Semigroup Theory Differential Equations Heat Equation
Semi-Groups of Bounded Linear Operators
Walton Green
Clemson University
MATH 9270
Final Project
December 9, 2016
Semigroup Theory Differential Equations Heat Equation
Table of Contents
1 Semigroup Theory
Motivation
Basic Properties
Hille-Yosida Theorem
Lumer-Phillips Theorem
2 Differential Equations
Abstract Problem
Proof of Theorem
3 Heat Equation
Semigroup Theory Differential Equations Heat Equation
Motivation
Motivation
d u(t) = Au(t) t > 0
dt (1)
u(0) = u
0
where A : D(A) → H for some Hilbert space H. Consider the
“solution operator” (if it exists)
S(t)u0 = u(t)
Desired Properties:
S is bounded and linear if (1) is well-posed
S(0)u0 = u0
S(t1 )S(t2 )u0 = S(t1 + t2 )u0
d
S(t)u0 = AS(t)u0 i.e. e tA u0 = S(t)u0
dt
Semigroup Theory Differential Equations Heat Equation
Basic Properties
Definitions
A parameter family T (t), t ≥ 0 of bounded linear operators
T (t) : H → H is a semigroup of bounded linear operators if
(i) T (0) = I
(ii) T (t + s) = T (t)T (s) for all t, s ≥ 0
T (t) is uniformly/strongly continuous if
lim ||T (t) − I || = 0 lim T (t)x − x = 0
t&0 t&0
Define A : D(A) → H, infinitesimal generator of T (t) by
T (t)x − x
Ax := lim
t&0 t
Semigroup Theory Differential Equations Heat Equation
Basic Properties
Proposition
Let T (t) be a strongly continuous semigroup. Then there exists
M, ω > 0 s.t.
||T (t)|| ≤ Me ωt ∀t ≥ 0
Pf. Claim: There exists τ > 0 s.t. T (t) is bounded for 0 ≤ t ≤ τ .
If this is false, then there exists
tn → 0 but {T (tn )} is unbounded
By Uniform Bounded Principle, {T (tn )x} cannot converge for all
x ∈ H. However, since T (t) is strongly continuous,
lim T (tn )x = x ∀x ∈ H
n→∞
which is a contradiction. Let M := sup ||T (t)||.
t≤τ
Semigroup Theory Differential Equations Heat Equation
Basic Properties
By the Division Algorithm, for any t ≥ 0,
t = nτ + r
for some n ∈ N, 0 ≤ r < τ .
T (t) = T (nτ + r ) = T (nτ )T (r ) = T (τ )n T (r )
Define ω := log M/τ . Then, e ωt = M t/τ .
||T (t)|| ≤ ||T (τ )||n · ||T (r )|| ≤ M n · M ≤ M t/τ · M = Me ωt
Corollaries
The mapping t 7→ T (t)x is continuous for all x ∈ H therefore
Z t+h
1
lim T (s)x ds = T (t)x ∀x ∈ H
h&0 h t
Semigroup Theory Differential Equations Heat Equation
Basic Properties
Theorem
Let T (t) be a strongly continuous semigroup with infinitesimal
generator A. Then,
d
(a) T (t)x = T (t)Ax = AT (t)x for all x ∈ D(A).
dt
(b) D(A) ⊆ H is dense.
(c) A is closed.
Pf. For x ∈ D(A) and h > 0,
T (t + h)x − T (t)x T (t)[T (h)x − x] T (h)x − x
= = T (t)
h h h
T (h) − I
= T (t)x
h
Take limits as h → 0 to acheive (a).
Semigroup Theory Differential Equations Heat Equation
Basic Properties
We also have the following important identity
Z t
A T (s)x ds = T (t)x − x (∗)
0
for all x ∈ D(A) and t ≥ 0. Indeed, let h > 0, x ∈ D(A)
T (h) − I t
Z Z t Z t
1
T (s)x ds = T (s + h)x ds − T (s)x ds
h 0 h 0 0
Z t+h Z t
1
= T (s)x ds − T (s)x ds
h h 0
1 t+h 1 h
Z Z
= T (s)x ds − T (s)x ds
h t h 0
→ T (t)x − x
as h → 0.
Semigroup Theory Differential Equations Heat Equation
Basic Properties
Z t
Therefore, T (s)x ds ∈ D(A) and
0
Z t
1
T (s)x ds → x ∈ H
t 0
thus we have D(A) ⊆ H is dense (b).
Semigroup Theory Differential Equations Heat Equation
Basic Properties
Z t
Therefore, T (s)x ds ∈ D(A) and
0
Z t
1
T (s)x ds → x ∈ H
t 0
thus we have D(A) ⊆ H is dense (b). To show A is closed, let
xn → x such that Axn → y . Then,
Z t Z t Z t
T (s)Axn ds − T (s)y ds ≤
||T (s)||·||Axn −y || ds → 0
0 0 0
Z t
T (t)x − x = T (s)y ds
0
t
T (t)x − x
Z
1
= T (s)y ds
t t 0
Ax = y
Semigroup Theory Differential Equations Heat Equation
Hille-Yosida Theorem
Hille-Yosida Theorem
Definition
T (t) is a semigroup of contractions if ||T (t)|| ≤ 1 for all t ≥ 0.
