A.2.1 Lp space, Lp norm
Let Rm be the m-dimensional Euclidean space, Ω be a Borel subset ofRm and µa positive Borel measure in Ω. Given a 1≤p <∞real numberLp(Ω, µ) is the space of allµ-measurable funtionsu in Ω with norm
kukp =kukLp(Ω,µ) =
for p = ∞ the Banach space L∞(Ω, µ) consists of all µ-measurable, µ-essentially bounded functions u with the norm
kuk∞ =kukL∞(Ω,µ) =µess sup{|u(x)|; x∈Ω}. (A.2.2) Strictly speaking the elements of Lp(Ω, µ) are not functions but equivalence classes of func-tions, the equivalence relation being equality almost everywhere with respect to µ. When µ is the Lebesgue measure we simply write Lp(Ω, µ) = Lp(Ω); when Ω is clear we further
TheLp(Ω, µ)ν space of vector-valued functions for finite integerνconsists of all vector-valued functions u(x) = (u1(x), ..., uν(x)) where each of the components is µ-measurable and
kukp = product can be defined similarly to (A.2.3).
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A.2.2 C(K) space of continuous functions
Another important space is that of continuous functions. LetK be a compact subset ofRm. The space C(K) consists of all continuous functions u in K endowed with the supremum norm
kukC = sup{|u(x)|; x∈K}. (A.2.4)
The so-definedC(K) is a Banach space. WhenK is closed but not compact we defineC0(K) to be the space of all continuous functions in K with
lim
|x|→∞u(x) = 0,
again endowed with the supremum norm
kukC0 = sup{|u(x)|; x∈K}. (A.2.5)
Both C(K) andC0(K) are separable.
For the Banach spaceC0(K)ν of all vector functionsu= (u1, ..., uν) where all components uj belong to C0(K) we define the supremum norm as
kukC0 = sup
1≤j≤ν
kujkC0. (A.2.6)
Ba differentiation monomial with Dj =∂/∂xj.
Let Ω ⊆ Rm be an arbitrary domain, j ≥ 0 integer and 1 ≤ p < ∞ real. The Sobolev space Wj,p(Ω) is defined as the space of all functions u∈Lp(Ω) such that
Dαu∈Lp(Ω)
for all|α|=α1+...+αm ≤j (the derivatives understood in the sense of distributions). The space Wj,p(Ω) is a Banach space with the norm
kukj,p=kukWj,p(Ω) =
For Ω = Rm and alternative characterization of Hj(Rm) can be given using the Fourier-Plancherel transform inL2(Rm), subindices to signal the change in variables). The inverse ofF is given by
u(x) =F−1u(x) = lim˜
These remarks make natural the introduction of the spaces Hs(Rm) – s now being an arbitrary
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nonnegative parameter – that consists of all u∈L2x such that (1 +|σ|2)s/2u˜ belongs to L2σ. These are Hilbert spaces under the scalar products
(u, v)s,2 = Z
Rm
(1 +|σ|2)su(σ)˜¯˜ v(σ)dσ (A.2.11) corresponding to the norm
kuks,2 =k(1 +|σ|2)s/2uk˜ L2(Rm). (A.2.12) The Hs(Rm)ν spaces of vector-valued functions can be introduced with the scalar product (u(σ), v(σ)) being in the integrand of (A.2.11).
A.2.4 Operators
An operatorAisclosedif and only if whenever {un}is a sequence inD(A) such thatun→u and Aun →v for someu, v ∈E, it follows that u∈D(A) and Au=v.
Theorem 3. Closed Graph Theorem. Let A be closed and everywhere defined. Then A is bounded.
A.2.5 Well posedness in the sense of Lax
Let A be a densely defined operator in an arbitrary Banach space E with a norm k.k.
Consider the equation
u0(t) =Au(t) (−∞< t <∞) (A.2.13) A solution of (A.2.13) is a function t→u(t) such that u(t) is continuously differentiable for
−∞< t <∞, u(t) is in the domain D(A) of A, and (A.2.13) is satisfied for−∞< t <∞.
