The Calder\'{o}n problem for the fractional wave equation: Uniqueness and optimal stability

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension $n\in \N$.

In this paper, we study an inverse problem for the fractional wave equation with a potential. The mathematical model for the fractional wave equation is formulated as follows. Let Ω ⊂ R n be a bounded Lipschitz domain, for n ∈ N. Given T > 0, s ∈ (0, 1) and q = q(x) ∈ L ∞ (Ω), consider the initial exterior value problem for the wave equation with the fractional Laplacian,      where (−∆) s is the standard fractional Laplacian 1 , and Ω e := R n \ Ω denotes the exterior domain. The fractional wave equation can be regarded as a special case of the peridynamics which models the nonlocal elasticity theory, see e.g. [Sil16].
Throughout this work, we assume that the (lateral) exterior data f is compactly supported in the set W T := W × (0, T ) ⊂ (Ω e ) T , where W ⊂ Ω e with W ∩ Ω = ∅ can be any nonempty open subset with Lipschitz boundary, and, to simplify our notations, we assume that both q and f are real-valued functions. Note that the initial boundary value problem (1.1) is a mixed local-nonlocal type equation. In order to study the inverse problem of (1.1), we will use the strong approximation property of (1.1), which is due to the nonlocality of the fractional Laplacian (−∆) s , for 0 < s < 1. Hence, by the well-posedness of (1.1) (see Theorem 2.1), one can formally define the associated Dirichlet-to-Neumann (DN) map Λ q (1.2) Λ q : C ∞ c ((Ω e ) T ) → L 2 (0, T ; H −s (Ω e )), Λ q : f → (−∆) s u| (Ωe)T , where u is the unique solution to (1.1). The precise definitions of the Sobolev spaces will be given in Section 2.1. Let us state the first main result of our work.
The proof of Theorem 1.1 is based on the qualitative form of the Runge approximation for the fractional wave equation: For any g ∈ L 2 (Ω T ), there exists a sequence of functions {f k } k∈N ∈ C ∞ c ((W 1 ) T ) such that u k → g in L 2 (Ω T ) as k → ∞, where u k is the solution to (1.1) with u k = f k in (Ω e ) T , for all k ∈ N. The preceding characterization can be regarded as an exterior control approach, in the sense that one can always control the solution by choosing appropriate exterior data.
The second main result of the paper is a quantitative version of Theorem 1.1, which provides a stability estimate for our fractional Calderón problem. Before we state the stability result, we introduce some notations.
To shorten our notations, we denote the operator norm as · * = · L 2 (0,T ;H 2s W )→L 2 (0,T ;H −2s (W )) , where the Sobolev space H 2s W will be described in Section 2.1. We are now ready to state the second main result of our work.
Inspired by Theorem 1.1, we will prove Theorem 1.2 by using a quantitative version of Runge approximation, which involves the well-known Caffarelli-Silvestre extension for the fractional Laplacian and the propagation of smallness. Moreover, Theorem 1.1 and Theorem 1.2 are satisfied for any spatial dimension n ∈ N.
The third main result of this work studies the exponential instability of the Calderón problem for the fractional wave equation. In other words, the stability result in Theorem 1.2 is optimal. For brevity, we denote the operator norm where B r with r > 0 stands for the ball of radius r centered at the origin.
For 1-dimensional case (n = 1), we can also establish the same estimate.
For the local counterpart, let us consider the following initial boundary value problem for the local wave equation: where q = q(x) ∈ L ∞ (Ω). It is known that (1.7) is well-posed (for example, see [Eva98]) with suitable compatibility conditions. Assuming the well-posedness of (1.7), the corresponding (hyperbolic) Neumann-to-Dirichlet map of (1.7) is defined by Λ q g := u| ∂Ω×[0,T ] for all g ∈ C ∞ c ((∂Ω) T ). In fact, Λ q : L 2 (∂Ω × (0, T )) → H 1 (0, T ; L 2 (∂Ω)) is a bounded linear operator, which can be proved by the energy estimate of (1.7), see e.g. [CP82, Section 6.7.5]. Now we assume Under assumption (1.8), in [RW88], they showed the global uniqueness result for time-independent potentials: In [Sun90], the author showed that, if (1.8) holds, under some apriori assumptions, the following estimate hold: for some constants C and α, where · L stands for the operator norm for the Neumann-to-Dirichlet map. A similar estimate also holds for the hyperbolic Dirichletto-Neumann map [AS90]. In other words, the stability of the inverse problem for the local wave equation is of Hölder-type. We also mention other related results of inverse problems for the local wave equation with potentials [Esk06,Esk07,Isa91,Kia17,RS91,Sal13]. Similar to the local version, we can prove the global uniqueness result for timeindependent potentials for the fractional wave equation (see Theorem 1.1). However, in the nonlocal counterpart of (1.9), we show that the stability of the inverse problem for the fractional wave equation is of (optimal) logarithmic-type in view of Theorem 1.2 and Theorem 1.3. We also want to point out that we do not need to assume the large influence time condition (1.8). One possible explanation is that while the speed of propagation of the local wave equation is finite, the speed of propagation of the fractional wave operator is infinite due the nonlocal nature of the fractional Laplacian (−∆) s , for 0 < s < 1. We will offer some detailed arguments in Section 2.
Before ending this section, we would like to discuss some interesting results for the time-harmonic wave equation. Consider the time-harmonic wave equation with a potential (a.k.a. Schrödinger equation): Ignoring the effect of the frequency κ > 0, Alessandrini [Ale88] proved the wellknown logarithmic stability estimate for the inverse boundary value problem of (1.10), and Mandache [Man01] established that this logarithmic estimate is optimal by showing that the inverse problem is exponentially unstable. Nonetheless, by taking the frequency into account, it was shown in [INUW14] that where C q1 , C q2 are the Cauchy data of the Schrödinger equation (1.10) corresponding to q 1 , q 2 , and dist (C q1 , C q2 ) is the Hausdorff distance between C q1 and C q2 . Isakov [Isa11] proved a similar estimate in terms of the DN maps.
The estimate (1.11) is shown to be optimal in the recent paper [KUW21]. The estimate (1.11) clearly indicates that the logarithmic part decreases as the frequency κ > 0 increases and the estimate changes from a logarithmic type to a Hölder type. This phenomena is termed as the increasing stability. It is interesting to compare the stability estimate (1.11) of the time-harmonic wave equation (1.10) with the stability estimate (1.9) of the local wave equation (1.7).
Similarly to the local wave equation, we consider the following time-harmonic fractional wave equation which is a fractional Schrödinger equation. Without considering the effect of the frequency κ > 0, Rüland and Salo [RS20] obtained a logarithmic type stability estimate for the inverse boundary value problem of the time-harmonic fractional wave equation (1.12) and, in [RS18], they proved that such logarithmic estimate is optimal by showing the exponential instability phenomenon. These results give rise to a natural question: in the inverse boundary value problem for (1.12), if we take the frequency κ into account, does the increasing stability estimate similar to (1.11) hold? In view of the optimal logarithmic stability results in Theorem 1.2 and Theorem 1.3, we have a strong reason to believe that the answer to this question is negative.
The paper is organized as follows. We discuss and prove the well-posedness of the fractional wave equation in Section 2 and in Appendix A, respectively. We then prove Theorem 1.1 in Section 3, and prove Theorem 1.2 in Section 4. The approach is mainly based on the qualitative and quantitative Runge approximation properties for the fractional wave equation. Finally, we prove Theorem 1.3 and Theorem 1.4 in Section 5.

