10.1 Properties of the Free Wave Propagator
55
c) W (t) is a strongly continuous group
Thanks to the group law, it is enough to show the strong continuity at t = 0, namely
that, for any bounded region V in R
S ,
lim
t→0
(W (t) − 1)u V = 0, ∀u ∈ X loc .
(10.10)
Equation (10.10) is obvious for u ∈ S × S (see (10.1)) and it can be extended to the
whole X loc by using (10.3). In fact, if u j ∈ S × S and u j → u in X loc , one has
(W (t) − 1)u V ≤ ≤(W (t) − 1)u j V + +(W (t) − 1)(u j − u) V .
By (10.3), the latter term is majorized by (e
1
2 + 1)u j − u Ω R , where Ω R is a sphere
such that Ω R−1 ⊃ V , and therefore can be made arbitrarily small.
One can show that the domain of the generator K is H
2
loc (R
S
) ⊕ H
1
loc (R
S
); in fact,
∀u ∈ X loc ⊂ S
× S
, in the distributional sense from (10.2) one has
K
ϕ
ψ
=
ψ
Δϕ
.
The condition that the r.h.s. belongs to X loc gives ψ ∈ H
1
loc (R
S
) and Δϕ ∈ L
2
loc (R
S
),
which is equivalent to ϕ ∈ H
2
loc (R
S
).
10.2 The Cauchy Problem for Small Times
Theorem 10.1 If f (u) satisfies a local Lipschitz condition, then properties 1), 2),
3), 4), listed in Chap. 4, hold.
Proof.
45 1) One has to check that W (t − s) f (u(s)) is an integrable function; it is
enough to show that it is a continuous function in the X loc topology. To this purpose,
we consider the inequality ( f s ≡ f (u(s)))
W (t − s) f s − W (t − s
) f s Ω R−t ≤
≤ ≤(W (t − s) − W (t − s
)) f s Ω R−t + +W (t − s
)( f s − f s ) Ω R−t . (10.11)
The first term on the right-hand side goes to zero as s
→ s as a consequence of
the strong continuity of W (t) on X loc (see Sect. 10.1, c)). The second term can be
estimated by using the hyperbolic character of W (t)
45 We essentially follow Ref. I. quoted in Chap. 3, footnote 4, to which we refer for a more detailed
and general discussion.
55
c) W (t) is a strongly continuous group
Thanks to the group law, it is enough to show the strong continuity at t = 0, namely
that, for any bounded region V in R
S ,
lim
t→0
(W (t) − 1)u V = 0, ∀u ∈ X loc .
(10.10)
Equation (10.10) is obvious for u ∈ S × S (see (10.1)) and it can be extended to the
whole X loc by using (10.3). In fact, if u j ∈ S × S and u j → u in X loc , one has
(W (t) − 1)u V ≤ ≤(W (t) − 1)u j V + +(W (t) − 1)(u j − u) V .
By (10.3), the latter term is majorized by (e
1
2 + 1)u j − u Ω R , where Ω R is a sphere
such that Ω R−1 ⊃ V , and therefore can be made arbitrarily small.
One can show that the domain of the generator K is H
2
loc (R
S
) ⊕ H
1
loc (R
S
); in fact,
∀u ∈ X loc ⊂ S
× S
, in the distributional sense from (10.2) one has
K
ϕ
ψ
=
ψ
Δϕ
.
The condition that the r.h.s. belongs to X loc gives ψ ∈ H
1
loc (R
S
) and Δϕ ∈ L
2
loc (R
S
),
which is equivalent to ϕ ∈ H
2
loc (R
S
).
10.2 The Cauchy Problem for Small Times
Theorem 10.1 If f (u) satisfies a local Lipschitz condition, then properties 1), 2),
3), 4), listed in Chap. 4, hold.
Proof.
45 1) One has to check that W (t − s) f (u(s)) is an integrable function; it is
enough to show that it is a continuous function in the X loc topology. To this purpose,
we consider the inequality ( f s ≡ f (u(s)))
W (t − s) f s − W (t − s
) f s Ω R−t ≤
≤ ≤(W (t − s) − W (t − s
)) f s Ω R−t + +W (t − s
)( f s − f s ) Ω R−t . (10.11)
The first term on the right-hand side goes to zero as s
→ s as a consequence of
the strong continuity of W (t) on X loc (see Sect. 10.1, c)). The second term can be
estimated by using the hyperbolic character of W (t)
45 We essentially follow Ref. I. quoted in Chap. 3, footnote 4, to which we refer for a more detailed
and general discussion.