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Hyperbolic systems of conservation laws in one space dimension

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Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations.
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arXiv:math/0212392v1 [math.AP] 1 Dec 2002
ICM 2002 · Vol. I · 159–178
Hyperbolic Systems of Conservation Laws
in One Space Dimension
Alberto Bressan*
Abstract
Aim of this paper is to review some basic ideas and recent developments
in the theory of strictly hyperbolic systems of conservation laws in one space
dimension. The main focus will be on the uniqueness and stability of entropy
weak solutions and on the convergence of vanishing viscosity approximations.
2000 Mathematics Subject Classification: 35L60, 35L65.
Keywords and Phrases: Hyperbolic system of conservation laws, Entropy
weak solution, Vanishing viscosity.
1. Introduction
By a system of conservation laws in m space dimensions we mean a first order
system of partial differe ntial equations in divergence for m:
t
U +
m
X
α=1
x
α
F
α
(U) = 0 , U IR
n
, (t, x) IR × IR
m
.
The components of the vector U = (U
1
, . . . , U
n
) are the conserved quantities. Sys-
tems of this type express the bala nce equations of continuum physics, when small
dissipation effects are neglected. A basic example is provided by the eq uations of
non-viscous gases, acc ounting for the conse rvation o f mass , momentum and ener gy.
The s ubject is thus very classical, having a long tradition which can be traced back
to Euler (1755) and includes contributions by Stokes, Riemann, Weyl and Von Neu-
mann, among several others. The continued attention of analysts and mathematical
physicists during the span of over two centuries, however, has not accounted for a
comprehensive mathematical theory. On the contrary, as remar ked in [Lx2], [D2],
[S2], the field is still replenished with challenging open problems. In several space
dimensions, not even the global existence of solutions is presently known, in any
* S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy. E-mail: bressan@sissa.it
160 A. Bressan
significant degree of generality. Until now, most of the analysis has been concerned
with the one-dimensional case, and it is only here that basic questions could b e
settled. In the re mainder of this pape r we shall thus consider systems in one space
dimension, referring to the books of Majda [M], Serre [S1] or Dafermos [D3] for a
discussion of the multidimensional case.
Toward a rigorous mathematical analysis of solutions, the main difficulty that
one encounters is the lack of regularity. Due to the strong nonlinearity of the
equations and the absence of diffusion terms with s moothing effect, solutions which
are initially smooth may become discontinuous within finite time. In the presence
of discontinuities, most of the classical tools of differe ntial calculus do not a pply.
Moreover, for general n ×n systems, the powe rful techniques of functional analysis
cannot be used. In particular, solutions cannot be re presented as fixed points of
a nonlinea r transformation, or in variational form as critical points of a suitable
functional. Dealing with vector valued functions, comparison arg uments based on
upper and lower solutions do not apply either. Up to now, the theory of conservation
laws has progre ssed largely by ad hoc methods. A survey of these techniques is the
object of the prese nt paper.
The Cauchy problem for a system of conservatio n laws in one space dimension
takes the form
u
t
+ f(u)
x
= 0, (1.1)
u(0, x) = ¯u(x). (1.2)
Here u = (u
1
, . . . , u
n
) is the vector of conserved quantities, while the c omponents
of f = (f
1
, . . . , f
n
) are the fluxes. We shall always assume that the flux function
f : IR
n
7→ IR
n
is smooth and that the system is strictly hyperbolic, i. e., at ea ch
point u the Jacobian matrix A(u) = Df(u) has n real, distinct eigenvalues
λ
1
(u) < ··· < λ
n
(u). (1.3)
As already mentioned, a distinguished feature of nonlinear hyperbolic systems is
the po ssible loss of regularity. Even with smooth initial data, it is well known that
the solution ca n develop shocks in finite time. Therefore, solutions defined globally
in time c an o nly be found within a space of discontinuous functions. The equation
(1.1) must then be interpreted in distributional sense. A vector valued function
u = u(t, x) is a weak solution of (1.1) if
ZZ
u φ
t
+ f(u) φ
x
dxdt = 0 (1.4)
for every test function φ C
1
c
, continuously differentiable with compact suppor t.
In particular, the piecewise constant function
u(t, x)
.
=
u
if x < λt ,
u
+
if x > λt ,
(1.5)
Hyperbolic Systems of Conservation Laws 161
is a weak solution of (1.1) if and only if the left and right states u
, u
+
and the
sp e e d λ satisfy the famous Rankine-Hugoniot equations
f(u
+
) f(u
) = λ (u
+
u
) . (1.6)
When discontinuities are prese nt, the weak so lution of a Cauchy problem may
not be unique. To single out a unique “good” solution, additional entropy conditions
are usually imposed along shocks [Lx1], [L3]. These conditions often have a physical
motivation, characterizing those solutions which can be recovered from higher order
models, letting the diffusion or dispersion coefficients approach zero (see [D3]).
In one space dimension, the mathematical theory of hyper bolic systems of
conservatio n laws has developed along two main lines.
1. The BV setting, pioneered by Glimm (1965). Solutions are here constructed
within a space of functions with bounded variation, controlling the BV norm by a
wave interaction potential.
2. The L
setting, introduced by DiPerna (1983), ba sed on weak convergence and
a compensated compactness argument.
