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
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