QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXII, NUMBER 3
SEPTEMBER 2004, PAGES 541-551
STABILITY OF THE RIEMANN SEMIGROUP
WITH RESPECT TO
THE KINETIC CONDITION
By
RINALDO M. COLOMBO (Department of Mathematics, University of Brescia, Italy)
AND
ANDREA CORLI (Department of Mathematics, University of Ferrara, Italy)
Abstract. This note deals with systems of hyperbolic conservation laws that are
endowed with a generalized kinetic relation and develop phase transitions. The L1-
Lipschitzean continuous dependence of the solution from the kinetic relation is proved.
Preliminarily, we rephrase several results known in the case of standard conservation laws
to the case comprising phase boundaries.
1. Introduction. This paper deals with conservation laws in presence of phase tran-
sitions. More precisely, we deal with the system
dtu + dx[f(u)]= 0 (1.1)
with t E [0, +oo[, x £ R, u G 12, /: i—> R™ and $7 C R™, under the assumption that SI
be the disjoint union of two open sets, which we refer to as phases, i.e.,
n = n0u fix. (1.2)
A phase transition is a jump discontinuity in a solution u to 1.1 between states u(t,x—)
and u(t, x+) belonging to different phases.
Physical models leading to this setting are provided by liquid - vapor phase transitions,
elastodynamics, or combustion models; see [2, 7, 8, 9, 19, 20] and the references therein.
Typically, in the case 1.2 the Riemann problem for 1.1 turns out to be underdetermined
and further conditions need to be supplemented. Physically, various criteria have been
devised: viscosity [20], viscocapillarity [19], or other kinetic conditions [2], From an
Received May 16, 2003.
2000 Mathematics Subject Classification. Primary 35L65, 82B26.
Key words and phrases. Hyperbolic Systems, Conservation Laws, Phase Transitions.
E-mail address: rinaldoSing.unibs.it
WWW address: http://dm.ing.unibs.it/rinaldo/
E-mail address: crl@unife.it
WWW address: http://utenti.unife.it/crl/
©2004 Brown University
541
542 RINALDO M. COLOMBO and ANDREA CORLI
analytical point of view, the above criteria can be described through the generalized
kinetic condition
"J/ (u(t,x—), u(t5 a;+)) = 0 (1.3)
for a given smooth function ^ having a suitable number of components.
When no phase transitions develop, 1.1 generates the so-called Standard Riemann
Semigroup, or SRS for short; see [5, 15] and the references therein. Phase transitions,
when present, depend on the particular admissibility criterion 1.3 chosen. Hence, the
solution operator generated by 1.1-1.3 is referred to as the *1'-Riemann Semigroup, or
^RS; see [7]. For the sake of completeness, we note here that 1.3 can be substituted by
constraints on the structure of the solution, as in [8].
The aim of this paper is first (in Sec. 2) to extend several results obtained in the case
of the SRS to the case of systems endowed with a generalized kinetic relation. Secondly,
in Sec. 3, we study the dependence of the solution u to 1.1-1.3 on the flow / and, in
particular, on the function 'P. We shall prove that the solution u in L1 is a Lipschitz
continuous function of / and ^ in C1. The last section is devoted to examples of possible
applications of these results.
2. Notations and Preliminary Results. On the system (1.1) we require that
(1): / is of class C3, the n x n matrix Df(u) is strictly hyperbolic both in Slo and in
Sli; i.e., Df has n real distinct eigenvalues and each characteristic field is either
genuinely nonlinear or linearly degenerate.
For i = 1 ,...,n and u G Q, denote by Ai(u) and r,(u) the i-th eigenvalue and the
corresponding right eigenvector of the n x n matrix A{u) = Df(u). The indexes are
chosen so that Ai_i(-u) < A;(it) for all u and i. If the i-th characteristic field is genuinely
nonlinear, the eigenvector is normalized so that VAi(u) ■ Ti(u) = 1. Denote by A an
upper bound for |Aj (u)l. for all i = 1,..., n and u £ Q. We refer to [5, 11] for the basic
definitions related to conservation laws. In particular, below we mean entropic in the
sense specified by Lax inequalities [5, Formula (4.38)].
Let u: [0,+oo[ x R i—> fl be a weak solution to 1.1, entropic both in fl0 and in 1
and such that u(t, ■) £ BV for all t. A Lipschitz-continuous curve x = p(t) is a phase
boundary if the traces
u{t,p{t)—) = lim u(t,x) and u(t,p(t)+) = lim u(t,x)
x—>p(t) — X—>p(t)+
are in different phases. The phase boundary x = p(t) is of type (j, h) [13] at time t if
(2.1)
Aj_i {u(t,p{t)-)) < p(t) < Aj(u{t,p(t)-)) ,
Ah(u{t,p(t)+)) < p(t) < Ah+i(u(t,p(t)+)) .
