6. Solution Theory for Evolutionary Equations

We define

and if d = 3, Furthermore, we put and

The relations and \( \operatorname {\mathrm {curl}}_{0}\) are all densely defined, closed linear operators.

The operators and \( \operatorname {\mathrm {curl}}_{\mathrm {c}}\) are densely defined by Exercise 6.3. Thus, and \( \operatorname {\mathrm {curl}}\) are closed linear operators by Lemma 2.​2.​7. Moreover, it follows from integration by parts that \( \operatorname {\mathrm {grad}}_{\mathrm {c}}\subseteq \operatorname {\mathrm {grad}}\), and \( \operatorname {\mathrm {curl}}_{\mathrm {c}}\subseteq \operatorname {\mathrm {curl}}\). Thus, and \( \operatorname {\mathrm {curl}}\) are also densely defined. This, in turn, implies that and \( \operatorname {\mathrm {curl}}_{\mathrm {c}}\) are closable by Lemma 2.​2.​7 with respective closures and \( \operatorname {\mathrm {curl}}_{0}\) by Lemma 2.​2.​4. □

If f ∈ L ₂( Ω) and g = (g _j)_{j ∈{1,…,d}}∈ L ₂( Ω)^d then the following statements hold:

If d = 3 and f, g ∈ L ₂( Ω)³ then the following statements hold:

We emphasise that \(H_0^1(\Omega )=\overline {C_{\mathrm {c}}^\infty (\Omega )}^{H^1(\Omega )}\subseteq H^1(\Omega )\) is a proper inclusion for many open Ω. The ‘0’ in the index is a reminder of ‘0’-boundary conditions. In fact, the only difference between these two spaces lies in the behaviour of their elements at the boundary of Ω. The space \(H_0^1\) signifies all H ¹-functions vanishing at ∂ Ω in a generalised sense. The corresponding statements are true for the inclusions and \(H_0( \operatorname {\mathrm {curl}},\Omega )\subseteq H( \operatorname {\mathrm {curl}},\Omega )\). The space describes -vector fields with vanishing normal component and to lie in \(H_0( \operatorname {\mathrm {curl}},\Omega )\) provides a handy generalisation of vanishing tangential component. We will anticipate these abstractions when we apply the solution theory of evolutionary equations for particular cases. In a later chapter we will come back to this issue when we discuss inhomogeneous boundary value problems.

Let d = 3. We have the following inclusions:

It is elementary to show that for given \(\psi \in C_{\mathrm {c}}^\infty (\Omega )^{3}\) and \(\phi \in C_{\mathrm {c}}^\infty (\Omega )\) we have as well as \( \operatorname {\mathrm {curl}}_{0} \operatorname {\mathrm {grad}}_{0}\phi =0\). Thus, we obtain and \(\operatorname {ran}( \operatorname {\mathrm {grad}}_{\mathrm {c}})\subseteq \operatorname {ker}( \operatorname {\mathrm {curl}}_{0})\). Since and \(\operatorname {ker}( \operatorname {\mathrm {curl}}_{0})\) are closed, and \(C_{\mathrm {c}}^\infty (\Omega )^{3}\) and \(C_{\mathrm {c}}^\infty (\Omega )\) are cores for \( \operatorname {\mathrm {curl}}_{0}\) and \( \operatorname {\mathrm {grad}}_{0}\) respectively, we obtain the first two inclusions. The last two inclusions follow from the first two by taking into account the orthogonal decompositions and which follow from Corollary 2.​2.​6. □

Let \(\nu _{0}\in \mathbb {R}\) and H be a Hilbert space. Let \(M\colon \operatorname {dom}(M)\subseteq \mathbb {C}\to L(H)\) be a material law with \(\mathrm {s}_{\mathrm {b}}\left ( M \right )<\nu _{0}\) and let \(A\colon \operatorname {dom}(A)\subseteq H\to H\) be skew-selfadjoint. Assume that

for some c > 0. Then for all \(\nu \geqslant \nu _{0}\) the operator ∂ _t,ν M(∂ _t,ν) + A is closable and

Furthermore, S _ν is causal and satisfies \(\left \Vert S_{\nu } \right \Vert { }_{L(L_{2,\nu })}\leqslant 1/c\) , and for all \(F\in \operatorname {dom}(\partial _{t,\nu })\) we have

Furthermore, for \(\eta ,\nu \geqslant \nu _{0}\) and \(F\in L_{2,\nu }(\mathbb {R};H)\cap L_{2,\eta }(\mathbb {R};H)\) we have that S _ν F = S _η F.

