18 2. FOUR IMPORTANT LINEAR PDE

2.1. TRANSPORT EQUATION

One of the simplest partial diﬀerential equations is the transport equation

with constant coeﬃcients. This is the PDE

(1) ut + b · Du = 0 in

Rn

× (0, ∞),

where b is a ﬁxed vector in

Rn,

b = (b1,... , bn), and u :

Rn

× [0, ∞) → R

is the unknown, u = u(x, t). Here x = (x1,... , xn) ∈

Rn

denotes a typical

point in space, and t ≥ 0 denotes a typical time. We write Du = Dxu =

(ux1 , . . . , uxn ) for the gradient of u with respect to the spatial variables x.

Which functions u solve (1)? To answer, let us suppose for the moment

we are given some smooth solution u and try to compute it. To do so, we

ﬁrst must recognize that the partial diﬀerential equation (1) asserts that a

particular directional derivative of u vanishes. We exploit this insight by

ﬁxing any point (x, t) ∈

Rn

× (0, ∞) and deﬁning

z(s) := u(x + sb, t + s) (s ∈ R).

We then calculate

˙(s) z = Du(x + sb, t + s) · b + ut(x + sb, t + s) = 0

·

=

d

ds

,

the second equality holding owing to (1). Thus z(·) is a constant function of

s, and consequently for each point (x, t), u is constant on the line through

(x, t) with the direction (b, 1) ∈

Rn+1.

Hence if we know the value of u at

any point on each such line, we know its value everywhere in

Rn

× (0, ∞).

2.1.1. Initial-value problem.

For deﬁniteness therefore, let us consider the initial-value problem

(2)

ut + b · Du = 0 in

Rn

× (0, ∞)

u = g on

Rn

× {t = 0}.

Here b ∈

Rn

and g :

Rn

→ R are known, and the problem is to compute

u. Given (x, t) as above, the line through (x, t) with direction (b, 1) is

represented parametrically by (x + sb, t + s) (s ∈ R). This line hits the

plane Γ :=

Rn

× {t = 0} when s = −t, at the point (x − tb, 0). Since u is

constant on the line and u(x − tb, 0) = g(x − tb), we deduce

(3) u(x, t) = g(x − tb) (x ∈

Rn,t

≥ 0).

So, if (2) has a suﬃciently regular solution u, it must certainly be given

by (3). And conversely, it is easy to check directly that if g is

C1,

then u

deﬁned by (3) is indeed a solution of (2).