Theorem (Hille-Yosida)
A linear (unbounded) operator A : D(A) → H is the generator of a
strongly continuous semigroup if and only if
(i) A is closed
(ii) D(A) ⊆ H is dense
(iii) For each λ > 0, λ ∈ ρ(A) and
1
||(λI − A)−1 || ≤
λ
Semigroup Theory Differential Equations Heat Equation
Hille-Yosida Theorem
Necessity. We already have (i) and (ii) by the previous work. To
show (iii), set Z ∞
R(λ)x := e −λt T (t)x dt
0
for all x ∈ H. We will show that R(λ) = (A − λI )−1 . First though,
we prove the estimate in part (iii):
Z ∞
1
||R(λ)x|| ≤ e −λt ||T (t)x|| dt ≤ ||x||
0 λ
Semigroup Theory Differential Equations Heat Equation
Hille-Yosida Theorem
Then, for h > 0,
T (h) − I 1 ∞ −λt
Z
R(λ)x = e [T (t + h)x − T (t)x] dt =
h h 0
e λh − 1 ∞ −λt e λh h −λt
Z Z
= e T (t)x dt − e T (t)x dt
h 0 h 0
→ λR(λ)x − x
as h → 0. And the LHS → AR(λ)x thus we have the right inverse:
AR(λ)x = λR(λ)x − x
(λI − A)R(λ)x = x
We have the left inverse if A and R(λ) commute which follows
from the fact that A and T (t) commute.
Semigroup Theory Differential Equations Heat Equation
Hille-Yosida Theorem
The proof for sufficiency is more interesting. We know how to get
A from T (t), but what about the other direction? One idea is that
T (t) = e tA , but how should this be understood?
∞
tA
X (tA)n
e =
n!
n=0
t −n
e tA = lim I − A
n→∞ n
Semigroup Theory Differential Equations Heat Equation
Hille-Yosida Theorem
The proof for sufficiency is more interesting. We know how to get
A from T (t), but what about the other direction? One idea is that
T (t) = e tA , but how should this be understood?
∞
tA
X (tA)n
e =
n!
n=0
t −n
e tA = lim I − A
n→∞ n
Introduce the Yosida approximation for λ > 0
Aλ := λ2 (λI − A)−1 − λI
Aλ is bounded, so we can define e tAλ and it can be shown
T (t)x = lim e tAλ x ∀x ∈ H
λ→∞
Semigroup Theory Differential Equations Heat Equation
Lumer-Phillips Theorem
Definition
A is dissipative if
RehAx, xi ≤ 0 ∀x ∈ D(A)
Theorem (Lumer-Phillips)
Let A be a densely defined linear operator. If A is dissipative and
ρ(A) 6= ∅, then A is the infinitesimal generator of a strongly
continuous semigroup of contractions T (t).
Pf. For all x ∈ D(A) and λ > 0,
||λx − Ax|| · ||x|| ≥ hλx − Ax|xi ≥ Rehλx − Ax|xi ≥ λ||x||2
1
Thus for all λ ∈ ρ(A), ||R(λ)|| ≤ . NTS ρ(A) ⊇ (0, ∞).
λ
Semigroup Theory Differential Equations Heat Equation
Lumer-Phillips Theorem
We know the resolvent set is open so Λ := ρ(A) ∩ (0, ∞) is as well.
Let λ ∈ Λ such that λn → λ > 0. Fix y ∈ H. Then there exists
xn ∈ D(A) such that
λn xn − Axn = y
{xn } is Cauchy so let x be its limit. Then, since A is closed
λx − Ax = y
so λ ∈ Λ. Therefore ρ(A) contains (0, ∞).
Applying Hille-Yosida, A is the infinitessimal generator of a
strongly continuous semigroup of contractions.
Semigroup Theory Differential Equations Heat Equation
Lumer-Phillips Theorem
Corollary
If both A and A∗ are dissipative, then A is the infinitesimal
generator of a strongly continuous semigroup of contractions.
It suffices to show Ran(I − A) = H. Let (xn − Axn ) → y . Since A
is dissipative,
||xn − Axn || ≥ ||xn ||
so xn is Cauchy and convergent to some x. Since A is closed,
x − Ax = y so y ∈ Ran(I − A). If Ran(I − A) 6= H, then there
exists non-zero z ∈ H such that
hz|x − Axi = 0 ∀x ∈ D(A)
=⇒ hz − A∗ z|xi = hz|xi − hA∗ z|xi = hz|xi − hz|Axi = 0
thus z − A∗ z = 0. Since A∗ is dissipative, ||z − A∗ z|| ≥ ||z|| thus
z = 0 which is a contradiction.