Both appro aches yield results on the global existence of weak solutions. How-
ever, it is only in the BV setting that the well posedness of the Cauchy problem
could recently be proved, as well as the stability and c onverge nce of vanishing vis-
cosity approximations. On the other hand, a counterexample in [BS] indicates that
similar results cannot be expected, in general, for solutions in L
. In the remainder
of this paper we thus concentrate on the theory of BV solutions, referring to [DP2]
or [S1] for the alternative approach based on compensated compactness.
We shall first review the main ideas involved in the construction of weak so-
lutions, based on the Riemann problem and the wave interaction functional. We
then present more recent results on stability, uniqueness and characterization of
entropy weak solutions. All this material can be found in the monograph [B3]. The
last section contains an outline of the latest work o n stability and convergence of
vanishing viscosity approximations.
2. Existence of weak solutions
Toward the construction of more general solutions of (1.1), the basic building
block is the Riemann problem, i.e. the initial value problem where the data are
piecewise constant, with a single jump at the origin:
u(0, x) =
u
if x < 0 ,
u
+
if x > 0 .
(2.1)
Assuming that the amplitude |u
+
u
| of the jump is small, this problem was
solved in a classical paper of Lax [Lx1], under the additional hypothesis
162 A. Bressan
(H) For each i = 1, . . . , n, the i- th field is either genuinely nonlinear, so that
Dλ
i
(u) · r
i
(u) > 0 for all u, or linearly degenerate, with Dλ
i
(u) · r
i
(u) = 0 for
all u.
The solution is self-similar: u(t, x) = U(x/t). It consists of n + 1 constant sta tes
ω
0
= u
, ω
1
, . . . , ω
n
= u
+
(see Fig. 1). Each couple of adiacent states ω
i1
, ω
i
is separated either by a shock (the thick lines in Fig. 1) satisfying the Rankine
Hugoniot equations, or else by a centered rarefaction. In this second case, the
solution u varies continuously between ω
i1
and ω
i
in a sector of the t-x-plane (the
shaded region in Fig. 1) where the gradient u
x
coincides with an i-eigenvector of
the matrix A(u).
2∆
2∆
ω =
0
u
ω =
3
u
+
x
0
t
t
x x
ω
1
t
ω
2
Figure 1 Figure 2
Approximate solutions to a more general Cauchy problem can be constructed
by patching together several solutions of Riema nn problems. In the Glimm scheme
(Fig. 2), one works with a fixed grid in the x-t plane, with mesh sizes x, t. At
time t = 0 the initial data is approximated by a piecewise constant function, with
jumps at grid points. Solving the correspo nding Riemann problems, a solution is
constructed up to a time t sufficiently s mall so that waves generated by different
Riemann problems do not interact. By a r andom sampling procedure, the solution
u(∆t, ·) is then approximated by a piecewise constant function having jumps only
at grid points. Solving the new Riemann problems at every o ne of these points, one
can prolong the solution to the next time interval [∆t, 2∆t], etc. . .
t
1
t
2
x
x
σ
σ
t
Figure 3
An alternative technique for co ntructing approximate solutions is by wave-
Hyperbolic Systems of Conservation Laws 163
front tracking (Fig. 3). This method was introduced by Dafermos [D1] in the scalar
case and later developed by various authors [DP1], [B1], [R], [BJ]. It now provides
an efficient tool in the study of general n ×n systems of c onservation laws, both for
theoretical and numerical purpo ses [B3], [HR].
The initial data is here approximated with a piecewise constant function, and
each Riemann problem is solved approximately, within the cla ss of piecewise con-
stant functions. In particula r, if the exact solution contains a centered rarefaction,
this must be approximated by a rarefaction fan, containing several small jumps. At
the fir st time t
1
where two fronts interact, the new Riemann problem is again ap-
proximately solved by a piecewise constant function. The solution is then prolonged
up to the second interaction time t
2
, where the new Riemann problem is solved,
etc. . . The main difference is that in the Glimm scheme one specifies a priori the
nodal points where the the Riema nn problems are to be solved. On the other hand,
in a solution constructed by wave-front tracking the locations of the jumps and of
the interaction points depend on the solution itself, and no restarting proce dure is
needed.
In the end, both algo rithms produce a sequence of approximate solutions,
whose convergence relies on a compactness argument based on uniform bounds on
the total variation. We sketch the main idea involved in these a priori BV bounds.
Consider a piecewise constant function u : IR 7→ IR
n
, say with jumps at points
x
1
< x
2
< ··· < x
N
. C all σ
α
the amplitude of the jump at x
α
. T he total s trength
of waves is then defined as
V (u)
.
=
X
α
|σ
α
|. (2.2)
Clearly, this is an equivalent way to measur e the total variation. Along a solution
u = u(t, x) constructed by front tracking, the quantity V (t) = V
u(t, ·)
may well
increase at interaction times. To provide global a priori bounds, following [G] one
introduces a wave interaction potential, defined as
Q(u) =
X
(α,β)∈A
|σ
α
σ
β
|, (2.3)
where the summation runs over the set A of a ll couples of approaching waves.
Roughly speaking, we say that two wave-fronts located at x
α
< x
β
are approaching
if the one at x
α
has a faster speed than the one at x
β
(hence the two fronts are
exp ected to collide at a future time). Now consider a time τ where two incoming
wave-fronts interact, say with strengths σ, σ
(for exa mple, take τ = t
1
in Fig . 3).