If \(F\in \operatorname {dom}(\partial _{t,\nu })\), then by Theorem 6.2.1. Since M(∂ _t,ν) leaves the space \(\operatorname {dom}(\partial _{t,\nu })\) invariant, this gives that \(M(\partial _{t,\nu })U\in \operatorname {dom}(\partial _{t,\nu })\) and thus, U solves the evolutionary equation literally; that is, while for \(F\in L_{2,\nu }(\mathbb {R};H)\), in general, we just have

Let H be a Hilbert space and T ∈ L(H). If T is selfadjoint, we write \(T\geqslant c\) for some \(c\in \mathbb {R}\) if Moreover, we define the real part of T by .

Let H ₀, H ₁ be Hilbert spaces.

The proof of the first statement can be done in two steps. First, notice that the inclusion \(\begin {pmatrix} 0 & B^{*}\\ C^{*} & 0 \end {pmatrix}\subseteq \begin {pmatrix} 0 & C\\ B & 0 \end {pmatrix}^{*}\)follows immediately. If, on the other hand, \(\begin {pmatrix}\phi \\\psi \end {pmatrix}\in \operatorname {dom}\left (\begin {pmatrix} 0 & C\\ B & 0 \end {pmatrix}^{*}\right )\) with \(\begin {pmatrix} 0 & C\\ B & 0 \end {pmatrix}^{*}\begin {pmatrix}\phi \\\psi \end {pmatrix}=\begin {pmatrix}\xi \\\zeta \end {pmatrix}\) we get for all \(x\in \operatorname {dom}(B)\) that Hence, \(\psi \in \operatorname {dom}(B^{*})\) and B ^∗ ψ = ξ. Similarly, we obtain \(\phi \in \operatorname {dom}(C^{*})\) and C ^∗ ϕ = ζ.

For the second statement, we compute for all ϕ ∈ H ₀ using the Cauchy–Schwarz inequality Thus, a is one-to-one. Since \(\operatorname {Re} a=\operatorname {Re} a^{*}\) it follows that a ^∗ is one-to-one, as well. Thus, we get that a has dense range by Theorem 2.​2.​5. The inequality implies that a ⁻¹ is bounded with \(\left \Vert a^{-1} \right \Vert \leqslant \frac {1}{c}\). Hence, as a ⁻¹ is closed, \(\operatorname {dom}(a^{-1})=\operatorname {ran}(a)\) is closed by Lemma 2.​1.​3 and hence, \(\operatorname {dom}(a^{-1})=H_0\); that is, a ⁻¹ ∈ L(H ₀). To conclude, let ψ ∈ H ₀ and put . Then \(\left \Vert \psi \right \Vert { }_{H_0}=\left \Vert aa^{-1}\psi \right \Vert { }_{H_0}\leqslant \left \Vert a \right \Vert \left \Vert a^{-1}\psi \right \Vert { }_{H_0}\) and so

□

The first example we will consider is the heat equation in an open subset \(\Omega \subseteq \mathbb {R}^d\). Under a heat source, \(Q\colon \mathbb {R}\times \Omega \to \mathbb {R}\), the heat distribution, \(\theta \colon \mathbb {R}\times \Omega \to \mathbb {R}\), satisfies the so-called heat flux balance Here, \(q\colon \mathbb {R}\times \Omega \to \mathbb {R}^{d}\) is the heat flux which is connected to θ via Fourier’s law where \(a\colon \Omega \to \mathbb {R}^{d\times d}\) is the heat conductivity, which is measurable, bounded and uniformly strictly positive in the sense that for all x ∈ Ω and some c > 0 in the sense of positive definiteness. Moreover, we assume that Ω is thermally isolated, which is modelled by requiring that the normal component of q vanishes at ∂ Ω; that is, (see Remark 6.1.3). Written as a block matrix and incorporating the boundary condition, we obtain

For all ν > 0, the operator

is densely defined and closable in \(L_{2,\nu }\left (\mathbb {R};L_2(\Omega )\times L_2(\Omega )^{d}\right )\) . The respective closure is continuously invertible with causal inverse being eventually independent of ν.

