us determine these equations for the Hamiltonian (11) corresponding to the
Lagrangian (6). We have

$\nabla_{\pi}\mathcal{H}=\frac{\pi}{\rho}-A\ .$ (20) $\nabla_{\nabla
x_{0}}\,\mathcal{H}=-2\,\nabla_{x_{0}}x\cdot\nabla_{C}\mathcal{H}\cdot C$

which, taking into account (2), (7) and (8), reduces to:

$\nabla_{\nabla
x_{0}}\,\mathcal{H}=-\left(\dfrac{1}{2\,\rho}\,\parallel\pi\parallel^{2}-\dfrac{\rho}{2}\,\parallel
A\parallel^{2}-\rho\,(\phi+e_{int})\right)\,\nabla_{x_{0}}x-2\,\rho\,\nabla_{x_{0}}x\cdot\nabla_{C}e_{int}\cdot
C$

Taking into account (12) and (20), we obtain

$\nabla_{\nabla x_{0}}\,\mathcal{H}\cdot\nabla
x_{0}-\left(\mathcal{H}-\nabla_{\pi}\mathcal{H}\cdot\pi\right)\,I+\nabla_{\pi}\mathcal{H}\otimes\pi=-\sigma_{R}+v\otimes\pi\
.$ (21)

Besides

$-\nabla\mathcal{H}=\rho\,\left((\nabla
A)\cdot\left(\frac{\pi}{\rho}-A\right)-\nabla\phi\right)=\rho\,\left((\nabla\,A)\cdot
v-\nabla\phi\right)\ .$ (22)

For the Lagrangian (6), we obtain

$\frac{dx}{dt}=\frac{\pi}{\rho}-A,\qquad-\frac{\partial\pi}{\partial
t}+\nabla\cdot(\sigma_{R}-v\otimes\pi)+\rho\,((\nabla A)\cdot v-\nabla\phi)=0$
(23)

where we can recognize the definition (10) of the linear momentum and the
equation (14) of balance of the linear momentum [11].

##  4 Symplectic Brezis-Ekeland-Nayroles principle for dissipative continua

To build a minimum principle for such continua as a viscous fluid, we adapt
the scheme proposed in [6]. The key-idea is a decomposition of $\zeta$ into
reversible and irreversible parts

$\zeta=\zeta_{R}+\zeta_{I},\qquad\zeta_{R}=X_{H},\qquad\zeta_{I}=\zeta-X_{H}$

We are interested in the material continua that are modelized by a
differentiable and convex potential of dissipation $\Phi$. We claim that the
law of dissipative yielding is given by

$\zeta_{I}=X_{\Phi}$ (24)

where the Hamiltonian vector field $X_{\Phi}$ is such that

$\forall\zeta^{\prime},\qquad\omega(X_{\Phi},\zeta^{\prime})=\lim_{\epsilon\to
0}\frac{1}{\epsilon}(\Phi(\zeta+\epsilon\,\zeta^{\prime})-\Phi(\zeta))$

Because of the convexity property [5]

$\forall\zeta^{\prime},\quad\Phi(\zeta+\zeta^{\prime})-\Phi(\zeta)\geq\omega(X_{\Phi},\zeta^{\prime})$

Next, we define the symplectic Fenchel polar (or conjugate) function
$\Phi^{*\omega}$ by

$\Phi^{*\omega}(\zeta_{I})=\sup_{\zeta}\;(\omega(\zeta_{I},\zeta)-\Phi(\zeta))$
(25)

As superior envelop of affine functions, it satifies the symplectic Fenchel
inequality

$\forall\zeta^{\prime},\forall\zeta^{\prime}_{I},\qquad\Phi(\zeta^{\prime})+\Phi^{*\omega}(\zeta^{\prime}_{I})-\omega(\zeta^{\prime}_{I},\zeta^{\prime})\geq
0$ (26)

In particular, when the dynamical dissipative constitutive law (24) is
satisfied, it can be proved the equality is reached [6]

$\zeta_{I}=X_{\Phi}\qquad\Leftrightarrow\qquad\Phi(\zeta)+\Phi^{*\omega}(\zeta_{I})-\omega(\zeta_{I},\zeta)=0$
(27)

We are interested during the interval $\left[0,T\right]$ by the evolution
paths $t\mapsto(\kappa_{t},\zeta)$ which are admissible in the sense that they
satisfy the boundary conditions and the initial conditions of the considered
problem (that we shall suppose well posed). Following pionnering works by
Brezis, Ekeland and Nayroles in 1976, Buliga and the author proposed in [6] a
symplectic version of the Brezis-Ekeland-Nayroles principle (in short SBEN) of
which the functional is built from three functions, the symplectic form (for
the dynamics), the dissipation potential (for the irreversible behavior) and
the Hamiltonian (for the reversible behavior) through the Hamiltonian vector
field:

SBEN principle: the natural evolution path $t\mapsto(\kappa_{t},\zeta)$
minimizes the functional

$\Pi[\kappa,\zeta]=\int^{T}_{0}\\{\Phi(\zeta)+\Phi^{*\omega}(\zeta-
X_{H})-\omega(\zeta-X_{H},\zeta)\\}\ \mbox{d}t$ (28)

among all the admissible evolution paths, and the minimum is zero.