The differe nce betwee n the outgoing waves emerging from the interaction and the
two incoming waves σ, σ
is of magnitude O(1) · |σσ
|. On the other hand, after
time τ the two incoming waves are no longer approaching. This accounts for the
decrease of the functional Q in (2.3) by the amount |σσ
|. Observing that the new
waves generated by the interaction could approach all other fronts, the change in
the functionals V, Q acr oss the interaction time τ is estimated as
V (τ) = O(1) · |σσ
|, Q(τ) = −|σσ
| + O(1 ) · |σσ
|V (τ).
164 A. Bressan
If the initial data has small total variation, for a suitable constant C
0
the quantity
Υ(t)
.
= V
u(t, ·)
+ C
0
Q
u(t, ·)
is monotone decreasing in time. This argument provides the uniform BV bounds on
all approximate solutions. Using Helly’s compactness theorem, one obtains the con-
vergence of a subsequence of approximate solutions, and hence the global existence
of a weak solution.
Theorem 1. Let the system (1.1) be strictly hyperbolic and satisfy the assumptions
(H). Then, for a sufficiently small δ > 0 the following holds. For every initial
condition ¯u with
k¯uk
L
< δ , Tot.Var.{¯u} < δ , (2.4)
the Cauchy problem has a weak solution, defined for all times t 0.
This result is based on careful analysis of solutions of the Riemann pr oblem
and on the use of a quadratic interaction functional (2 .3) to control the creation of
new waves. Thes e techniques als o provided the basis for subsequent inves tigations
of Glimm and L ax [GL] and Liu [L2] on the asymptotic behavior of weak s olutions
as t .
3. Stability
The pre vious existence result relied on a compactness argument which, by
itself, does not provide informations on the uniqueness of solutions. A first under-
standing of the dependence of weak solutions on the initial data was provided by
the analysis of front tracking approximations. The idea is to perturb the initial
data by s hifting the position of one of the jumps, say from x to a nearby po int x
(see Fig. 3). B y carefully estimating the corresponding shifts in the positions of
all wave-fronts at a later time t, one obtains a bound on the L
1
distance between
the original and the per turbed approximate solution. After much technical work,
this appr oach yielded a proof of the Lipschitz continuous dependence of solutions
on the initial data, first in [BC1] for 2 ×2 systems, then in [BCP] for general n ×n
systems.
Theorem 2. Let the system (1.1) be strictly hyperbolic and satisfy the assumptions
(H). Then, for every initial data ¯u satisfying (2.4) the weak solution obtained as
limit of Glimm or front tracking approximations is unique and depends Lipschitz
continuously on the initial data, in the L
1
distance.
These weak solutions can thus be written in the fo rm u(t, ·) = S
t
¯u, as tra-
jecories of a semigroup S : D × [0, [ 7→ D on some domain D containing all func-
tions with sufficiently small total variatio n. For some Lipschitz constants L, L
one
has
S
t
¯u S
s
¯v
L
1
L k¯u ¯vk
L
1
+ L
|t s|, (3.1)
Hyperbolic Systems of Conservation Laws 165
for all t, s 0 and initial data ¯u, ¯v D.
An alternative proof of Theorem 2 was later achieved by a technique introduced
by Liu and Yang in [LY] and pr e sented in [BLY] in its final form. The heart of the
matter is to construct a nonlinear functional, equivalent to the L
1
distance, which
is decreasing in time along every pair of solutions. We thus seek Φ = Φ(u, v) and a
constant C such that
1
C
·
v u
L
1
Φ(u, v) C ·
v u
L
1
, (3.2)
d
dt
Φ
u(t), v(t)
0. (3.3)
v
u
= u(x)
0
ω
ω
1
2
ω
3
ω
= v(x)
x
x
α
q
1
3
q
α
σ
Figure 4
In connection with piecewise constant functions u, v : IR 7→ IR
n
generated by
a front tracking algorithm, this functional can be defined as follows (Fig. 4). At
each point x, we connect the states u(x), v(x) by mea ns of n shock curves. In
other words, we construct intermediate states ω
0
= u(x), ω
1
, . . . , ω
n
= v(x) such
that each pair ω
i1
, ω
i
is connected by an i-shock. These states can b e uniquely
determined by the implicit function theorem. Call q
1
, . . . , q
n
, the strengths of these
shocks. We rega rd q
i
(x) as the i-th scalar component of the jump
u(x), v(x)
. For
some constant C
, one clearly has
1
C
·
v(x) u(x)
n
X
i=1
q
i
(x)
C
·
v(x) u (x)
. (3.4)
The functional Φ is now defined as
Φ(u, v)
.
=
n
X
i=1
Z
−∞
W
i
(x)
q
i
(x)
dx, (3.5)
where the weights W
i
take the form
W
i
(x)
.
= 1 + κ
1
·
total strength o f waves in u and in v
which approa ch the i- wave q
i
(x)
+ κ
2
·
wave interaction potentials of u and of v
.
= 1 + κ
1
V
i
(x) + κ
2
Q(u) + Q(v)
(3.6)
166 A. Bressan
for suitable constants κ
1
, κ
2
. Notice that, by construction, q
i
(x) re presents the
strength of a fictitious shock wave located at x, travelling with a speed λ
i
(x) de-
termined by the Rankine-Hugoniot equations. In (3.6), it is thus meaningful to
consider the quantity
V
i
(x)
.