The assertion follows from Theorem 6.2.1 applied to Note that M is a material law with \(\mathrm {s}_{\mathrm {b}}\left ( M \right )=0\) by Example 5.​3.​1. Moreover, for (x, y) ∈ L ₂( Ω) × L ₂( Ω)^d and \(z\in \mathbb {C}_{\operatorname {Re}\geqslant \nu }\) with ν > 0 we estimate where we have used Proposition 6.2.3 (b) in the first inequality. Moreover, A is skew-selfadjoint by Proposition 6.2.3 (a). □

Assume that \(Q\in \operatorname {dom}(\partial _{t,\nu })\). It then follows from Theorem 6.2.1 that Then, as in Remark 6.2.2, it follows that θ and q satisfy the heat flux balance and Fourier’s law in the sense that \(\theta \in \operatorname {dom}(\partial _{t,\nu })\cap \operatorname {dom}( \operatorname {\mathrm {grad}})\) and and This regularity result is true even for \(Q\in L_{2,\nu }(\mathbb {R};L_2(\Omega ));\) see [88] and Chap. 15, Theorem 15.​2.​3.

The classical scalar wave equation in a medium \(\Omega \subseteq \mathbb {R}^{d}\) (think, for instance, of a vibrating string (d = 1) or membrane (d = 2)) consists of the equation of the balance of momentum where the acceleration of the (vertical) displacement, \(u\colon \mathbb {R}\times \Omega \to \mathbb {R}\), is balanced by external forces, \(f\colon \mathbb {R}\times \Omega \to \mathbb {R}\), and the divergence of the stress, \(\sigma \colon \mathbb {R}\times \Omega \to \mathbb {R}^{d}\), in such a way that The stress is related to u via the following so-called stress-strain relation (here Hooke’s law) where the so-called elasticity tensor, \(T\colon \Omega \to \mathbb {R}^{d\times d}\), is bounded, measurable, and satisfies for some c > 0 uniformly in x ∈ Ω. The quantity \( \operatorname {\mathrm {grad}} u\) is referred to as the strain. We think of u as being fixed at ∂ Ω (“clamped boundary condition”). This is modelled by \(u\in \operatorname {dom}( \operatorname {\mathrm {grad}}_{0})\).

Let \(\Omega \subseteq \mathbb {R}^{d}\) be open, and T as indicated above. Then, for all ν > 0,

is densely defined and closable in \(L_{2,\nu }\left (\mathbb {R};L_2(\Omega )\times L_2(\Omega )^{d}\right )\) . The respective closure is continuously invertible with causal inverse being eventually independent of ν.

We apply Theorem 6.2.1 to , which is skew-selfadjoint by Proposition 6.2.3 (a), and \(M(z)=\begin {pmatrix} 1 & 0\\ 0 & T^{-1} \end {pmatrix}\), which defines a material law with \(\mathrm {s}_{\mathrm {b}}\left ( M \right )=-\infty \). The positive definiteness constraint needed in Theorem 6.2.1 is satisfied by Proposition 6.2.3 (b) on account of the selfadjointness of T, which implies the same for T ⁻¹. Indeed, for ν ₀ > 0 and \(z\in \mathbb {C}_{\operatorname {Re}\geqslant \nu _0}\) we estimate for each (x, y) ∈ L ₂( Ω) × L ₂( Ω)^d, where we used the selfadjointness of T ⁻¹ in the second line. □

Let \(f\in L_{2,\nu }(\mathbb {R};L_2(\Omega ))\), ν > 0, and define By Theorem 6.2.1, we obtain Hence, we have or Thus, formally, after another time-differentiation and the setting of \(\sigma =\partial _{t,\nu }\widetilde {\sigma }\) we obtain a solution of the wave equation, (u, σ). Notice, however, that differentiating cannot be done without any additional knowledge of the regularity of \(\widetilde {\sigma }\). In fact, in order to arrive at the balance of momentum equation, one would need to have . However, one only has . It is an elementary argument, see [110, Lemma 4.6], that we in fact have which suggests that, in general, , see Exercise 6.6.

