1.1. The method of characteristics 9

Gronwall’s lemma implies ii.

Proof of Theorem 1.1.1. The change of variable shows that it suﬃces to

consider the case of A = diag(c1,...,cN ).

The solution u is constructed as a limit of approximate solutions

un.

The solution

u0

is defined as the solution of the initial value problem

∂tu0

+ A

∂xu0

= f,

u0|t=0

= g .

The solution is explicit by the method of characteristics. It is

Ck

with

bounded derivatives on [0,T] × R, so

(1.1.10) ∃C1, ∀t ∈ [0,T],

mk(u0,t)

≤ C1 .

For n 0 the solution

un

is again explicit by the method of character-

istics in terms of

un−1,

(1.1.11)

∂tun

+ A

∂xun

+ B

un−1

= f ,

un−1|t=0

= g .

Using (1.1.10) and Haar’s inequality yields,

(1.1.12) ∃C2, ∀t ∈ [0,T],

mk(u1,t)

≤ C2 .

For n ≥ 2 estimate

un

−

un−1

by applying Haar’s inequality to

˜

L

(un

−

un−1)

+ B

(un−1

−

un−2)

= 0,

(un

−

un−1)

t=0

= 0 ,

to find

(1.1.13)

mk(un

−

un−1,t)

≤ C

t

0

mk(un−1

−

un−2,σ)

dσ .

For n = 2, this together with (1.1.10) and (1.1.12) yields

mk(u2

−

u1,t)

≤ (C1 + C2)Ct .

Injecting this in (1.1.13) yields

mk(u3

−

u2,t)

≤ (C1 +

C2)C2t2/2

.

Continuing yields

(1.1.14)

mk(un

−

un−1,t)

≤ (C1 +

C2)Cn−1tn−1/(n

− 1)! .

The summability of the right-hand side implies (Weierstrass’s M-test)

that

un

and all of its partials of order ≤ k converge uniformly on [0,T] × R.

The limit u is

Ck

with bounded partials. Passing to the limit in (1.1.11)

shows that u solves the initial value problem.

To prove uniqueness, suppose that u and v are solutions. Haar’s inequal-

ity applied to u − v implies that u − v = 0.