=
X
α∈A
i
(x)
|σ
α
|,
where the summation extends to all wave-fronts σ
α
in u and in v which are ap-
proaching the i-shock q
i
(x). From (3.4) and the boundedness of the weights W
i
,
one ea sily derives (3.2). By ca reful estimates on the Riemann problem, one can
prove that also (3.3) is approximately satisfied. In the end, by taking a limit of
front tracking approximations, one obta ins Theor e m 2.
For general n × n systems, in (3.1) one finds a Lipschitz cons tant L > 1.
Indeed, it is only in the scalar case that the semigro up is contractive and the theor y
of accretive operators and abstract evolution equations in Banach spaces can be
applied, see [K], [C]. We refer to the flow generated by a system of conservation
laws as a Riemann semigroup, beca use it is entirely determined by specifying how
Riemann problems are solved. As proved in [B2], if two s emigroups S, S
yield the
same solutions to all Riemann problems, then they coincide, up to the choice of
their domains.
From (3.1) one can deduce the error bound
w(T ) S
T
w(0)
L
1
L ·
Z
T
0
(
lim inf
h0+
w(t + h) S
h
w(t)
L
1
h
)
dt , (3.7)
valid for every Lipschitz continuous map w : [0, T ] 7→ D taking values inside the
domain of the semigroup. We can think of t 7→ w(t) as an approximate s olution of
(1.1), while t 7→ S
t
w(0) is the exact solution having the same initial data. According
to (3.7), the distance at time T is bounded by the integral of an instantaneous error
rate, amplified by the Lipschitz constant L of the semigr oup.
Using (3.7), one can estimate the distance between a front tracking approxima-
tion and the corresponding exact solution. Fo r approximate solutions constructed
by the Glimm scheme, a direct application of this same formula is not possible
because of the additional erro rs introduced by the restarting procedures a t times
t
k
.
= k t. However, relying on a careful a nalysis of Liu [L1], one can construct a
front tracking approximate solution having the same initial and terminal values a s
the Glimm solution. By this technique, in [BM] the authors prove d the estimate
lim
x0
u
Glimm
(T, ·) u
exact
(T, ·)
L
1
x · |ln x|
= 0 . (3.8)
In other words, letting the mesh sizes x, t 0 while keeping their ratio x/t
constant, the L
1
norm of the error in the Glimm approximate s olution tends to zero
at a rate slightly slower than
x.
Hyperbolic Systems of Conservation Laws 167
4. Uniqueness
The uniquenes s and stability results stated in Theorem 2 refer to a special
class of weak solutions: those obtained as limits of Glimm or front tracking ap-
proximations. For several applications, it is desirable to have a uniqueness theorem
valid for general weak solutions, without reference to any particular constructive
procedure. Results in this direction were proved in [BLF], [BG], [BLe]. They are
all based on the error formula (3.7). In the proo fs, one considers a weak solution
u = u (t, x) of the Cauchy problem (1.1)–(1.2). Assuming that u satisfies suitable
entropy and regularity conditions, one shows that
lim inf
h0+
u(t + h) S
h
u(t)
L
1
h
= 0 (4.1)
at almost every time t. By (3.7), u thus coincides with the semig roup tr ajectory
t 7→ S
t
u(0) = S
t
¯u. Of course, this implies uniqueness. As an ex ample, we state
below the result of [BLe]. Consider the following assumptions:
(A1) (Conservation Equations) The function u = u(t, x) is a weak solution of
the Cauchy problem (1.1)–(1.2), taking va lues within the domain D of the
semigroup S. More precisely, u : [0, T ] 7→ D is co ntinuous w.r.t. the L
1
distance. The initial condition (1.2) holds, together with
ZZ
u φ
t
+ f (u) φ
x
dxdt = 0
for every C
1
function φ with compact support contained inside the open strip
]0, T [ ×IR.
(A2) (Lax Entropy Condition) Let u have an approximate jump discontinuity at
some point (τ, ξ) ]0, T [×IR. In other words, assume that there exists states
u
, u
+
and a speed λ IR such that, calling
U(t, x)
.
=
u
if x < ξ + λ(t τ),
u
+
if x > ξ + λ(t τ),
(4.2)
there holds
lim
ρ0+
1
ρ
2
Z
τ +ρ
τ ρ
Z
ξ+ρ
ξρ
u(t, x) U (t, x)
dxdt = 0. (4.3)
Then, for some i {1, . . . , n}, o ne has the entropy inequality:
λ
i
(u
) λ λ
i
(u
+
). (4.4)
(A3) (Bounded Variation Condition) The function x 7→ u
τ(x), x) has bounded
variation along every Lipschitz continuous space-like curve
t = τ(x)
, which
satisfies |/dx| < δ a.e., for some constant δ > 0 small enough.
168 A. Bressan
Theorem 3. Let u = u(t, x) be a weak solution of the Cauchy problem (1.1)–(1.2)
satisfying the assumptions (A1), (A2) and (A3). Then
u(t, ·) = S
t
¯u (4.5)
for all t. In particular, the solution that satisfies the three above conditions is unique.
An additional characterization o f these unique solutions, based on local integral
estimates, was given in [B2]. The underlying idea is as follows. In a forward
neighborhood of a point (τ, ξ) where u has a jump, the weak solution u behaves
much in the same way as the solution of the corresponding Riemann problem. On
the other hand, on a region where its total variation is small, our solution u can be
accurately approximated by the solution of a linear hyperbolic system with constant
coefficients.