The final example in this chapter forms the archetypical evolutionary equation—Maxwell’s equations in a medium \(\Omega \subseteq \mathbb {R}^{3}\). In order to identify the particular choices of M(∂ _t,ν) and A in the present situation (and to finally conclude the 2 × 2-block matrix formulation historically due to the work of [59, 64, 102]), we start out with Faraday’s law of induction, which relates the unknown electric field, \(E\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\), to the magnetic induction, \(B\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\), via We assume that the medium is contained in a perfect conductor, which is reflected in the so-called electric boundary condition which asks for the vanishing of the tangential component of E at the boundary. This is modelled by \(E\in \operatorname {dom}( \operatorname {\mathrm {curl}}_{0})\). The next constituent of Maxwell’s equations is Ampère’s law which relates the unknown electric displacement, \(D\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\), (free) current (density), \(j_{c}\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\), and magnetic field, \(H\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\), to the (given) external currents, \(j_{0}\colon \mathbb {R}\times \Omega \to \mathbb {R}^{3}\). Maxwell’s equations are completed by constitutive relations specific to each material at hand. Indeed, the (bounded, measurable) dielectricity, \(\varepsilon \colon \Omega \to \mathbb {R}^{3\times 3}\), and the (bounded, measurable) magnetic permeability, \(\mu \colon \Omega \to \mathbb {R}^{3\times 3}\), are symmetric matrix-valued functions which couple the electric displacement to the electric field and the magnetic field to the magnetic induction via Finally, Ohm’s law relates the current to the electric field via the (bounded, measurable) electric conductivity, \(\sigma \colon \Omega \to \mathbb {R}^{3\times 3}\), as All in all, in terms of (E, H), Maxwell’s equations read For the time being, we shall assume that there exist c > 0 and ν ₀ > 0 such that for all \(\nu \geqslant \nu _{0}\) we have in the sense of positive definiteness. Note that the latter condition allows particularly for ε = 0 on certain regions, if \(\operatorname {Re}\sigma \) compensates for this. To approximate small ε by 0 is referred to as the eddy current approximation in these regions. With the above preparations at hand, we may now formulate the well-posedness result concerning Maxwell’s equations.

Let \(\Omega \subseteq \mathbb {R}^{3}\) be open and \(\nu \geqslant \nu _{0}\) . Then

is densely defined and closable in \(L_{2,\nu }\left (\mathbb {R};L_2(\Omega )^{3}\times L_2(\Omega )^{3}\right )\) . The respective closure is continuously invertible with causal inverse being eventually independent of ν.

The assertion follows from Theorem 6.2.1 applied to the material law and the skew-selfadjoint operator

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In the physics literature (see e.g. [40, Chapter 18]), Maxwell’s equations are usually complemented by Gauss’ law, as well as the introduction of the charge density, , and the current, j = j ₀ − j _c, by the continuity equation We shall argue in the following that these equations are automatically satisfied if (E, H) is a solution to Maxwell’s equation. Indeed, assuming \(j_{0}\in \operatorname {dom}(\partial _{t,\nu })\), then, as a consequence of Theorem 6.2.1, for

we observe \(\begin {pmatrix} E\\ H \end {pmatrix}\in \operatorname {dom}\left (\partial _{t,\nu }\right )\cap \operatorname {dom}\left (\begin {pmatrix} 0 & - \operatorname {\mathrm {curl}}\\ \operatorname {\mathrm {curl}}_{0} & 0 \end {pmatrix}\right )\). Reformulating the latter equation yields Since \( \operatorname {\mathrm {curl}}_{0}E\in \operatorname {ran}( \operatorname {\mathrm {curl}}_{0})\), we have by Proposition 3.​1.​6(b) that \(\partial _{t,\nu }^{-1} \operatorname {\mathrm {curl}}_{0}E\in \operatorname {\overline {ran}}( \operatorname {\mathrm {curl}}_{0})\). Thus, by Proposition 6.1.4, we obtain Similarly, we deduce that If, in addition, we have that , we recover the continuity equation. In general, the continuity equation is satisfied in the integrated sense just derived.

