THE LEGACY OF OLGA OLEǏNIK IN
HYPERBOLIC CONSERVATION LAWS
BARBARA LEE KEYFITZ
Systems of conservation laws are partial differential equations that model
the dynamics of continua — fluids, plasmas or elastic solids — by means of
the fundamental principles of conservation of mass, momentum, and energy,
supplemented by constitutive relations, typically based in thermodynamics
or other classical physics. The standard form of a conservation law system,
in one space dimension, is Ut + F (U )x = 0, where U ∈ Rn is a vector of
states, or conserved quantities, and F is a (linear or nonlinear) flux function.
The familiar wave equation, (ρut )t = (T ux )x , for the vertical displacement
u of a stretched string of constant linear density ρ and constant tension T ,
expresses conservation of momentum and assumes the standard form if we
define c2 = T /ρ, and
¶
µ
µ ¶
0 c
ut
U.
,
F (U ) = AU =
U=
c 0
cux
The multidimensional analogue, utt − c2 ∆u = 0, or utt − ∇ · (c2 ∇u) = 0,
models the vibrations of a membrane or a solid and can also be written as a
system. The system is nonlinear if c = c(u), for example.
One of the most widely studied examples of a system of conservation laws
is given by the equations of adiabatic, compressible, ideal gas dynamics.
They express conservation of mass, momentum, and energy; in two space
dimensions, they take the form
ρt + (ρu)x + (ρv)y
(ρu)t + (ρu2 + p)x + (ρuv)y
(ρv)t + (ρuv)x + (ρv 2 + p)y
(ρE)t + (ρuH)x + (ρvH)y
=
=
=
=
0
0
0
0.
2000 Mathematics Subject Classification. Primary: 35L65, 35J70, 35R35; Secondary:
35M10, 35J65, 76J20.
Key words and phrases. Olga Oleı̆nik, hyperbolic conservation laws, entropy conditions,
two-dimensional Riemann problems.
Research supported by the Department of Energy, National Science Foundation, and
NSERC of Canada.
37
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WOMEN IN MATHEMATICS: MAY 18–20, 2006
The basic variables are density ρ, velocity (u, v), and pressure p; thermodynamic principles give the energy equation with
E=
1 p 1 2
+ (u + v 2 ),
γ−1ρ 2
H=
γ p 1 2
+ (u + v 2 ).
γ−1ρ 2
One of the paradoxes of multidimensional conservation laws is that although
they are important in applications and form the object of numerous computational simulations, there is no existence theory, even for small data, in any
regime that contains interesting behavior.
The underlying reason is that smooth data lead to discontinuous solutions in finite time, and hence, one needs to study weak solutions. However,
discontinuities in quasilinear equations propagate on shocks, not on characteristics. Even for linear equations, characteristics in higher dimensions
are complicated (the theory of wave front sets was designed to study this
question), and a satisfactory way to study nonlinear discontinuities has not
been found yet.
In this talk, I shall say a little about the pioneering work of Olga Oleı̆nik,
who was among the first to give a theory for a scalar conservation law in
one space dimension, and then will describe recent work that my co-authors
and I have done to analyse some self-similar problems in multidimensional
conservation laws. Oleı̆nik’s major paper [29] is the main reference for her
work in this area.
Weak solutions, well known for linear equations, can be defined for quasilinear equations in divergence form (that is, for conservation laws). We say
that ∇ · F (U ) = 0 is satisfied in the weak sense if
ZZ
F · ∇θ dx dt = 0,
for all test functions θ. For linear equations, the theory of weak solutions
is usually posed for U in a Sobolev space (or in the space of distributions);
linear hyperbolic equations are well-posed in Lp or W m,p . A standard approach to existence theorems is to enlarge the class of solutions in this way
and then prove regularity. In fact, it is standard that for linear elliptic equations, weak solutions, under reasonable hypotheses, turn out to be classical
solutions. For hyperbolic equations, by contrast, under reasonable conditions, there are weak solutions that are not differentiable. This is plausible
from the characteristic structure of hyperbolic equations: one does not expect to see any smoothing of solutions. In fact, in higher dimensions, it is
well known that there is a loss of regularity owing to focusing when waves
interact.