The idea is that the functional is non negative because of (26) and vanishes
if and only if the dynamical dissipative constitutive law (24) is satisfied
almost eveywhere because of (27).

##  5 SBEN principle for compressible Navier-Stokes equation

With the notations of the previous section, the canonical equations (23) lead
to

$\zeta_{I}=\zeta-X_{H}=(v_{I},\pi_{I})$

with

$v_{I}=v-\frac{\pi}{\rho}+A,\quad\pi_{I}=\frac{\partial\pi}{\partial
t}-\nabla\cdot(\sigma_{R}-v\otimes\pi)-\rho\,((\nabla A)\,v-\nabla\phi)$

With the same classical simplifications used to transform (14) into (17), we
have for a barotropic fluid

$\pi_{I}=\rho\,\frac{Dv}{Dt}+\nabla p-\rho\,\left(g-2\,\Omega\times v\right)$
(29)

To apply the very general formalism of the previous section to Navier-Stokes
equation, we need two additional hypotheses. The first one claims that the
convex smooth potential $\Phi$ depends explicitly only on $v$
($\partial\pi/\partial t$ is ignorable):

$\Phi(\zeta)=\varphi(v)$ (30)

that reads, taking into account (18) and (30):

$\Phi^{*\omega}(\zeta_{I})=\sup_{\zeta}\;\left\\{\int_{\Omega_{t}}\left(v_{I}\cdot\frac{\partial\pi}{\partial
t}-v\cdot\pi_{I}\right)d^{3}x-\varphi(v)\right\\}$

where $\zeta=(v,\partial\pi/\partial t)$. Hence, it holds:

$\Phi^{*\omega}(\zeta_{I})=\sup_{v}\;\left\\{\int_{\Omega_{t}}v\cdot(-\pi_{I})\,d^{3}x-\varphi(v)\right\\}+\sup_{\tiny{\partial\pi/\partial
t}}\;\int_{\Omega_{t}}v_{I}\cdot\frac{\partial\pi}{\partial t}\,d^{3}x$

where the value of the last term is zero if $v_{I}=0$ and $+\infty$ otherwise.
Then the symplectic Fenchel polar function has a finite value

$\Phi^{*\omega}(\zeta_{I})=\Phi^{*\omega}(v_{I},\pi_{I})=\varphi^{*}(-\pi_{I})$

if $v_{I}=0$, that is (10), where $\varphi^{*}$ is the classical Fenchel polar
function of $\varphi$ [15]. As in the SBEN principle the minimum value of the
functional is zero then finite, we suppose in the sequel that (10) is a priori
satisfied. Moreover, owing to (18), the last term in the functional (28)
becomes

$-\omega(\zeta-X_{H},\zeta)=\int_{\Omega_{t}}\left(\pi_{I}\cdot
v-v_{I}\cdot\frac{\partial\pi}{\partial
t}\right)\mbox{d}^{3}x=\int_{\Omega_{t}}\pi_{I}\cdot v\,\mbox{d}^{3}x$

Then the SBEN functional becomes

$\Pi[\kappa,\zeta]=\int^{T}_{0}\\{\varphi(v)+\varphi^{*}(-\pi_{I})+\int_{\Omega_{t}}\pi_{I}\cdot
v\,\mbox{d}^{3}x\\}\ \mbox{d}t$

Owing to (29), the functional (28) can be recast, that leads to

SBEN principle for compressible Navier-Stokes equation:
the natural evolution path $t\mapsto(\kappa_{t},v)$ minimizes the functional

$\displaystyle\Pi[\kappa,v]$ $\displaystyle=$
$\displaystyle\int^{T}_{0}\\{\varphi(v)+\varphi^{*}\left(-\rho\,\frac{Dv}{Dt}-\nabla
p+\rho\,\left(g-2\,\Omega\times v\right)\right)$ (31)
$\displaystyle+\int_{\Omega_{t}}\left[\rho\,\frac{Dv}{Dt}+\nabla
p-\rho\,g\right]\cdot v\,\mbox{d}^{3}x\\}\ \mbox{d}t$

among all the admissible evolution paths, and the minimum is zero.

It is worth to remark that
$\Phi(\zeta)+\Phi^{*\omega}(\zeta_{I})=\varphi(v)+\varphi^{*}(-\pi_{I})$ is an
anti-selfdual Lagrangian, a tool proposed by Ghoussoub [22] to study among
others the solutions of Navier-Stokes equation. This reveals the symplectic
origin of the structure of such self-antidual Lagrangians. On the other hand,
while the last term of Ghoussoub’s functional is depending on specific
boundary value problems, ours is general and does not depend on the boundary
conditions that are only considered as constraints in the minimization.