Let H be a Hilbert space and \(B\colon \operatorname {dom}(B)\subseteq H\to H\) densely defined and closed. Assume there exists c > 0 such that

Then B ⁻¹ ∈ L(H) and \(\left \Vert B^{-1} \right \Vert \leqslant 1/c\).

Since B is not necessarily bounded here, the present argument requires a refinement of the one in Proposition 6.2.3. In fact, the first assumed inequality implies closedness of the range of B as well as continuous invertibility with \(B^{-1}\colon \operatorname {ran}(B)\to H\). The fact that \(\operatorname {ran}(B)\) is dense in H follows from the second inequality. □

In the proof of Theorem 6.2.1, we will apply Proposition 6.3.1 in a situation, where \(\operatorname {dom}(B^*)\subseteq \operatorname {dom}(B)\). In this case, the condition readily implies

Let \(\nu \geqslant \nu _0\) and \(z\in \mathbb {C}_{\operatorname {Re}\geqslant \nu }\). Define . Since M(z) ∈ L(H) it follows from Theorem 2.​3.​2 that \(B(z)^{*}=\left (zM(z)\right )^{*}-A\) and \(\operatorname {dom}(B(z))=\operatorname {dom}(B(z)^{*})=\operatorname {dom}(A)\). Moreover, for all \(\phi \in \operatorname {dom}(A)\) we have due to the skew-selfadjointness of A. Thus, by Proposition 6.3.1 (see also Remark 6.3.2) applied to B(z) instead of B, we deduce that is bounded and assumes values in L(H) with norm bounded by 1∕c. By Exercise 6.5, we have that S is holomorphic. Thus, S is a material law and \(\left \Vert S(\partial _{t,\nu }) \right \Vert \leqslant 1/c\) by Proposition 5.​3.​2. Moreover, Theorem 5.​3.​6 implies that S(∂ _t,ν) is independent of ν and causal.

Next, if \(f\in \operatorname {dom}(\partial _{t,\nu }),\) it follows that \(\left (\mathrm {i}\mathrm {m}+\nu \right )\mathcal {L}_{\nu }f\in L_2(\mathbb {R};H)\). Hence, for all \(t\in \mathbb {R}\) we obtain Thus, by the boundedness of M and S, we deduce \(S(\mathrm {i}\cdot +\nu )\mathcal {L}_{\nu }f\in L_2(\mathbb {R};\operatorname {dom}(A))\). This implies \(S(\partial _{t,\nu })f\in L_{2,\nu }(\mathbb {R};\operatorname {dom}(A))\) by Exercise 6.2. Similarly, but more easily, it follows that \(\left (\mathrm {i}\cdot +\nu \right )S(\mathrm {i}\cdot +\nu )\mathcal {L}_{\nu }f\in L_2(\mathbb {R};H)\) also, which shows \(S(\partial _{t,\nu })f\in \operatorname {dom}(\partial _{t,\nu })\).

We now define the operator B(im + ν) by and Then one easily sees that B(im + ν) = S(im + ν)⁻¹ and since S(im + ν) is closed, it follows that B(im + ν) is closed as well. Moreover and hence, the operator (im + ν)M(im + ν) + A is closable, which also yields the closability of ∂ _t,ν M(∂ _t,ν) + A by unitary equivalence. To complete the proof, we have to show that as this equality implies \(S(\partial _{t,\nu })=\big (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\big )^{-1}\) by unitary equivalence. For showing the asserted equality, let \(f\in \operatorname {dom}(B(\mathrm {i}\mathrm {m}+\nu ))\). For \(n\in \mathbb {N}\) we define . Then \(f_n\in \operatorname {dom}(\mathrm {i}\mathrm {m}+\nu )\cap \operatorname {dom}(A) \subseteq \operatorname {dom}\big ((\mathrm {i}\mathrm {m}+\nu )M(\mathrm {i}\mathrm {m}+\nu )+A\big )\) for each \(n\in \mathbb {N}\) and by dominated convergence, we have that f _n → f as n →∞ as well as n →∞. This shows that \(f\in \operatorname {dom}\big (\overline {(\mathrm {i}\mathrm {m}+\nu )M(\mathrm {i}\mathrm {m}+\nu )+A}\big )\) and hence, the assertion follows. □