The above inequalities mean that the characteristics entering into the phase boundary
are precisely those numbered by j, j +1,..., n on the left and 1,2,... ,h on the right. The
usual Lax shocks of the A;-th characteristic family behave as a (fc, k) phase boundary. The
stability of (2,1) phase boundaries is considered in [7] in the framework of elastodynamics
and liquid-vapor systems.
STABILITY WITH RESPECT TO THE KINETIC CONDITION 543
/ ^1
h + 1
h+ 1
ul
(a) (b)
Fig. 1. (a) Characteristics and a (j, h) phase boundary, (b) Solution
to a Riemann problem. The phase boundary is represented by a thick
line, the other waves by thin lines.
The requirement that a solution u with a phase transition between the states ul G fio
and ur G be a weak solution to 1.1, implies that the Rankine-Hugoniot conditions [5,
§4.2] between ul, ur and the speed p of the transition be satisfied.
In 2.1 we neglect the sonic case in which one of the inequalities is replaced by an
equality. This situation can be treated, for example, as in [8, 9].
2.1. The Riemann Problem. Let ul G and ur G and assume that the Riemann
problem
dtu + dx [/(«)] = 0
ul if x < 0 (2-2)
ur if x > 0
admits a weak solution consisting of a (j, h) phase boundary x = At that satisfies the
Rankine - Hugoniot condition
f(u1)- f(ur) = A- (ul-ur). (2.3)
If j > h, attempting to solve any small perturbation of 2.2 leads to an underdetermined
problem; see [13]. Indeed, further j — h conditions need to be supplemented through the
introduction of an admissibility function : (f^o x U x fl0) i—> R-5-'1. We assume
throughout that ^ is of class C2 and, in order to respect the x —> —x symmetry of 1.1,
we also require that
V(ul,ur) = V{ur,ul). (2.4)
Definition 2.1. Given problem 1.1 together with an admissibility function : (f2o x
fil) U (fii x fig) * R-7'-'1, a phase transition in a weak entropic solution u to 1.1 is
admissible if 1.3 is satisfied at almost every point of the phase boundary. A -admissible
solution to the Riemann problem
dtu + dx [/(u)] = 0
ul if x < 0 (2.5)
ur if x > 0
u(0, x) =
u( 0, x) =
with data in different phases is a self-similar weak solution to 1.1 consisting, from left to
right,
(1) if ul e fio and ur G fii, of j — 1 Lax waves in f2o, a ^-admissible phase boundary,
and n — h Lax waves in fix;
544 RINALDO M. COLOMBO AND ANDREA CORLI
(2) if ul 6 fix and ur E Ooi of n — h Lax waves in a ^-admissible phase boundary,
and j — 1 Lax waves in Qo-
Above, by Lax waves we mean the usual (possibly null) simple waves that constitute the
Lax [16] solution to Riemann problems.
The local well-posedness of the Riemann problem 2.5 near y} and ur requires suitable
compatibility conditions between / and \I>. A sufficient condition, obtained in [10], is
(2): iP: (fio x Qi) U (fij x f20) > ~R,J~h is of class C1, satisfies 2.4 and the matrix
(A-Aiki (A-A [u] (A-A h+1)rh+1 ... (A - A„)r„
D^r:i ... D1frj_1 0 -D2Vrh+1 ... -D2^rn
is invertible.
Above, Di'i' (resp. D2^) is the (j — h) x n matrix of the partial derivative of VP with
respect to the first (resp. second) argument ul (resp. ur), evaluated at (ul,ur). Similarly,
Ai,..., Aj_i and the corresponding eigenvectors are computed at u , while A^+i,..., Xn
as well as their related eigenvectors are evaluated at ur.
A direct application of the implicit function theorem leads to the following proposition.
Proposition 2.2. Let assumption (1) hold. Fix two states ul & fio and ur € such
that (2) holds. Then, for all it', ur in suitable neighborhoods of ul and ur, the Riemann
problem 2.5 admits a unique ^-admissible solution in the sense of Definition 2.1.
