Chapter 2
First-Order Differential
Equations
In this chapter, we discuss initial-value problems for first-order PDEs. Main
topics include noncharacteristic conditions, methods of characteristics and
a priori estimates in
L∞-norms
and in
L2-norms.
In Section 2.1, we introduce the basic notion of noncharacteristic hyper-
surfaces for initial-value problems. In an attempt to solve initial-value prob-
lems, we illustrate that we are able to compute all derivatives of solutions
on initial hypersurfaces if initial values are prescribed on noncharacteris-
tic initial hypersurfaces. For first-order linear PDEs, the noncharacteristic
condition is determined by equations and initial hypersurfaces, independent
of initial values. However, for first-order nonlinear equations, initial values
also play a role. Noncharacteristic conditions will also be introduced for
second-order linear PDEs in Section 3.1 and for linear PDEs of arbitrary
order in Section 7.1, where multi-indices will be needed.
In Section 2.2, we solve initial-value problems by the method of charac-
teristics if initial values are prescribed on noncharacteristic hypersurfaces.
For first-order homogeneous linear PDEs, special curves are introduced along
which solutions are constant. These curves are given by solutions of a system
of ordinary differential equations (ODEs), the so-called characteristic ODEs.
For nonlinear PDEs, characteristic ODEs also include additional equations
for solutions of PDEs and their derivatives. Solutions of the characteristic
ODEs yield solutions of the initial-value problems for first-order PDEs.
In Section 2.3, we derive estimates of solutions of initial-value problems
for first-order linear PDEs. The
L∞-norms
and the
L2-norms
of solutions
9
http://dx.doi.org/10.1090/gsm/120/02
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