The second additional hypothesis claims that $\varphi$ depends on $v$ through
its symmetric gradient $D=\mathcal{D}(v)=\nabla_{s}v=1/2\,(\nabla v+(\nabla
v)^{T})$ and is quadratic with respect to $v$ of the form

$\varphi(v)=\int_{\Omega_{t}}W(\mathcal{D}(v))\,d^{3}x=\int_{\Omega_{t}}\mu\,\left[Tr(D^{2})-\frac{1}{3}\,(Tr(D))^{2}\right]\,\mbox{d}^{3}x$

then the viscous part of the stress tensor is traceless (Stokes hypothesis)

$\sigma_{I}=\nabla_{D}W(\mathcal{D}(v))=2\mu\,\left(D-\frac{1}{3}\,Tr(D)\,I\right)$
(32)

As $\varphi$ is a functional, its variational derivative is

$\nabla_{v}\varphi(v)=-\nabla\cdot\sigma_{I}$

Now, let us prove that the variational principle of minimum restitues Navier-
Stokes equation. Indeed, if the minimum equal to zero is reached, we have
almost everywhere in $\left[0,T\right]$

$\varphi(v)+\varphi^{*}(-\pi_{I})+\int_{\Omega_{t}}\pi_{I}\cdot
v\,\mbox{d}^{3}x=0$

that is equivalent to the dynamical dissipative law

$-\pi_{I}=\nabla_{v}\varphi(v)=-\nabla\cdot\sigma_{I}$

Owing to (29) and (32), we recover Navier-Stokes equation

$\rho\,\frac{Dv}{Dt}=-\nabla p+\mu\,\triangle
v+\frac{\mu}{3}\,\nabla\,(\nabla\cdot v)+\rho\,\left(g-2\,\Omega\times
v\right)\ .$

For a flow, the term of the last line in the functional (31) is the sum of the
velocity head, pressure head and elevation head losses due to dissipation
during the interval from $0$ to $T$. For the limit case of inviscid flows, the
potential of dissipation $\varphi$ vanishes and its polar function
$\varphi^{*}$ has a finite value equal to zero if $\pi_{I}=0$, i.e. Euler’s
equations (17), then the SBEN principle claims that the total head loss is
zero, that is the expression of Bernoulli’s principle. It is worth to notice
that in this limit case the SBEN principle does not degenerate into Hamilton’s
principle.

Likewise an argument developped in Proposition $2$ of [3] for a simpler case
where $\mathcal{D}$ is the gradient of a scalar field, we can show that, if
$W$ is quadratic and if the velocity or the dissipative stress vector is null
on the boundary, Fenchel polar function of $\varphi$ is:

$\varphi^{*}(f)=\int_{\Omega_{t}}W(\mathcal{D}(K^{-1}(f)))\,d^{3}x\ .$

where the linear operator $K$ is define by

$f=K(v)=-\nabla\cdot\left(\nabla_{D}W(\mathcal{D}(v))\right)$ (33)

The proof is given in Appendix D.

Although we study dissipative systems, we do not use concepts of the
Thermodynamics. The framework of thermodynamic forces and fluxes, introduced
by Onsager [45] and used in their variational formulation by Glansdorff and
Prigogine [24], Lebon and Lambermont [33], is not used explicitly in our
formalism but it is not far behind. In fact, the dissipation potential depends
on $v$ through its symmetric gradient $D$ of which the components can be taken
as thermodynamic fluxes, then the corresponding thermodynamic forces are the
components of dissipative stress tensor $\sigma_{I}$ given by (32).

##  6 SBEN principle for incompressible Navier-Stokes equation

For this limit case, $\nabla\cdot v=0$ and the pressure $p$ becomes a free
variable independent of $\kappa$. Navier-Stokes equation is reduced to:

$\rho\,\frac{Dv}{Dt}=-\nabla p+\mu\,\triangle v+\rho\,\left(g-2\,\Omega\times
v\right)\ .$

To obtain the corresponding SBEN principle, we proceed as follows. The
internal energy is cancelled in (6) and (11). The incompressibility condition
is introduced as a constraint in the minimization. The pressure disappears of
the functional and reappears as a Lagrange multiplier of this constraint in
the equation characterizing the minimizers. Then we state

SBEN principle for incompressible Navier-Stokes equation:
the natural evolution path $t\mapsto(\kappa_{t},v)$ minimizes the functional

$\displaystyle\Pi[\kappa,v]=\int^{T}_{0}\left\\{\varphi(v)+\varphi^{*}\left(-\rho\,\frac{Dv}{Dt}+\rho\,\left(g-2\,\Omega\times
v\right)\right)+\int_{\Omega_{t}}\rho\,\left(\frac{Dv}{Dt}-g\right)\cdot
v\,\mbox{d}^{3}x\right\\}\ \mbox{d}t$

among all the admissible evolution paths such that $\nabla\cdot v=0$, and the
minimum is zero.

