General distribution solutions
1. Something worse can happen
We are concerned here with the solution of initial value problems for nonlinear,
first order systems of conservation laws, of the form
(1.1) wt + q(w)x = Q, xeR,t0
(1.2) w(x, t) G D C
a.e. in R x R
qeC2 :D-*Rn
with D a given open set, and initial data
(1.4) w(-,0) :R-*£.
For w(-,0) smooth and bounded, and bounded uniformly away from dD, the
systems we consider admit unique smooth solutions, continuously dependent on
the initial data, for some finite time depending on the initial data. In particular
the characteristic speeds, the eigenvalues of qw(w), are real for all w G D. It is
well-known that in general for q nonlinear, the continuation of a solution depends
on the acceptance of weak solutions, involving a loss of regularity in w and a related
fundamental uniqueness question.
A natural question is whether anything worse can happen. In his signal pa-
per [G], James Glimm found weak solutions of (1.1) as discontinuous functions of
bounded variation in x, pointwise in £, satisfying (1.1) in the space of measures on
R x R
, and continuing indefinitely in t. He suggested: "Presumably nothing worse
can happen. Apparently the concept of weaker solutions, e.g. distributions, has no
meaning because q is not linear."
When initial data of large variation is permitted, however, the lack of a max-
imum principle for systems (1.1) indicates that worse things can indeed happen.
Consider a system (1.1) with initial data satisfying
wi(x,0) ^ i Q , x e R
and admitting a unique solution, either a classical solution or an entropy weak
solution, up to time 0 but such that for some t G (0, £i] wi(-,t) assumes
negative values on a set of finite measure in R. This is very mild requirement,
taking linear combinations of the equations in (1.1) and adding constants to w as
necessary. We now regard x as a "Lagrangian" space coordinate, and introduce an
"Eulerian" space coordinate Y determined from
(1.5) dY = w\dx qidt.
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