8 1. INTRODUCTION

the scalar conservation law

ut + F (u)x = 0.

We will see in §3.4 that this PDE governs various one-dimensional phenom-

ena involving fluid dynamics, and in particular models the formation and

propagation of shock waves. Now a shock wave is a curve of discontinuity

of the solution u; and so if we wish to study conservation laws, and recover

the underlying physics, we must surely allow for solutions u which are not

continuously diﬀerentiable or even continuous. In general, as we shall see,

the conservation law has no classical solutions but is well-posed if we allow

for properly deﬁned generalized or weak solutions.

This is all to say that we may be forced by the structure of the par-

ticular equation to abandon the search for smooth, classical solutions. We

must instead, while still hoping to achieve the well-posedness conditions (i)–

(iii), investigate a wider class of candidates for solutions. And in fact, even

for those PDE which turn out to be classically solvable, it is often most

expedient initially to search for some appropriate kind of weak solution.

The point is this: if from the outset we demand that our solutions be very

regular, say k-times continuously diﬀerentiable, then we are usually going

to have a really hard time ﬁnding them, as our proofs must then necessarily

include possibly intricate demonstrations that the functions we are building

are in fact smooth enough. A far more reasonable strategy is to consider as

separate the existence and the smoothness (or regularity) problems. The idea

is to deﬁne for a given PDE a reasonably wide notion of a weak solution, with

the expectation that since we are not asking too much by way of smoothness

of this weak solution, it may be easier to establish its existence, uniqueness,

and continuous dependence on the given data. Thus, to repeat, it is often

wise to aim at proving well-posedness in some appropriate class of weak or

generalized solutions.

Now, as noted above, for various partial diﬀerential equations this is

the best that can be done. For other equations we can hope that our weak

solution may turn out after all to be smooth enough to qualify as a classical

solution. This leads to the question of regularity of weak solutions. As we

will see, it is often the case that the existence of weak solutions depends

upon rather simple estimates plus ideas of functional analysis, whereas the

regularity of the weak solutions, when true, usually rests upon many intricate

calculus estimates.

Let me explicitly note here that once we are past Part I (Chapters 2–4),

our eﬀorts will be largely devoted to proving mathematically the existence

of solutions to various sorts of partial diﬀerential equations, and not so much