To state the result more precisely, we introduce some notations. Given a
function u = u(t, x) and a point (τ, ξ), we denote by U
(u;τ)
the solution of the
Riemann problem with initial da ta
u
= lim
xξ
u(τ, x), u
+
= lim
xξ+
u(τ, x). (4.6)
In addition, we define U
(u;τ)
as the solution of the linear hype rbolic Cauchy prob-
lem with constant coe fficients
w
t
+
b
Aw
x
= 0, w(0, x) = u(τ, x). (4.7)
Here
b
A
.
= A
u(τ, ξ)
. Observe that (4.7) is obtained from the quasilinear system
u
t
+ A(u)u
x
= 0 (A = Df ) (4.8)
by “freezing” the coefficients of the matrix A(u) at the point (τ, ξ) a nd choosing
u(τ) as initial data. A new notion o f “good solution” can now be introduced, by
locally comparing a function u with the self-similar solution of a Riemann problem
and with the solution of a linear hyperbolic system with constant coefficients. More
precisely, we say that a function u = u(t, x) is a viscosity solution of the system
(1.1) if t 7→ u(t, ·) is continuous as a map with values into L
1
lo c
, and moreover the
following integral estimates hold.
(i) At e very point (τ, ξ), for every β
> 0 one has
lim
h0+
1
h
Z
ξ+β
h
ξβ
h
u(τ + h, x) U
(u;τ)
(h, x ξ)
dx = 0. (4.9)
(ii) There exist c onstants C, β > 0 such that, for every τ 0 and a < ξ < b, one
has
lim sup
h0+
1
h
Z
bβh
a+βh
u(τ + h, x) U
(u;τ)
(h, x)
dx C ·
Tot.Var.
u(τ); ]a, b[
2
.
(4.10)
Hyperbolic Systems of Conservation Laws 169
As proved in [B2], this concept of viscosity solution completely characterizes
semigroup trajectories.
Theorem 4. Let S : D × [0, [×D be a semigroup generated by the system of
conservation laws (1.1). A function u : [0, T ] 7→ D is a viscosity solution of (1.1) if
and only if u(t) = S
t
u(0) for all t [0, T ].
5. Vanishing viscosity approximations
A natural conjecture is tha t the entropic solutions of the hyperbolic system
(1.1) actually coincide with the limits of solutions to the parabolic s ystem
u
ε
t
+ f (u
ε
)
x
= ε u
ε
xx
, (5.1)
letting the viscosity coefficient ε 0. In view of the previous uniqueness results,
one expects that the vanishing viscosity limit should single out the unique “good”
solution of the Cauchy problem, satisfying the appropriate entropy c onditions. In
earlier literature, results in this directio n were based on three main techniques:
1 - Comparison principles for parabolic equations. For a scalar conservation
law, the existence, uniqueness and global s tability of vanishing visc osity solutions
was first established by Oleinik [O] in one space dimension. The famous paper by
Kruzhkov [K] covers the more general class of L
solutions and is also valid in
several space dimensions.
2 - Singular perturbations. Let u be a piecewise smooth solution of the n × n
system (1 .1), with finitely many non-interacting, entropy admissible shocks. In
this special case, using a singular perturbation technique, Goodman and Xin [GX]
constructed a family of solutions u
ε
to (5.1), with u
ε
u as ε 0.
3 - Compensated compactness. If, instea d of a BV bound, only a uniform
bound on the L
norm of solutions of (5.1) is available, one can still construct a
weakly convergent subsequence u
ε
u. In general, we canno t expect that this weak
limit sa tisfies the nonlinear equations (1.1). However, for a class of 2 × 2 systems,
in [DP2] DiPerna showed that this limit u is indeed a weak solution of (1.1). The
proof relies on a compensated compac tness argument, based on the re presentation
of the weak limit in terms of Young measures, which must reduce to a Dirac mass
due to the presence of a lar ge family of entropies.
Since the main existence and uniqueness results for hyperbolic systems of
conservatio n laws are valid within the space of B V functions, it is natural to seek
uniform BV bounds also for the vis c ous approximations u
ε
in (5.1). This is indeed
the main goal accomplished in [BB]. As soon as these BV bounds are established, the
existence of a vanis hing viscosity limit follows by a standard compactness argument.
The uniqueness of the limit can then be deduced from the uniqueness theorem in
[BG]. By further analysis , o ne can also prove the continuous dependence on the
170 A. Bressan
initial data for the viscous approximations u
ε
, in the L
1
norm. Remarkably, these
results a re valid for general n × n strictly hyperbolic systems, not necess arily in
conservatio n form.
Theorem 5. Consider the Cauchy problem for a strictly hyperbolic system with
viscosity
u
ε
t
+ A(u
ε
)u
ε
x
= ε u
ε
xx
, u
ε
(0, x) = ¯u(x) . (5.2)
Then there exist constants C, L , L
and δ > 0 such that the following holds. If
Tot.Var.{¯u} < δ ,
¯u(x)k
L
< δ , (5.3)
then for each ε > 0 the Cauchy problem (5.2) has a unique solution u
ε
, defined for
all t 0. Adopting a semigroup notation, this will be written as t 7→ u
ε
(t, ·)
.
= S
ε
t
¯u.
