QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXII, NUMBER 1
MARCH 2004, PAGES 163-179
STABILITY OF CONSTANT EQUILIBRIUM STATE
FOR DISSIPATIVE BALANCE LAWS SYSTEM WITH A CONVEX
ENTROPY
By
TOMMASO RUGGERI (Department of Mathematics and Research Center of Applied
Mathematics (C.I.R.A.M.), University of Bologna, Via Saragozza 8, 40123 Bologna, Italy)
DENIS SERRE (UMPA (UMR CNRS/ENS Lyon 5669) Ecole Normale Superieure de Lyon, 46,
allee d'ltalie, 69364 Lyon Cedex 07, France)
Abstract. For a one-dimensional system of dissipative balance laws endowed with a
convex entropy, we prove, under natural assumptions, that a constant equilibrium state
is asymptotically L2-stable under a zero-mass initial disturbance. The technique is based
on the construction of an appropriate Liapunov functional involving the entropy and a
so-called compensation term.
1. Introduction. Recently, nonequilibrium theories and, in particular, the Extended
Thermodynamics [16] have generated a new interest in quasi-linear hyperbolic systems
of balance laws with dissipation due to the presence of production terms (systems with
relaxation). On this subject, it is very important to find connections between properties
of the full system and the associated subsystem obtained when certain parameters (re-
laxation coefficients) are just equal to zero. Mathematical examples on this topic were
developed in the linear case by Whitham [21] and in the nonlinear case by Liu [14] and by
Chen, Levermore, and Liu [5]. The requirement that the system of balance laws satisfies
an entropy principle with a convex entropy density gives several strong restrictions. In
fact, as is well known, starting from the observation of Godunov [9], it was shown that the
entire system of balance laws can be put in a very special hyperbolic symmetric system
providing the introduction of main field variables [2], [17]. As was observed by Boillat
and Ruggeri [3], the main field also permits us to recognize that these nonequilibrium
systems have a structure of nesting theories. In fact, it is possible to define the principal
subsystems that are obtained by freezing some components of the main field that have the
properties that preserve the existence of a convex entropy law and the spectrum of the
Received July 24, 2002.
2000 Mathematics Subject Classification. Primary 35L65.
This research work was done when Denis Serre was visiting C.I.R.A.M. in April/May 2002.
©2004 Brown University
163
164 TOMMASO RUGGERI and DENIS SERRE
characteristic eigenvalues is contained into the one of the full system (sub-characteristic
conditions). A particular subsystem is the equilibrium one.
Our goal in this paper is to prove that, under natural assumptions, a constant equilib-
rium state of such balance laws is asymptotically L2-stable. Though the technique em-
ployed here may look rather classical, involving an "energy" (actually entropy) estimate,
plus a compensation term as introduced by Kawashima et al. [13] for other purposes, it
has the nice feature that it applies to weak entropy solutions. It is therefore valid in the
presence of shock waves. For some natural reason, due to the finite propagation velocity
of the support of a solution, it is natural to assume that the total mass of the conserved
components of the unknown vanish. Under this condition, we find a i"1/2 decay rate of
the L2-norm of the solution, though the decay could not be better than £-1/4 in general,
that is, when a non-zero mass is present at initial time. Though we were able to get rid
of this zero-mass assumption in the linear case, to the price of the loss of the decay rate,
we did not succeed in overcoming this restriction in the nonlinear case.
The plan is as follows. The next section recalls a few important facts about dissipative
balance laws endowed with a convex entropy. In particular, we emphasize the solution
of equilibrium subsystem. Then we turn to the analysis of the linear case, where we
construct an appropriate quadratic Liapunov function whose dissipation rate is positive
definite. In the last section, we deal with the more involved nonlinear case, where a few
additional ideas are needed. Our main result is Theorem 8.
We should mention here a companion paper [19], where the Instability is proved for
the Jin-Xin relation model (see [11]) in the presence of a positively invariant domain.
There, the linearity of the principal part allows a treatment by compensated compactness,
which yields a similar but weaker result, without the zero-mass assumption.
At last, we must emphasize our smallness hypothesis, which constrained not only the
data but the solution itself. We feel that this is reasonable, especially in the light of the
global existence of smooth solutions for small data, obtained recently by Hanouzet and
Natalini [10].
2. Balance laws, systems, entropy, and generators. Let us consider a general
hyperbolic system of N balance laws:
daFa(u) = F(u), (1)
where the densities F°, the fluxes Fl and the productions F are IR^-column vectors
depending on the space variables xl (i = 1,2,3) and the time t = x°, (a = 0,1,2,3; da —
d/dxa) through the field u = u(xa) G HiN.
Now we suppose, as is usual in the applications, that the system (1) satisfies an entropy
principle; i.e., there exists an entropy density h = h°(u) and an entropy flux hl(u) such
that for every solution of (1) we have a nonpositive-entropy production1 £(u):
daha = E < 0. (2)
thermodynamics, —h°, —hl, and —£ have the physical meaning of entropy density, entropy flux,
and entropy production, respectively. But, by mathematical tradition, it is common to use the word
convex entropy density instead of concave entropy density and improperly call entropy the negative
entropy.