Definition 6. We say that the Cauchy problem for (A.2.13) is well posed in the sense of Lax (or simply well posed) in−∞< t <∞ if the following two assumptions hold:
(1) Existence of solutions for sufficiently many initial data: There exists a dense subspace D of E such that, for any u0 ∈ D, there exists a solution u(.) of (A.2.13) in −∞< t < ∞ with
u(0) =u0. (A.2.14)
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(2) Continuous dependence of solutions on their initial data: There exists a function C(t) defined for −∞< t <∞ such that C(t) and C(−t) are nondecreasing, nonnegative, and
||u(t)|| ≤C(t)||u(0)|| (−∞< t <∞) (A.2.15) for any solution of (A.2.13).
Condition (2) can be given an equivalent (but more palpable) formulation as:
(2’) Let {un(.)} be a sequence of solutions of (A.2.13) with un(0) → 0. Then un(t) → 0 uniformly on compacts of −∞< t <∞.
Note that we treat initial value problems and mixed initial-boundary value problems within the same abstract framework. Boundary values, if given, are incorporated as restric-tions onE or on the domain D(A) of the operator A.
In this framework the distribution-theoretic idea of a weak solution of a differential equation can be formulated as follows. (Motivation: an otherwise appropriate u(t) might not belong to the domain ofA.) LetA be closed and letu(t) be a locally integrable function in −∞ < t < ∞. Denote the adjoint of A by A∗. We say that u(t) is a weak solution of (A.2.13) and (A.2.14) if and only if, for every u∗ ∈ D(A∗) and for every Schwartz test functionϕ ∈ D(consisting of all infinitely differentiable functions whose support is compact and contained in Rm) we have
Z ∞ 0
hA∗u∗, u(t)iϕ(t)dt=− Z ∞
0
hu∗, u(t)iϕ0(t)dt− hu∗, u0iϕ(0) . (A.2.16)
A.2.6 Propagators
A useful consequence of the definition of well posedness is the existence of the so-called propagator. Let us assume that equation (A.2.13) is well posed in−∞< t <∞, letu0 ∈D, and define the operator valued function S(.), for all such t, by
S(t)u0
=. u(t), (A.2.17)
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whereu(.) is the only solution of (A.2.13) withu(0) =u0. Due to condition (A.2.15)S(t) is a bounded operator inD, and since Dis dense inE we can extendS(t) to a bounded operator S(t) in¯ E. This extended function ¯S(.) is called the propagator of equation (A.2.13).
Note that due to the extension to E the E-valued function ¯S(t)v0 makes sense for all v0 ∈E; writing now
v(t) .
= ¯S(t)v(0) (−∞< t <∞) (A.2.18) where v(0) = v0 we arrive at the notion of a generalized solution of equation (A.2.13). A generalized solution is identical to a solution when the latter exists, but it does not need to be a genuine solution of (A.2.13) i.e. when v0 does not belong to the dense subset D on which we assumed the existence of a solution.
The notion of a generalized solution is equivalent with the notion of a weak solution in the sense defined above, see (Fattorini; 1983, p. 30) and onwards.
Definition 7. We call agroupany functions S(.)with values in the set of bounded operators over E defined in −∞< t <∞ which satisfy the equations
S(0) = I (A.2.19)
S(s+t) = S(s)S(t) (A.2.20)
for −∞< s, t <∞.
Definition 8. A group S(.) is strongly continuous if
∀u0 ∈E : kS(t)u0−u0k →0 as t→0 . (A.2.21) The propagator of equation (A.2.13) for which the Cauchy problem is well posed is a strongly continuous group (see (Fattorini;1983, p. 81)), and in general any strongly contin-uous group is called apropagator. The converse relationship between well posedness and the existence of a propagator also holds:
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Theorem 4. Let S(.) be a strongly continuous group. Then there exists a (unique) closed, densely defined operator A for which the Cauchy problem of (A.2.13) is well posed in the sense of Lax in −∞< t <∞ such that S is the propagator of this equation (A.2.13).