When ul and ur are in the same phase, a Lax solution to 2.2 may not necessarily exist,
for the jump ||u( — wr|| may well be large. It is then natural to look for a solution to 2.2
containing two different phase transitions. Such a solution models the nucleation of two
phase boundaries. A solution to 2.2, and hence also of its perturbation 2.5, is obtained
by gluing solutions of type (1) and (2) above. Note that the two phase boundaries need
not be of the same type (j, h); hence, a wide variety of cases may appear. The extension
of the Proposition above to the case of nucleation (and even to the case of several phase
boundaries) follows easily, simply assuming the stability condition (2) on each of the
phase boundaries in the solution to the unperturbed problem 2.2; see [7, 17, 18].
2.2. The -Riemann Semigroup. In this first part, most of the proofs are omitted, for
they usually follow from the corresponding ones related to the SRS and full references
are provided. In this section we consider (j, h) phase boundaries, for fixed j and h, j > h.
Definition 2.3. System 1.1 and condition 1.3, with the admissibility function "I*: (f2ox
f^i) U (fii x n0) i—► satisfying 2.4, generate a $-Riemann Semigroup ("fRS)
S: [0, +oo[ x V h-> D if the following holds:
(1) V is a non-trivial domain in BV(R);
(2) 5 is a semigroup: So = Id and StoSs = St+s;
(3) S is L1-Lipschitzean: there exists a positive L such that for all u, w £ V,
||Stii - S*H|l1 < L ' (llu - HIll + \f - sl) >
(4) if it £ P is piecewise constant with jumps at, say, Xj, j = 1,..., m, then for t
small, Stu coincides with the gluing of the ^-admissible solutions to the Riemann
problem 2.5 with ul — u(xj-) and ur = u(xj+).
STABILITY WITH RESPECT TO THE KINETIC CONDITION 545
In the case of the Standard Rieraann Semigroup (SRS) (see [5]), (1) above amounts to
asking that V contains all functions with suitably small total variation. In [7], this condi-
tion is replaced by the assumption that V contains, at least, all BV-small perturbations
of the Riemann data 2.2. Note also that, due to the uniform continuity implied by (3),
S can be uniquely extended to the L1 closure of V.
A first result towards the construction of a 'I'RS was obtained in [7] in the case n = 2.
In the general case n > 2, the techniques used in [7, 17, 18] allow to prove the following
result. Denote by M. the set of smooth increasing diffeomorphisms Rm R,
Theorem 2.4. Assume that /: Ho Ufii h Rn satisfies (1). Fix j, h £ {1,... ,n} with
j > h and let $: (fio x ^i) U (f2i x Qq) i—> RJ~'1, ul £ Hq and ur £ such that (2)
holds. Let u be the 'F-admissible solution to 2.2.
Then, the problem 1.1-1.3 generates a ^RS S: [0, +oo[ x T> i-> V with the following
properties:
(1) t i—* Stu is a weak solution to 1.1-1.3;
(2) there exists a S > 0 such that V contains all u: R i—> 0, for which 3/x £ M. with
||u(-) - u(l,/x(-))||Li < °°, TV{u(-) - u{l,n(-))} < 6; (2.6)
(3) for every u £ V, Stu is the limit of front tracking approximations;
(4) there exists a map p: [0,+oo[ h-> R whose graph x = p(t) supports the phase
boundary, p is Lipschitzean, and p has bounded variation.
We remark that the above theorem also ensures the structural stability of the phase
boundary with respect to all perturbations having suitably small total variation.
Note also that in the case of nucleation, i.e., of two phase boundaries in the solution
to 2.2, Theorem 2.4 still holds, provided the so-called strong non-resonance conditions
are imposed on /, and on the data ul, ur; see [6, Formula (2.13)], [7, Formula (2.12)],
[17, Formulas (1.11)—(1.12)], and [18, Formulas (2.12)-(2.13)].
Essentially, as in the case of the SRS, when Theorem 2.4 applies, then the ^RS is
unique and its orbits yield solution to 1.1—1.3.
Theorem 2.5. With the same assumptions of Theorem 2.4 above, call S: [0,+oo[x£> h-»
V the ^RS constructed therein. Call S: [0,+oo[ xPhP another *I/RS, with PDP,
Then, for all u £ V,
Stu = Stu for all t > 0.
The proof follows the lines of [5, Theorem 9.1],
Once a 'I'RS is constructed as limit of wave front tracking approximations, the problem
of characterizing the 'I'RS can also be solved, again as in the case of the SRS. Indeed,
consider a trajectory t i—> Sf.u of the *I/RS S. Fix a point (r, £) and denote =
limx_,£-j- u(t, x). We define:
• the function U^u.T ^ is the "f-admissible solution to the Riemann problem with
initial data (m~,m+);