STABILITY OF CONSTANT EQUILIBRIUM STATE 165
The compatibility between (1) and (2) implies the existence of a main field u' of
Lagrange multipliers such that
daha -E = u' ■ (daFa - F). (3)
As a consequence of this identity, we have
dha = u' ■ dFa, E = u'-F<0. (4)
As it has been well known since the pioneering papers of Godunov [9] and Friedrichs and
Lax [8], it is possible to show [2, 17] that, because of (4)i, there exist four potentials h'a:
h'a = u'■ Fa - ha, (5)
such that
dh'a
(6)
It follows that, upon selecting the main field as the field variables, the original system
(1) can be written with Hessian matrices in the symmetric form
provided that h = h° is a convex function of u = F° (or, equivalently, the Legendre
transform h! = h'° is a convex function of the dual field u'). This form is called canonical
by Dafermos [6].
3. Principal subsystems. We split the main field u! € into two parts: u! =
(v',w'),v' G Mm,uj' € RN~M (0 < M < N); and the system (7) with F = (/, 7r), reads
»«(^) =/(»>')■ W
(f^) = (9)
Given some assigned value w'^x01) of w' (in the usual case w'M — const.), Boillat and
Ruggeri [3] call a principal subsystem of (8) and (9) the system
9a (^7= /KX). (10)
By construction, the solutions of the principal subsystems satisfy also a supplementary
entropy law with convex density and the spectrum of the characteristic eigenvalues is
included in the spectrum of the full system (sub-characteristic conditions) [3].
4. Equilibrium subsystem. In examples from physics involving nonequilibrium
processes such as extended thermodynamics [16], M < N equations of (1) represent
conservation laws while the remaining ones contain productions responsible for the dis-
sipative mechanism. In this case, / = 0 in (8). Recalling that a thermodynamical
equilibrium state is such that the production of the entropy E in (2) vanishes and attains
its maximum value:
/ qy, \ ( d2T, \
Eli! = 0' WJE = 0' (aSwJ,, 'S "egative semi-definite, (11)
166 TOMMASO RUGGERI and DENIS SERRE
it is easy to prove [4]:
Theorem 1. In an equilibrium state, under the assumption of dissipative productions,
i.e., if
dir ( dir \ ^ 1
0^ + | is negative definite, (12)
then the production vanishes and the main field components vanish except for the first
M ones. Thus
tt|e = 0, w'\e = 0. (13)
These results reveal another privilege of the main field. In fact, while the field of
densities u are in general all different from zero in equilibrium, the components of the
main field are all zero except for the first M ones. Therefore the infinite equilibrium
states are the ones belonging to the M-dimensional manifold
w'(v, w) = 0 (14)
as solution of the N — M equations (14) (we recall the global univalence between w
and w' from the concavity assumption), it is possible to write u>\e as function of v.
The principal system of M conservation law with w = w \e is the equilibrium principal
subsystem associated to the system (8) and (9).
Another important characteristic property of the equilibrium state is put in evidence
from the following theorem:
Theorem 2. In equilibrium the entropy density h is minimal; i.e.,
h > H\e Vm ^ u\e,
where
h\E = h{v,w\E{v)).
The proof follows immediately, taking into account the convexity of h:
W = h\E - h + u'\e • {u - u\e) < 0 Vu^u\e, (15)
and therefore choosing
U ). (16)
wj Vw|£(v)/
The last term in (15) is null due to the fact that u'\e and u — u\e are orthogonal (see
(16) and (13)2) and therefore
W = h\s - h < 0 Vu 7^ u\E. (17)
In this manner we have confirmed that the entropy density reaches its minimum value
in the equilibrium state.
We would stress the fact that these results are a consequence not only of the convexity
of the entropy density but also of the dissipation condition (12). In the mathematical
literature there exist several definitions of dissipation. The one most used is due to
Dafermos [7].
STABILITY OF CONSTANT EQUILIBRIUM STATE 167
Now we consider the one-dimensional case and, taking into account (13), we assume,
without loss of generality, that n — —ffw'; i.e.:
rdth'v, +dxk'v, = 0
(dt/w + dxkw> = -g(v',w')w',
with D2h! > 0 and E := tw'g(v', w')w'.
5. The linear problem. In a first instance, we restrict to the linear case:
• h! is a positive definite quadratic form,
• k' is a quadratic form,
• g is a constant matrix.
Lemma 3. Without loss of generality, we may suppose that h = |(|i/|2 + |u/|2). Therefore
v = v'\ w = w', and the system rewrites
dtv + dx(K0v + Kiw) = 0,
dtw -I- dx(K^v + K2w) = -Gw,
where K0, K2 are symmetric, and twGw > 0 for w ^ 0.
Proof. We have
ti = ^v'Hqv' + tv'Hlw' + ^tw'H2w'
and
h'v, = H0v' + HlW'.
Since Ho > 0, we introduce the square root ho = \fHa and define y := h0v' + Hq lH\w' =
o:
/in 1h',. Then we have
h' = \\V\2 + - HlHo1H1)w'.
By Schur's complement theorem (see, for instance, [20]), H2 — Hj Hq1Hi is positive.
Let hi be its square root, and define z := h\w'. Then
h = \{\y? + \z\2).
In variables w, z, the system still writes
dtV + dx (linear flux) = 0,
dtz + c?x(linear flux) = —gz.
Lastly, since h is an entropy of the problem, the fluxes must be given by a potential,
obviously a quadratic one. The general form of this potential is
^yKoy + lyKiz + ilzK2z.
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