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

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http://dx.doi.org/10.1090/gsm/120/02