We turn now to quasilinear equations and look at the very simplest scalar
equation model, Burgers equation,
ut + (u2 /2)x = 0.
THE LEGACY OF LADYZHENSKAYA AND OLEINIK
39
The formula
u(x0 + u0 (x0 )t, t) = u0 (x0 )
defines implicitly a function u(x, t) that satisfies the Cauchy data u(x, 0) =
u0 (x). As the characteristic curves of Burgers equation are intergral curves
of dx/dt = u(x, t), it can be seen that characteristics approach each other
(and eventually intersect) if they originate in an interval where u00 < 0, while
“weak” initial data containing simple jump discontinuities with u0 (x−) <
u0 (x+) result in regions of the x − t plane that are not reached by characteristics emanating from the initial line, and hence, the solution is not
completely determined by the initial data. That is, the Cauchy problem is
ill-posed as stated.
The intersecting characteristics problem is resolved by the introduction
of weak solutions, as just defined. However, Burgers equation (and other
quasilinear hyperbolic conservation laws) support discontinuities, not along
characteristics, but along shocks (or shock surfaces in higher dimensions)
given, for Burgers equation, by
dx
u(x+, t) + u(x−, t)
=
.
dt
2
The problem of uniqueness is more difficult to resolve. In the region that
is not reached by characteristics from the initial line, a weak solution can be
defined in an infinite number of ways.
In a seminal paper, Oleı̆nik studied a scalar equation in the form of a
balance law (a generalization of the notion of a conservation law) in one
space dimension:
∂u ∂φ(t, x, u)
+
+ ψ(t, x, u) = 0
∂t
∂x
with an important convexity assumption φuu ≥ 0. Her first insight was what
she termed the entropy inequality,
u(t, x1 ) − u(t, x2 )
2E
<
,
x1 − x2
t
implies uniqueness.
She was then able to approximate solutions in two different ways, using
either finite differences or “vanishing viscosity,” to show that both of these
respect the entropy inequality and to show the convergence of the approximations. Hence, Oleı̆nik was the first to prove a rigorous well-posedness
theorem in conservation laws. The most general theorem for a scalar equation was given by her student Kružkov [23].
Since then, an essentially complete theory for small data (close to a constant) has been developed for systems of conservation laws in one space
dimension, through the work of Lax [24], [25], [26], [27], Glimm [17], Liu
[28], Dafermos [15], Smoller [31], DiPerna [16], Bressan [1], and others. It is
interesting to note that there are obstructions to developing a general theory
for arbitrary data, although for important examples (such as isentropic gas
40
WOMEN IN MATHEMATICS: MAY 18–20, 2006
dynamics), large-data results have been proved. It is also important to note
that even for data close to constant, linearizing the equations is not a useful
way to proceed. As suggested earlier, the linear notion of characteristics is
inadequate, and a much more useful notion is that of a Riemann problem:
resolution of the discontinuity in self-similar data consisting of two constant
states. In one space dimension, solutions of Riemann problems have become
the building blocks of the theory and even of some successful computational
methods.
The Riemann problem for a system of conservation laws in one space
dimension is the Cauchy problem
½
UL , x < 0
Ut + F (U )x = 0,
U (x, 0) =
.
UR , x > 0
It is interesting to observe how the problem looks in terms of the similarity
variable ξ = x/t, because this is the route we followed in looking at twodimensional self-similar problems. We have the ordinary differential equation
system (with A = dF )
−ξU 0 + A(U )U 0 = 0
or
(−ξI + A)U 0 = 0,
U (−∞) = UL , U (∞) = UR .
One sees there are two types of non-constant solutions. Suppose ξ = λ(U ),
U 0 = ~r(U ) for an eigenvalue and eigenvector of the flux matrix A, and that
λ increases with U along the integral curve of ~r(U ); then one obtains a
local solution of the equation that is continuous except at the origin; this is
called a rarefaction wave. Alternatively, the equation may hold weakly, at a
discontinuity at ξ = s. Then,
¡
¤s+
− ξU + F (U ) s− = 0 or s[U ] = [F (U )],
and this is a shock. To rule out the obvious failure of uniqueness stemming
from constructing a shock when a rarefaction is available, one needs to impose an entropy condition that generalizes to systems Oleı̆nik’s condition. It
corresponds roughly to having λ decrease across the discontinuity.