In addition, one has:
BV bounds : Tot.Var.
S
ε
t
¯u
C Tot.Var.{¯u}. (5.4)
L
1
stability :
S
ε
t
¯u S
ε
t
¯v
L
1
L
¯u ¯v
L
1
, (5.5)
S
ε
t
¯u S
ε
s
¯u
L
1
L
|t s| +
εt
εs
. (5.6)
Convergence. As ε 0+, the solutions u
ε
converge to the trajectories of a
semigroup S such that
S
t
¯u S
s
¯v
L
1
L k¯u ¯vk
L
1
+ L
|t s|. (5.7)
These vanishing viscosity limits can be regarded as the unique vanishing viscosity
solutions of the hyperbolic Cauchy problems
u
t
+ A(u)u
x
= 0, u(0, x) = ¯u(x) . (5.8 )
In the conservative case where A(u) = Df(u) for some flux function f, the
vanishing viscosity solution is a weak solution of
u
t
+ f (u)
x
= 0, u(0, x) = ¯u(x) , (5.9)
satisfying the Liu admissibility conditions [L3]. Moreover, the vanishing viscosity
solutions are precisely the same as the viscosity solutions defined at (4.9)–(4.10) in
terms of local int egral estimates.
The key step in the proo f is to establish a priori bounds on the total variation
of solutions of
u
t
+ A(u)u
x
= u
xx
(5.10)
uniformly valid for all times t [0, [ . We outline here the main ideas.
Hyperbolic Systems of Conservation Laws 171
(i) At each point (t, x) we decompose the gradient along a suitable basis of unit
vectors ˜r
i
, say
u
x
=
X
v
i
˜r
i
. (5.11)
(ii) We then derive an equation describing the evolution of these gradient compo-
nents
v
i,t
+ (
˜
λ
i
v
i
)
x
v
i,xx
= φ
i
. (5.12)
(iii) Finally, we show that all source terms φ
i
= φ
i
(t, x) are integrable. Hence, for
all τ > 0,
v
i
(τ, ·)
L
1
v
i
(0, ·)
L
1
+
Z
0
Z
IR
φ
i
(t, x)
dxdt < . (5.13)
In this connection, it seems natural to decompose the gradient u
x
along the
eigenvecto rs of the hyperbolic matrix A(u). This approach however does NOT work.
In the c ase where the solution u is a travelling viscous shock profile, we would obtain
source terms which are not identically zer o. Hence they are certainly no t integrable
over the domain
t > 0 , x IR
.
An alternative approa ch, proposed by S. Bianchini, is to decompose u
x
as a
sum of gradients of viscou s t ravelling waves. By a viscous trave lling i-wave we mean
a solution of (5.10) having the form
w(t, x) = U(x σt) , (5.14)
where the sp e e d σ is clo se to the i-th eigenvalue λ
i
of the hyper bolic matrix A.
Clearly, the function U must provide a solution to the second order O.D.E.
U
′′
=
A(U) σ
U
. (5.15)
The underlying idea for the decomposition is as follows. At each point (t, x), given
(u, u
x
, u
xx
), we seek travelling wave profiles U
1
, . . . , U
n
such that
U
i
(x) = u(x), i = 1, . . . , n , (5.16)
X
i
U
i
(x) = u
x
(x) ,
X
i
U
′′
i
(x) = u
xx
(x) . (5.17)
In ge neral, the system of algebraic e quations (5.16)–(5.17) admits infinitely many
solutions. A unique so lution is singled out by considering only those travelling
profiles U
i
that lie on a suitable center m anifold M
i
. We now call ˜r
i
the unit vector
parallel to U
i
, so that U
i
= v
i
˜r
i
for some scalar v
i
. The decomposition (5.11) is
then obtained from the fir st equation in (5.17).
Toward the BV estimate, the second part of the proof consists in deriving the
equation (5.12) and estimating the integrals of the source terms φ
i
. Here the main
172 A. Bressan
idea is that these source terms can be regarded as ge nerated by wave interactions.
In analogy with the hyperbolic case considered by Glimm [G], the total amount of
these interactions can be controlled by suitable Lyapunov functionals. We describe
here the main ones.
1. Consider first two independent, scalar diffusion equations with stric tly different
drifts:
(
z
t
+
λ(t, x)z
x
z
xx
= 0 ,
z
t
+
λ
(t, x)z
x
z
xx
= 0 ,
assuming that
inf
t,x
λ
(t, x) sup
t,x
λ(t, x) c > 0 .
We regard z as the density of waves with a slow speed λ a nd z
as the density of
waves with a fast sp e e d λ
. A transversal interaction potential is defined as
Q(z, z
)
.
=
1
c
ZZ
IR
2
K(x
2
x
1
)
z(x
1
)
z
(x
2
)
dx
1
dx
2
, (5.18)
K(y)
.
=
e
cy/2
if y > 0 ,
1 if y 0 .
(5.19)
One can show that this functional Q is monoto nically decreasing along every couple
of solutions z, z
. The total amount of interaction between fast and slow waves can
now be e stimated as
Z
0
Z
IR
z(t, x)
z
(t, x)
dxdt
Z
0
d
dt
Q
z(t), z
(t)
dt
Q
z(0 ), z
(0)
1
c
Z
IR
z(0 , x)
dx ·
Z
IR
z
(0, x)
dx .