Semigroup Theory Differential Equations Heat Equation
Abstract Problem
d u(t) = Au(t) t > 0
dt (2)
u(0) = x
Under what conditions does this problem have a unique solution?
Theorem
Let A be a densely defined linear operator with ρ(A) 6= ∅. (2) has
a unique continuously differentiable solution u(t) for every
x ∈ D(A) if and only if A is the infinitesimal generator of a
strongly continuous semigroup.
If A generates T (t), then clearly T (t)x is a solution for every
x ∈ D(A). Now we show uniqeuness.
Semigroup Theory Differential Equations Heat Equation
Proof of Theorem
Let u(t) be a solution with u(0) = 0. We will show u(t) = 0.
d
R(λ)u(t) = R(λ)Au(t) = λR(λ)u(t) − u(t)
dt
Z t
R(λ)u(t) = e λ(t−s) u(s) ds
0
So, Z t
lim e λ(t−s) u(s) ds = 0
λ→∞ 0
therefore u(t) = 0.
Semigroup Theory Differential Equations Heat Equation
Proof of Theorem
As a result of uniqueness, we have that any solution to (2) has the
semigroup property:
d u(t) = Au(t) t > 0
d v (t) = Av (t) t > 0
dt dt
u(0) = x
v (0) = u(t )
1
Semigroup Theory Differential Equations Heat Equation
Proof of Theorem
As a result of uniqueness, we have that any solution to (2) has the
semigroup property:
d u(t) = Au(t) t > 0
d v (t) = Av (t) t > 0
dt dt
u(0) = x
v (0) = u(t )
1
Then, for w (t) = v (t) − u(t + t1 ),
d w (t) = Aw (t) t > 0
dt
w (0) = 0
so w (t) = 0 thus v (t) = u(t + t1 ).
Semigroup Theory Differential Equations Heat Equation
Proof of Theorem
Suppose A is densely defined and (2) has a unique continuously
differentiable solution u(t) for all x ∈ D(A). We show A generates
a strongly continuous semigroup. D(A) is Banach space equipped
with the norm
||x||D := ||x||H + ||Ax||H
since A is closed (which follows from ρ(A) 6= ∅). Fix t0 > 0 and
define the solution operator
S : D(A) → C [0, t0 ], D(A)
x 7→ u(·)
S is closed and linear therefore bounded by the Closed Graph
Theorem. For 0 ≤ t ≤ t0 define
T (t) : D(A) → D(A)
x 7→ u(t)
Semigroup Theory Differential Equations Heat Equation
Proof of Theorem
T (t) has the desired semigroup property since it solves (2).
Moreover, T (t) is uniformly bounded in || · ||D
sup ||T (t)x||D = ||Sx||sup ≤ ||S|| · ||x||D
0≤t≤t0
which can be extended to
||T (t)x||D ≤ Me ωt ||x||D
for any t > 0. Since ρ(A) 6= ∅, (λI − A)−1 exists for some λ. For
x ∈ D(A2 ), set y = (λI − A)−1 x. Then,
||T (t)x||H = ||(λI − A)T (t)y ||H ≤ C ||T (t)y ||D ≤ C2 e ωt ||x||H
Thus T (t) can be extended to H since D(A2 ) ⊆ H is dense.
Finally, it can be shown that A is the infinitesimal generator of
T (t).
Semigroup Theory Differential Equations Heat Equation
Heat Equation
d u(t) = Au(t) t > 0
dt (3)
u(0) = u
0
Pn 2
where A = ∆ = i=1 ∂x . Let H = L2 (Ω), where Ω ⊆ Rn and ∂Ω
is smooth. Let
D(A) = C0∞ (Ω)
With these specifications, the heat equation takes the form
u (x, t) = ∆u(x, t) x ∈ Ω t > 0
t
u(x, 0) = u0 (x) x ∈Ω (4)
u(x, t) = 0 x ∈ ∂Ω
Semigroup Theory Differential Equations Heat Equation
If we can show ∆ is dissipative then we can apply the corollary to
Lumer Phillips along with the previous Differential Equation result
to see that the Heat Equation has a unique solution. For
f ∈ C0∞ (Ω),
Z Z
h∆f , f i = ∆f · f = − (∇f )2 ≤ 0
Ω Ω
Then, ∆ is dissipative. It can be easily shown that ∆ is
self-adjoint. so its adjoint is also dissipative. Therefore the
conditions are met for (4) to have a unique solution.
Semigroup Theory Differential Equations Heat Equation
References
Pazy, A. Semigroups of Linear Operators and Applications to
Partial Differential Equations. Springer-Verlag, New York.
1983
Lax, Peter D. Functional Analysis. Wiley. 2002.
LeVarge, S. L. Semigroups of Linear Operators. 2003. link