About a decade ago, Sunčica Čanić and I began a program to apply this
self-similar approach to systems of conservation laws in two space dimensions. We have gained some insight into the behavior of multidimensional
systems of conservation laws. We have also found a number of unexpected
phenomena. An overview appears in [21]. See also [34] for background and
for related work by Yuxi Zheng and co-authors.
First, even the definition of a “two-dimensional Riemann problem” is not
completely straightforward. We examined only the case of sectorially constant Riemann data. (Other authors have restricted their attention even
further to data constant in quadrants; however, that precludes the study of
interesting bifurcation problems like shock reflection. See [9], [32], for example.) The salient fact about sectorially constant data is that sufficiently far
from the origin, the problem is a one-dimensional Riemann problem. Hence,
THE LEGACY OF LADYZHENSKAYA AND OLEINIK
41
the question is reduced to examining interactions of a finite number of waves.
The issue of how more general self-similar data behave is completely open.
Second, the conservation law in self-similar coordinates changes type, for
a large class of equations that are like gas dynamics, in a way that somewhat
resembles steady transonic flow, [2], [3], [8]. Far from the origin, the reduced
equation is hyperbolic, but a pair of characteristics becomes complex near
the origin, (the so-called subsonic region), leading to a mixed-type equation.
For some simple models that we have studied, one can separate the mixed
system into a quasilinear elliptic equation and a system of transport equations. For gas dynamics, the system is coupled in a complicated way that
we are beginning to unravel. However, the point is that degenerate elliptic
equations of a particular kind play a strong part in this self-similar study
[22], [33]. Here, Oleı̆nik’s research also played a role in our initial investigations. Setting up a line of research to solve some very different questions in mechanics, Oleı̆nik and Radkevič studied a class of equations with
“non-negative characteristic form” – that is, degenerate elliptic equations,
[30]. Although they looked only at linear problems, they made considerable
progress in ascertaining which were the correctly posed boundary value problems and in defining weighted Sobolev spaces in which to study solutions.
We used their formulation of the problem in the first existence theorems we
proved on self-similar problems [2], [3].
Third, the position of a shock appears in a natural way as a free boundary
problem in the reduced system. In the simplest examples we have studied,
the solution is completely known (in fact, constant) on one side of the unknown boundary, and the boundary position is coupled to the elliptic or
mixed-type system in the subsonic region. We have succeeded in solving
some of these free boundary problems, at least locally, using the classical
technique of applying the Schauder fixed-point theorem in weighted Hölder
spaces. The basic approach was developed with Gary Lieberman for a steady
transonic example [7], and has now been extended to several model systems
in work with Eun Heui Kim [4], [5], [6] and Katarina Jegdić [20], [19], and
most recently to the isentropic gas dynamics equations with Jegdić [18].
There is other current work in this direction; Chen and Feldman have also
studied steady shock problems [10], [11] for transonic flow and have extended
their approach to some self-similar problems [12]. One difference between
our approach and theirs is that they have used a potential formulation that
is suitable for irrotational flows, and that leads in a natural way to an elliptic
equation. I should also mention work of Shuxing Chen and co-authors, for
example [13], [14]. Chen’s main interest is in steady perturbations of some
classical symmetric solutions in transonic gas dynamics rather than in selfsimilar problems. This approach also leads to free boundary problems and
to degenerate elliptic equations; Chen has developed a novel approach, the
“partial hodograph method,” to handle these problems.
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WOMEN IN MATHEMATICS: MAY 18–20, 2006
Oleı̆nik was one of the pioneers in the analytic study of hyperbolic conservation laws. It is a tribute to her research strengths and to her imagination
that so many of the techniques she developed during her career – some for
quite different uses – still find application in current research in conservation
laws.
References
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Boston, MA: Birkhäuser, 2001.
Fields Institute for Research in Mathematical Sciences; Toronto, ON M5T
3J1 Canada
Department of Mathematics; University of Houston; Houston, Texas 772043008 USA
E-mail address: [email protected]