By means of Lyapunov functionals of this type one can control all s ource terms in
(5.12) due to the interaction of waves of different families.
2. To control the interactions between waves of the same family, we seek functionals
which are decreasing along every solution of a scalar viscous conservation law
u
t
+ g(u)
x
= u
xx
. (5.20)
For this purpose, to a scalar function x 7→ u(x) we a ssociate the curve in the plane
γ
.
=
u
g(u) u
x
=
conserved quantity
flux
. (5.21)
In connection with a s olution u = u(t, x) of (5.20), the curve γ evolves according to
γ
t
+ g
(u)γ
x
= γ
xx
. (5.22)
Hyperbolic Systems of Conservation Laws 173
Notice that the vector g
(u)γ
x
is parallel to γ, hence the presence of this term in
(5.22) only amounts to a re parametrization of the curve, and does not affect its
shape. The curve thus evolves in the direction of curvature. An obvious Lyapunov
functional is the length of the curve. In terms of the variables
γ
x
=
v
w
.
=
u
x
u
t
, (5.23)
this length is given by
L(γ)
.
=
Z
|γ
x
|dx =
Z
p
v
2
+ w
2
dx . (5.24)
We can estimate the rate of decrease in the length as
d
dt
L
γ(t)
=
Z
IR
|v|
(w/v)
x
2
1 + (w/v)
2
3/2
dx
1
(1 + δ
2
)
3/2
Z
|w/v|≤δ
|v|
(w/v)
x
2
dx ,
(5.25)
for any given constant δ > 0. This yields a useful a priori estimate on the integral
on the right hand side of (5.25).
3. In connection with the same cur ve γ in (5.21), we now introduce another func-
tional, defined in terms of a wedge product.
Q(γ)
.
=
1
2
ZZ
x<x
γ
x
(x) γ
x
(x
)
dx dx
. (5.26)
For any curve that moves in the plane in the direction o f curvature, one can show
that this functional is monotone decreasing and its decrease bounds the area swept
by the curve: |dA| dQ.
Using (5.22)–(5.23) we now compute
dQ
dt
dA
dt
=
Z
|γ
t
γ
x
|dx =
Z
|γ
xx
γ
x
|dx =
Z
|v
x
w vw
x
|dx .
Integrating w.r.t. time, we thus obtain another useful a priori bound:
Z
0
Z
|v
x
w vw
x
|dx dt
Z
0
dQ
γ(t)
dt
dt Q
γ(0)
.
Together, the functionals in (5.24) and (5.26) allow us to estimate all source terms
in (5.12) due to the interaction of waves of the same family.
This yields the L
1
estimates on the source terms φ
i
, in (5.12), proving the
uniform bounds on the total variation of a solution u of (5.10). See [BB] for details.
Next, to prove the uniform stability of all solutions of the parabolic system
(5.10) having small total variation, we consider the linearized system describing the
174 A. Bressan
evolution of a first order var iation. Inserting the formal expansion u = u
0
+ǫz+O(ǫ
2
)
in (5.10), we obta in
z
t
+
DA(u) · z
u
x
+ A(u)z
x
= z
xx
. (5.27)
Our basic goal is to prove the bound
z(t)
L
1
L
z(0 )
L
1
, (5.28)
for some constant L and all t 0 and every so lution z of (5.27). By a sta ndard
homotopy argument, from (5.28) one easily deduces the Lipschitz continuity of the
solution of (5.8) on the initial data. Namely, for every couple of solutions u, ˜u with
small total variation one ha s
u(t) ˜u(t)
L
1
L
u(0) ˜u(0)
L
1
. (5.29)
To prove (5.28) we decompose the vector z as a sum o f scalar components: z =
P
i
h
i
˜r
i
, write an evolution equation for these components:
h
i,t
+ (
˜
λ
i
h
i
)
x
h
i,xx
=
ˆ
φ
i
,
and show that the source ter ms
ˆ
φ
i
are integrable on the doma in {t > 0 , x IR }.
For every initial data u(0, ·) = ¯u with small total variation, the previous argu-
ments yield the existence of a unique global solution to the para bolic system (5.8),
depending Lipschitz continuously on the initial data, in the L
1
norm. Perform-
ing the rescaling t 7→ t/ε, x 7→ x/ε, we immediately obtain the same results for
the Cauchy problem (5.2). Adopting a semigroup notation, this solution can be
written as u
ε
(t, ·) = S
ε
t
¯u. Thanks to the uniform bounds on the total variation, a
compactness argument yields the e xistence of a stro ng limit in L
1
lo c
u = lim
ε
m
0
u
ε
m
(5.30)
at least for some s ubse quence ε
m
0 . Since the u
ε
depend continuously on the
initial data, with a unifor m Lipschitz constant, the same is true of the limit solution
u(t, ·) = S
t
¯u. In the conservative case where A(u) = Df(u), it is not difficult to
show that this limit u actually provides a weak solution to the Cauchy problem
(1.1)–(1.2).
The o nly remaining issue is to show that the limit in (5.30) is unique, i.e. it
does not depend on the s ubsequence {ε
m
}. In the standard conservative case, this
fact can already be deduced from the uniqueness result in [BG]. In the general case,
uniqueness is pr oved in two steps. First we show that, in the special case of a
Riemann problem, the solution obta ined as vanishing viscosity limit is unique and
can be completely characterized. To conclude the proof, we then rely on the same
Hyperbolic Systems of Conservation Laws 175
general argument a s in [B2]: if two Lipschitz semigroups S, S
provide the same
solutions to all Riemann problems, then they must coincide. See [BB] for details.