If the fluid is homogeneous, the mass density $\rho$ is constant and the
functional no longer depends on the motion map $\kappa$.

##  7 Conclusions and perspectives

In this work, the emphasis was laid on the geometric aspects thanks to the use
of differential geometry tools in order to give a strong foundation to the
formulation of variational principles for the dynamics of dissipative
continuous media in spatial or Eulerian representation. The present
variational approach covers a large class of problems including Navier-Stokes
equations for compressible and incompressible flows. The weak regularity of
the potential of dissipation adopted in [6] allows to encompass nonsmooth
dissipative constitutive laws such as the one of Bingham fluids. Also, it is
worth to observe that the expression of the functional is independent of the
boundary conditions that appear only as constraints of the minimization.
Moreover, the functional is not convex with respect to the unknown fields but
there is (at least partial) convexity, that is favourable for the convergence
of the minimization procedure.

The present work paves the way to provide numerical approximations of the
solutions. Many numerical schemes fail to correctly predict the expected
conservation of mechanical quantities such as the energy, as the time
increases. Geometric integrators appear to be robust for large time simulation
because they are based on the underlying geometric structure of the equations.
Among them, we can quote the symplectic integrators ([58], [27], [50], [13]),
the variational integrators ([38], [19], [50], [51]) and Dirac integrators
[51]. Another idea is to develop symplectic integrators to respect the
structure of the canonical equation of Section 3. Another perspective is to
construct variational schemes based on the Lagrangian of the SBEN principles
proposed in Sections 5 and 6. Also we think that the SBEN principle could be a
new tool in theory of turbulence to propose variational-based models for Large
Eddy Scale simulations. Moreover, when the convergence of the minimization
algorithm is reached, the value of the functional of SBEN principle for the
numerical solution can be interpreted as an indicator of the error in
constitutive law (or in dissipation) introduced by Ladevèze [32].

Finally a minimum variational principle may constitute an interesting means to
study the existence and smoothness of the solutions of Navier-Stokes equations
in the spirit of Ghoussoub and Moameni papers ([21], [22], [23]).

Acknowledgements

The author would like to thank Maurice Rossi and Djimedo Kondo for their
comments and suggestions that allowed to improve the paper and the Agence
Nationale de la Recherche (ANR) for the financial support of the project
BIpotentiels Généralisés pour le principe variationnel de Brezis-Ekeland-
Nayroles en mécanique (BigBen) that has enabled to perform this work.

Author ORCID.

G. de Saxcé http://orcid.org/0000-0002-0961-0513.

Appendix A

Owing to (4), the variation of the action (3) reads:

$\delta\alpha=\int_{\Omega^{\prime}}\ [\left(Tr\ \left(P\cdot\delta\
\left(\nabla_{Y}x_{0}\cdot\nabla_{X}Y\right)\right)-f\cdot\delta
x_{0}+h\cdot\delta X\right)\ \det\ \left(\nabla_{Y}X\right)$ $+\mathcal{L}\
\delta\ \left(\det\ \left(\nabla_{Y}X\right)\right)]\,\mbox{d}^{4}Y\
.\qquad\qquad$ (34)

First of all, we calculate the variation of the field derivative in terms of
the derivative of its variation:

$\delta\left(\nabla_{Y}x_{0}\cdot\nabla_{X}Y\right)=\delta\left(\nabla_{Y}x_{0}\right)\cdot\nabla_{X}Y+\nabla_{Y}x_{0}\cdot\delta\
\left(\nabla_{Y}X\right)^{-1}$
$\delta\left(\nabla_{Y}x_{0}\cdot\nabla_{X}Y\right)=\nabla_{Y}(\delta
x_{0})\cdot\nabla_{X}Y-\nabla_{Y}x_{0}\cdot\nabla_{X}Y\cdot\nabla_{Y}(\delta
X)\cdot\nabla_{X}Y\ .$ (35)

Incidently, it is worth noting that when $X=Y$:

$\delta\left(\nabla_{X}x_{0}\right)=\nabla_{X}(\delta
x_{0})-\nabla_{X}x_{0}\cdot\nabla_{X}(\delta X)\ .$

This formula shows that, unlike the usual rule used in the classical calculus
of variation, the derivative symbols $\nabla_{X}$ and $\delta$ may not be
permuted in the present approach. Next, using the adjugate of a matrix
$\mbox{adj}(M)=\det(M)\,M^{-1}$, one has:

$\delta\left(\det\ \left(\nabla_{Y}X\right)\right)=Tr\ \left(\nabla_{Y}(\delta
X)\cdot adj\left(\nabla_{Y}X\right)\right)$ (36)

Introducing the expressions (35) and (36) into the variation of the action
(34) gives:

$\delta\alpha=\int_{\Omega^{\prime}}\ [Tr\ \left(P\cdot\nabla_{Y}(\delta
x_{0})\cdot\det\ \left(\nabla_{Y}X\right)\ \nabla_{X}Y\right)-\det\
\left(\nabla_{Y}X\right)\ (f\cdot\delta x_{0}-h\cdot\delta X)$ $+Tr\
\left(\mathcal{L}\ \nabla_{Y}(\delta X)\cdot
adj\left(\nabla_{Y}X\right)-P\cdot\nabla_{X}x_{0}\cdot\nabla_{Y}(\delta
X)\cdot\det\ \left(\nabla_{Y}X\right)\ \nabla_{X}Y\right)]\ \mbox{d}^{4}Y\ .$
(37)

Taking into account Cramer’s rule and using the definition (4), the variation
of the action (37) becomes:

$\delta\alpha=\int_{\Omega^{\prime}}\ [\ Tr\ \left(adj\
\left(\nabla_{Y}X\right)\cdot P\cdot\nabla_{Y}(\delta x_{0})\right)\ -\det\
\left(\nabla_{Y}X\right)\ (f\cdot\delta x_{0}-h\,\delta X)$ $-Tr\ \left(adj\
\left(\nabla_{Y}X\right)\cdot T\cdot\nabla_{Y}(\delta X)\ \right)]\
\mbox{d}^{4}Y\ .$ (38)

Integrating by part and taking into account the fact that the values of
$x_{0}$ and $X$ are imposed on the boundary, the surface integrals vanish and
we obtain:

$\delta\alpha=\int_{\Omega^{\prime}}\ (\ -\left[\ \nabla_{Y}\cdot\left(adj\
\left(\nabla_{Y}X\right)\cdot P\right)+\det\
\left(\nabla_{Y}X\right)\,f\right]\cdot\delta x_{0}$
$+\left[\nabla_{Y}\cdot\left(adj\ \left(\nabla_{Y}X\right)\cdot T\right)+\det\
\left(\nabla_{Y}X\right)\,h\right]\cdot\delta X)\ \mbox{d}^{4}Y\ ,$

Finally, considering the particular case where $X=Y$, the variational
principle reads:

$\delta\alpha=\int_{\Omega}\ (\ -\left[\nabla_{X}\cdot P+f\right]\cdot\delta
x_{0}+\left[\nabla_{X}\cdot T+h\right]\cdot\delta X)\ \mbox{d}^{4}X=0\ .$

Then we obtain Euler-Lagrange equations of variation (5).

Appendix B

Developing the balance of linear momentum (14), we have:

$-\frac{\partial}{\partial t}(\rho\,(v+A))+\nabla\cdot\sigma_{R}-div\,(\rho
v)\,(v+A)-\rho\,\nabla(v+A)\cdot v+\rho\,((\nabla A)\cdot v-\nabla\phi)=0\ ,$

or, expanding the first term:

$-\rho\ \frac{\partial}{\partial t}(v+A)-\left(\frac{\partial\rho}{\partial
t}+div\,(\rho v)\right)\,(v+A)-\rho\,\nabla(v+A)\cdot
v+\nabla\cdot\sigma_{R}+\rho\,((\nabla A)\cdot v-\nabla\phi)=0\ .$

Taking into account the balance of mass (9), it holds:

$-\rho\ \left(\frac{\partial v}{\partial
t}+(v\cdot\nabla)v\right)+\nabla\cdot\sigma_{R}-\rho\left(\nabla\phi+\frac{\partial
A}{\partial t}\right)+\rho v\cdot\left(\nabla A-(\nabla A)^{T}\right)=0\ .$

that, owing to (16), leads to the new form of the balance of linear momentum
(15).

Appendix C

For sake of easiness, we introduce

$V=\nabla_{\pi}\mathcal{H},\qquad h^{*}=-\nabla\mathcal{H},\qquad
P^{*}_{x}=\nabla_{\nabla x_{0}}\mathcal{H}$

The variation of the action (19) with respect to the canonical variables $x$
and $\pi^{\prime}$ reads:

$\delta H=\int_{\Omega^{\prime}}\ [\left(Tr\ \left(P^{*}_{x}\cdot\delta\
\left(\nabla_{y}x_{0}\cdot\nabla
y\right)\right)+V\cdot\delta\left(\det\left(\nabla y\right)\,\left(\nabla
y\right)^{T}\pi^{\prime}\right)-h^{*}\cdot\delta x\right)$ $\ \det\
\left(\nabla_{y}x\right)+\mathcal{H}\ \delta\ \left(\det\
\left(\nabla_{y}x\right)\right)]\,d^{3}y\ .$ (39)