Note that Theorem 6.2.1 can partly be generalised in the following way (with the same proof). Let \(M\colon \mathbb {C}_{\operatorname {Re}>\nu _0}\to L(H)\) be holomorphic and A a closed, densely defined operator in H such that zM(z) + A is boundedly invertible for all \(z\in \mathbb {C}_{\operatorname {Re}>\nu _0}\) and that \(\sup _{z\in \mathbb {C}_{\operatorname {Re}>\nu _0}}\left \Vert (zM(z)+A)^{-1} \right \Vert { }_{L(H)}<\infty \). Then \(S_\nu \in L(L_{2,\nu }(\mathbb {R};H))\) is causal and eventually independent of ν.

As the proof of Theorem 6.2.1 shows, for \(\nu \geqslant \nu _0\) we have that \(S\colon \mathbb {C}_{\operatorname {Re}\geqslant \nu }\ni z\mapsto (zM(z)+A)^{-1}\in L(H)\) is a material law and S _ν = S(∂ _t,ν). Thus, the solution operator is a material law operator, and by Remark 5.​3.​3 applied to S and \(z\mapsto \frac {1}{z}1_{H}\) we obtain

Let \(\left (\Omega ,\Sigma ,\mu \right )\) be a σ-finite measure space and let H ₀, H ₁ be Hilbert spaces. Let \(A\colon \operatorname {dom}(A)\subseteq H_0\to H_1\) be densely defined and closed. Show that the operator is densely defined and closed. Moreover, show that \(\left (A_{\mu }\right )^{*}=\left (A^{*}\right )_{\mu }\).

In the situation of Exercise 6.1, if ( Ω₁, Σ₁, μ ₁) is another σ-finite measure space and \(\mathcal {F}\colon L_2(\mu )\to L_2(\mu _{1})\) is unitary, show that for j ∈{0, 1} there exists a unique unitary operator \(\mathcal {F}_{H_j}\colon L_2(\mu ;H_j)\to L_2(\mu _{1};H_j)\) such that Furthermore, prove that

Show that for \(\Omega \subseteq \mathbb {R}^{d}\) open, the set \(C_{\mathrm {c}}^\infty (\Omega )\subseteq L_2(\Omega )\) is dense.

Prove Theorem 6.1.2.

Let H be a Hilbert space, \(A\colon \operatorname {dom}(A)\subseteq H\to H\) skew-selfadjoint, and c > 0. Moreover, let \(M\colon \operatorname {dom}(M)\subseteq \mathbb {C}\to L(H)\) be holomorphic with Show that \(\operatorname {dom}(M)\ni z\mapsto \left (M(z)+A\right )^{-1}\) is holomorphic.

Let \(C\colon \operatorname {dom}(C)\subseteq H_0 \to H_1\) be a densely defined and closed linear operator acting in Hilbert spaces H ₀ and H ₁. For ν > 0 show that Hint: Apply Exercise 6.2 and show \(\overline {(\mathrm {i}\mathrm {m}+\nu )^{-1}C}=C(\mathrm {i}\mathrm {m}+\nu )^{-1}\) with a suitable approximation argument.

Let \(\Omega \subseteq \mathbb {R}^d\) be open.

The regularity assumption in (b) always holds and is known as Weyl’s Lemma, see e.g. [45, Corollary 8.11], where the more general situation of an elliptic operator with smooth coefficients is treated. See also [32, p.127], where the regularity is shown for harmonic distributions.