6. Concluding remarks
1. A classical tool in the a nalysis of first order hyperbolic systems is the method of
characteristics. To study the system
u
t
+ A(u)u
x
= 0 ,
one decomposes the solution along the e igenspaces of the matrix A(u). The evo-
lution of these components is then described by a family of O.D.E’s along the
characteristic curves. In the t-x plane, these are the curves which satisfy dx/dt =
λ
i
u(t, x)
. The lo c al dec omposition (5.16)–(5.17 ) in terms of viscous travelling
waves makes it possible to implement this “hyperbolic” approach also in connection
with the parabolic system (5.10). In this case, the projections are taken along the
vectors ˜r
i
, while the characteristic curves are defined as dx/dt = σ
i
, where σ
i
is the
sp e e d of the i-th travelling wave. Notice that in the hyperbolic case the projections
and the wave speeds depend only on the sta te u , through the eigenvectors r
i
(u) and
the eigenvalues λ
i
(u) of the matrix A(u). On the o ther hand, in the parabolic case
the construction involves the derivatives u
x
, u
xx
as well.
2. In nearly all previous works on BV solutions for systems of conservation laws,
following [G] the basic estimates on the total variation were obtained by a careful
study of the Riemann problem and of elementary wave interactions. The Riemann
problem also takes the center stage in all earlier proofs of the stability of solutions
[BC1], [BCP], [BLY]. In this connection, the hypothesis (H) introduced by Lax [Lx 1]
is widely adopted in the literature. It guarantees that solutions of the Riemann
problem have a simple structure, co nsisting of at most n elementary waves (shocks,
centered rarefactions or contact discontinuities). If the assumption (H) is dropped,
some res ults on global existence [L3], and continuous dependence [AM] are still
available, but their proofs become far more technical. On the other hand, the
approach introduced in [BB ] marks the first time where uniform BV estimates are
obtained without a ny reference to Riemann problems. Global exis tence and stability
of weak solutions are obtained for the whole class of strictly hyperbolic s ystems,
regardless of the hypothesis (H).
3. For the viscous system of conservation laws
u
t
+ f (u)
x
= u
xx
,
previous re sults in [L4], [SX], [SZ], [Yu] have established the stability of special types
of solutions, for example travelling viscous shocks or viscous r arefactions. Taking
ε = 1 in (5.2), from Theorem 5 we obtain the uniform Lipschitz stability (w..r.t. the
L
1
distance) of ALL viscous solutions with sufficiently small total variation. An
176 A. Bressan
interesting alternative technique for proving stability of visc ous solutions, based on
sp e c tral methods, was recently developed in [HZ].
4. In the present survey we only considered initial data with small total variation.
This is a convenient setting, adopted in much of the curre nt literature, which guar-
antees the global existence of BV s olutions of (1.1) and captures the main features
of the problem. A rece nt example cons tructed by Jenssen [J] shows that, for initial
data with large total variation, the L
norm of the solution can blow up in finite
time. In this more general setting, o ne expects tha t the existence and uniqueness
of weak solutions, together with the convergence of vanishing viscosity approxima-
tions, should hold locally in time as long as the total variation remains bounded.
For the hyperbolic system (1.1), results on the local existence and stability of solu-
tions with large BV data can be found in [Sc] and [BC2], re spectively. Because of
the counterexample in [BS], on the other hand, similar well posedness results are
not expected in the general L
case.
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... Then for any source e j (0), j = 1, . . . , s, see Subsection 2.1, the right hand side of (18c) vanishes and consequently u j (0, t) = 0 for all t > 0. Note that due to the tree structure and recursive procedure, we can choose such an order of edges that before calculating the solution on k-th edge we have values of all u j (x, t) for j ∈ J such that b kj > 0, see equation (4). Consequently, the system of conservation laws on network transforms into a sequence of initial-boundary-value problems of a form ...
... Think about the following initial configuration on the line. u| t=0 = χ [−4,−3] − χ [3,4] . ...
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The paper deals with the analysis of Burgers' equation on acyclic metric graphs. The main goal is to establish the existence of weak solutions in the $TV$ -- class of regularity. A key point is transmission conditions in vertices obeying the Kirchhoff law. First, we consider positive solutions at arbitrary acyclic networks and highlight two kinds of vertices, describing two mechanisms of flow splitting at the vertex. Next we design rules at vertices for solutions of arbitrary sign for any subgraph of hexagonal grid, which leads to a construction of general solutions with $TV$ -- regularity for this class of networks. Introduced transmission conditions are motivated by the change of the energy estimation.
... , n, where the dependent variables are chosen to be densities of conserved quantities, and the functions φ i (v) are the corresponding densities of fluxes (see, e.g., [9] regarding the physical applications of such systems). The relationships between solutions of the perturbed and unperturbed systems have been extensively studied for the case of dissipative perturbations of spatially one-dimensional systems of conservation laws (see, e.g., [6] and the references therein). Our strategic goal is the study of Hamiltonian perturbations of hyperbolic PDEs. ...
... The canonical coordinate is u = w, and the constant c is the central invariant. Up to terms of order 6 , the reducing transformation [16] is given by 4 ...
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