Chapter 6 Partial Differential Equations

6.2. LINEAR AND QUASILINEAR EQUATIONS OF FIRST ORDER 11
Definition 6.2.5. A hypersurface S is called characteristic for L at x ∈ S if the normal
vector ν(x) to S at x is in Char x (L), and S is called non-characteristic if it is not charateristic
at any x in S.
We now turn to the development for real first order systems. First recall the basic problem
in ODEs is the IVP:
Given a function F on say R 3 and (t 0 ,u 0 ) ∈ R 2 , find a function u(t) defined
in a neighborhood of t 0 such that F (t, u, u ′ )=0and u(t 0 )=u 0 .
In this disscussion we will consider the analog of this which is the initial value problem
for a first order partial differential equation. We will focus on the linear and quasi-linear
cases.
Let us first consider the linear equation
n∑
a j ∂ j u + bu = f(x). (6.2.1)
j=1
where a j , b and f are assumed to be C 1 functions of x. If we denote by A the vector field
then we have
A(x) =(a 1 (x), ··· ,a n (x),
Char x (L) ={ξ ≠0:A(x) · ξ =0}.
That is, Char x (L) ∪{0} is the hyperplane orthogonal to A(x). From this we see that:
A hypersurface S is characteristic at x if and only if A(x) is tangent to S at x.
INITIAL VALUE PROBLEM: Find a solution u to (6.2.1) with given initial values
u = ϕ on a given hypersurface S.
If S is characteristic at a point x 0 , then the quantity ∑ a j (x 0 )∂ j u(x 0 ) is completely
determined as a certain directional derivative of ϕ along S at x 0 . For this reason it may
not be possible to make it equal to f(x 0 ) − b(x 0 )u(x 0 ). As an example, if the equation is
∂ 1 u = 0 and S is the hyperplane x n = 0, we cannot have u = ϕ on S unless ∂ 1 ϕ =0.
Namely, consider the case of R 2 . The general solution is given by u(x 1 ,x 2 )=f(x 2 ) where f
is arbitrary. But if S corresponds to x 2 = 0 then the solution must satisfy u(x 1 , 0) = φ(x 1 )
and the only choice is that ϕ ≡ 0.
Thus to make the initial value problem well behaved, we must assume that S is noncharacteristic.
It turns out that to solve for u it is useful to compute the integral curves of
the vector field A(x).