CHAPTER 1

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

Rn

a.e. in R x R

+

(1.3)

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 t± 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.