First of all, we calculate the variation of the field derivative:

$\delta\left(\nabla_{y}x_{0}\cdot\nabla y\right)=-\nabla_{y}x_{0}\cdot\nabla
y\cdot\nabla_{y}(\delta x)\cdot\nabla y\ .$ (40)

Once again, using the adjugate of a matrix $\mbox{adj}(M)=\det(M)\,M^{-1}$,
one has:

$\delta\left(\det\ \left(\nabla y\right)\right)=Tr\ \left(\nabla_{y}(\delta
x)\cdot adj\left(\nabla_{y}x\right)\right)\ .$ (41)

Besides, the variation of the momentum $\pi$ reads:

$\displaystyle\delta\left(\det\left(\nabla y\right)\,\left(\nabla
y\right)^{T}\cdot\pi^{\prime}\right)$ $\displaystyle=$
$\displaystyle-\left(\det\left(\nabla
y\right)\right)^{2}Tr\left(\nabla_{y}(\delta x)\cdot
adj\left(\nabla_{y}x\right)\right)\cdot\left(\nabla
y\right)^{T}\cdot\pi^{\prime}$ $\displaystyle+\det\left(\nabla
y\right)\,\left(\nabla
y\right)^{T}\cdot\left(\delta\pi^{\prime}-\left(\nabla_{y}(\delta
x)\right)^{T}\cdot\left(\nabla y\right)^{T}\cdot\pi^{\prime}\right)$

Introducing the expressions (40), (41) and the latter one into the variation
of the Hamiltonian (39) and using Cramer’s rule gives:

$\displaystyle\delta H$ $\displaystyle=$ $\displaystyle\int_{\Omega^{\prime}}\
[V\cdot\left(\left(\nabla
y\right)^{T}\cdot\delta\pi^{\prime}\right)-(h^{*}\cdot\delta x)\,\det\
\left(\nabla_{y}x\right)$ $\displaystyle-Tr\
\\{(adj\left(\nabla_{y}x\right)\cdot\left(P^{*}_{x}\cdot\nabla
x_{0}-\left(\mathcal{H}-V\cdot\left(\det\,\left(\nabla y\right)\,\left(\nabla
y\right)^{T}\cdot\pi^{\prime}\right)\right)\,I\right)$ $\displaystyle+\nabla
y\cdot(V\otimes\pi^{\prime})\cdot\nabla y)\cdot\nabla_{y}(\delta x)\\}\ ]\
d^{3}y\ .$

Integrating by parts the last term with vanishing surface integrals, we
obtain:

$\delta H=\int_{\Omega^{\prime}}\ [V\cdot\left(\left(\nabla
y\right)^{T}\cdot\delta\pi^{\prime}\right)-(h^{*}\cdot\delta x)\,\det\
\left(\nabla_{y}x\right)$
$+\nabla_{y}\cdot\\{adj\left(\nabla_{y}x\right)\cdot\left(P^{*}_{x}\cdot\nabla
x_{0}-\left(\mathcal{H}-V\cdot\left(\det\,\left(\nabla y\right)\,\left(\nabla
y\right)^{T}\cdot\pi^{\prime}\right)\right)\,I\right)$ $+\nabla
y\cdot(V\otimes\pi^{\prime})\cdot\nabla y\\}\cdot\delta x]\ d^{3}y\ .$

Considering the particular case where $x=y$ (then $\pi=\pi^{\prime}$), the
variational principle reads:

$\delta H=\int_{\Omega}\ [V\cdot\delta\pi-h^{*}\cdot\delta x+(\nabla\cdot
T^{*})\cdot\delta x]\ d^{3}x\ ,$ (42)

introducing the $3\times 3$ matrix:

$T^{*}=P^{*}_{x}\cdot\nabla
x_{0}-\left(\mathcal{H}-V\cdot\pi\right)\,I+V\otimes\pi$ (43)

Appendix D

Fenchel polar function of $\varphi$ is [15]:

$\varphi^{*}(f)=\sup_{v}\,\int_{\Omega_{t}}[f\cdot
v-W(\mathcal{D}(v))]\,d^{3}x$ (44)

Denoting $\mathcal{L}_{\varphi}$ the integrand in the above expression, the
supremum is reached when Euler-Lagrange equation is satisfied:

$\nabla_{v}\mathcal{L}_{\varphi}=\nabla\cdot\left(\nabla_{\nabla
v}\mathcal{L}_{\varphi}\right)$

The right hand member defines a linear map $K$ from the velocity field $v$
onto a vector field $f$ of which the value is the internal force (by unit
volume) given by (33). If the associated boundary value problem is well posed,
the map is regular and $v=K^{-1}(f)$. Integrating by parts, it holds:

$\int_{\Omega_{t}}\nabla\cdot\left(\nabla_{D}W(\mathcal{D}(v))\cdot
v\right)\,d^{3}x=\int_{\Omega_{t}}\left[Tr\,\left\\{(\nabla_{D}W(\mathcal{D}(v))\cdot\mathcal{D}(v)\right\\}-f\cdot
v\right]\,d^{3}x\ .$

Applying the divergence formula to the first term of the right hand member and
taking into account that the velocity or the dissipative stress vector
$\sigma_{I}\cdot n$ vanishes on the boundary of $\Omega_{t}$ of unit normal
$n$, the left hand member is null. Then Fenchel polar function (44) becomes:

$\varphi^{*}(f)=\int_{\Omega_{t}}\left[Tr\,\left\\{\nabla_{D}W(\mathcal{D}(v))\cdot\mathcal{D}(v)\right\\}-W(\mathcal{D}(v))\right]\,d^{3}x$

Since $W$ is quadratic and $v=K^{-1}(f)$ the proof is achieved. [Batchelor59]

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00footnotetext: Received xx xx 2016

# Measurement Of Energy Resolution For TES Microcalorimeter With Optical
Pulses††thanks: CAS Strategic Priority A(Grant No. XDA040762)

Shuo Zhang 3;1) Qing-Ya Zhang 1,2;1)
Jianshe Liu 1,2 CongZhan Liu 3 Wenhui Dong 1,2 Wei Chen 1,2<EMAIL_ADDRESS><EMAIL_ADDRESS>1 Tsinghua National Laboratory for Information
Science and Technology, Tsinghua University, Beijing 100084, China 2 Institute
of Microelectronics, Department of Micro/Nanoelectronics, Tsinghua University,
Beijing 100084, China 3 Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy Of Sciences, Beijing, 100049, China

###### Abstract

Energy resolution is an important figure of merit for TES microcalorimeter. We
propose a laser system to measure the energy resolution of TES
microcalorimeter with a 1550 nm laser source. Compared to method that
characterizes the performance by irradiating the detector using X-ray photons
from a radioactive source placed inside the refrigerator, our system is safer
and more convenient. The feasibility of this system has been demonstrated in
the measurement of an Al/Ti bilayer TES microcalorimeter. In this experiment,
the tested detector showed a energy resolution of 72 eV in the energy range
from 0.2 keV to 0.9 keV.

###### keywords:

TES , Microcalorimeter , Energy Resolution , Laser Pulse

###### pacs:

2

9,42

00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  1 Introduction

TES (Transition Edge Sensor) microcalorimeter, as a very sensitive temperature
sensor, has been widely used in radiation detection. Compared to the
traditional semi-conductor detectors, TES microcalorimeter has excellent
detection efficiency and much higher energy resolution. TES Microcalorimeter
has been playing an important role in the field of X-ray astronomy [2, 3, 4,
5], nuclear energy spectrum[6], X-ray spectroscopy analysis[7], elemental
analysis[7], neutrino mass measurement[8], dark matter detection and so on.

Generally, the energy resolution of TES microcalorimeter is measured by
radioactive sources whose energy are limited and discrete. If we need to
measure the energy resolution at other energy, we should interrupt the
operation of the refrigerator and change the radioactive sources. This process
take a lot of time and will change the test environment.

Microcalorimeter detects X-ray base on the thermal signals, so we can use a
thermal source replace radioactive source. We use a 1550 nm laser source to
replace the radioactive source as the energy impulse for the test of TES. In
comparing with the radioactive source, the equivalent energy and the frequency
of laser pulse is adjustable. We can therefore use the laser to find the
proper work point of the TES and get the energy resolution, linear dynamic
range, characteristic shape of signal pulse very quickly. In this paper, we
will introduce the system, and demonstrate the feasibility of this system by
measuring the energy resolution of an experimental TES microcalorimeter we
fabricated.

###  1.1 Introduction to the laser test system

As shown in Figure 1.1, a 1550 nm laser source with a constant output power
$P_{0}$ is used to mimic the TES microcalorimeter, the pulse width $t$ and
frequency $f$ modulated by a function generator. Current change signal
$\delta{I}$ is converted to a voltage signal by SQUID amplifier, after
filtered by pre-amplifier and Liner amplifier the signal is analysed by a
Multi-channel analyser, then we got the energy spectrum of the laser pulse.

The bias voltage for the TES is provided by a room temperature current source
$I_{b}$ and a 8.1 m$\Omega$ resistor ($R_{b}$) at 2.5 K. The TES branch has a
parasitic resistance $R_{p}$ = 44m$\Omega$ from the wirings. The input
inductance of SQUID amplifier ($L_{in}$) is about 600 nH.

The diagram of TES testing system using optical laser pulses.

###  1.2 analysis on energy spectrum

As shown in Eq (1), keeping $P_{0}$ constant we can adjust photon number
$n_{t}$ and average energy of laser pulse $E(\overline{n_{t}})$ by controlling
$t$, here $E_{\lambda}$ is energy of single photon, for laser with wavelength
of 1550nm, $E_{\lambda}=0.8eV$.

$E(\overline{n_{t}})=\overline{n_{t}}*E_{\lambda}=k*P_{0}*t$ (1)

Compare to radiation sources, energy of laser pulse is not constant. Full
width of half hight $FWHM_{\overline{n_{t}}}$ from Multi-channel analyser have
two components fluctuations caused by input photon number
$\delta_{\lambda}^{2}$ and fluctuation caused by energy resolution of TES
microcalorimeter itself $\delta_{r}^{2}$.

$FWHM_{n_{t}}=2.35*\delta_{n_{t}}=2.35*\sqrt{\overline{n_{t}}*\delta_{\lambda}^{2}+\delta_{r}^{2}}$
(2)

Although it is not constant, photon number in a laser pulse obey Poisson
distribution $\pi(\overline{n_{t}})$, since the energy of laser photon is
pretty small compare to X-ray source, $\overline{n_{t}}$ should be a very big
number so we can simplify it to a Gauss distribution
$N(\overline{n_{t}},\overline{n_{t}}*\delta_{\lambda}^{2})$ where
$\delta_{\lambda}$ is a constant very near 1

As a thermal based signal detector, energy resolution of TES is almost
constant
$FWHM_{r}=2.35*\delta_{r}=2.35*\sqrt{4k_{B}T_{0}^{2}C/\alpha_{I}\sqrt{n/2}}$
if TES work in linear dynamic range [9]. That is, $\delta_{r}^{2}$ is not
changed with $\overline{n_{t}}$.

So $a=\delta_{\lambda}^{2}$ and $b=\delta_{r}^{2}$ in Eq (2) is constant and
we can simplify Eq (2):

$\delta_{n_{t}}^{2}=(FWHM_{\overline{n_{t}}}/2.35)^{2}=a*\overline{n_{t}}+b$
(3)

According to Eq (3) we can measure $\delta_{n_{t}}^{2}$ at different
$\overline{n_{t}}$ linear fitting it square root the intercept and multiply
2.35, then we get the energy resolution of TES.

##  2 energy resolution measurement

###  2.1 TES test

The TES resistance as a function of temperature, called RT curve is shown in
Fig 2.1. From Fig 2.1 we can get width of RT curve is about 3mK transition
temperature is about 542mK.

Ti/Al TES RT curve.

###  2.2 pulse hight vs $\overline{n_{t}}$

We measured pulse hightes at different $\overline{n_{t}}$ and they show
excellent linearity that’s mean TES is working at a linear work point of. By
pulse hight and shape, we can get the TES current and resistance. Then we can
calculate the energy of each pulse by integral the power change and time.

pulse hight at different $\overline{n_{t}}$

###  2.3 energy spectrum under different input energy

We measured energy spectrum under different $\overline{n_{t}}$ we can see the
$FWHM_{\overline{n_{t}}}$ of the spectrum increase as $\overline{n_{t}}$
increasing, agree with the theory prediction.

energy spectrum under different $\overline{n_{t}}$

##  3 Result and Discussion

###  3.1 square of energy resolution vs $\overline{n_{t}}$

By analysing the energy spectrum we can get ${FWHM_{\overline{n_{t}}}}^{2}$
under different $\overline{n_{t}}$, the result is shown in Fig 3.1, Eq (3)
agree with the result very well. We get $\delta_{\lambda}^{2}=0.992\pm 0.022$
agree with the theory prediction $\delta_{\lambda}^{2}=1$.
$\delta_{r}=38.295\pm 0.488$ corresponding to a $\delta{E}=72.30\pm 0.39eV$
energy resolution.

$\delta{{\overline{n_{t}}}^{2}}$ under different $\overline{n_{t}}$

###  3.2 future work

The experiment results agree very well with the theory predictions, but 72eV
is not a very good energy resolution. The main reason is as the following
description:

First energy resolution is affected wiby temperature,
$FWHM_{r}=2.35*\delta_{r}=2.35*\sqrt{4k_{B}T_{0}^{2}C/\alpha_{I}\sqrt{n/2}}$.
Our He3 refrigerator can only work on temperature higher than 400mK. If we can
reach lower temperature, we may get better resolution.

Secondly, we use Multichannel Signal Analyzer (MSA) to get the energy spectrum
directly, this is not the best method to get the energy resolution, so in the
future, we need design a new data acquisition system to improve the energy
resolution.

##  4 acknowledgment.

Fangjun Lu give a lot of instruction in the statistical method, We acknowledge
the support of CAS Strategic Priority A(Grant No. XDA040762), State Key
Development Program for Basic Research of China(Grant No.2011CBA00304) and
Tsinghua University Initiative Scientific Research Program(Grant
No.20131089314)

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