The forward problems for the fractional wave equation
In this section, we provide all preliminaries that we need in the rest of the paper. Let us first recall (fractional) Sobolev spaces and prove the well-posedness of the fractional wave equation (1.1).
2.1. Sobolev spaces. Let F , F −1 be Fourier transform and its inverse, respectively. For s ∈ (0, 1), the fractional Laplacian is defined via where H s (R n ) stands for the L 2 -based fractional Sobolev space (see [DNPV12,Kwa17,Ste16]). The space H a (R n ) = W a,2 (R n ) denotes the (fractional) Sobolev space equipped with the norm for any a ∈ R, where ξ = (1 + |ξ| 2 ) 1 2 . It is known that for s ∈ (0, 1), · H s (R n ) has the following equivalent representation Given any open set O of R n and a ∈ R, let us define the following Sobolev spaces, It is not hard to see that H a (O) ⊆ H a 0 (O), and that H a O is a closed subspace of H a (R n ). We also denote H −s (O) to be the dual space of H s (O). In fact, H −s (O) has the following characterization: see e.g. [GSU20, Section 2.1], [McL00, Chapter 3], or [Tri02] for more details about the fractional Sobolev spaces. Moreover, we will use in the rest of this paper, for any set A ⊂ R n .
2.2. The forward problem. We first state the well-posedness of the fractional wave equation. As above, let Ω ⊂ R n be a bounded Lipschitz domain with n ∈ N.
Given T > 0, s ∈ (0, 1), and q = q(x) ∈ L ∞ (Ω), consider the initial exterior value problem for the fractional wave equation for some open set with Lipschitz boundary W ⊂ Ω e satisfying W ∩ Ω = ∅, ϕ ∈ H s (Ω), and ψ ∈ L 2 (R n ) with supp (ψ) ⊂ Ω. We want to show the well-posedness of (2.1). Setting v := u − f , we then consider the fractional wave equation with zero exterior data . Hence, we simply denote the initial data as (ϕ, ψ) in the rest of the paper. Now it suffices to study the well-posedness of (2.1). Let us introduce the following notations. Define Similarly, the function F : [0, T ] → L 2 (Ω) can be defined analogously by With these notations at hand, we can define the weak formulation for the fractional wave equation. Let φ ∈ H s (Ω) be any test function, multiplying (2.1) with φ gives is a weak solution of the initial exterior value problem (2.2) if , for all φ ∈ H s (Ω), and for 0 ≤ t ≤ T a.e. (2) v(0) = ϕ and v ′ (0) = ψ.
Theorem 2.1 (Well-posedness). For any F ∈ L 2 (0, T ; L 2 (Ω)), ϕ ∈ H s (Ω), and ψ ∈ L 2 (R n ) with supp ( ψ) ⊂ Ω, there exists a unique weak solution v to (2.2). Moreover, the following estimate holds: v L ∞ (0,T ; H s (Ω)) + ∂ t v L ∞ (0,T ;L 2 (Ω)) ≤ C F L 2 (0,T ;L 2 (Ω)) + ϕ H s (Ω) + ψ L 2 (Ω) . (2.3) Corollary 2.2. Let Ω ⊂ R n be a bounded Lipschitz domain for n ∈ N, and W ⊂ Ω e be any open set with Lipschitz boundary satisfying W ∩ Ω = ∅. Then for any , and ψ ∈ L 2 (R n ) with supp (ψ) ⊂ Ω, there exists a unique weak solution u = v + f of (2.1), where v ∈ L 2 (0, T ; H s (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) is the unique weak solution of (2.2). Furthermore, we have the following estimate The proof of Theorem 2.1 is similar to the well-posedness of the classical wave equation (i.e., s = 1) and, for the sake of completeness, we will give a comprehensive proof in Appendix A. In this article, we only consider the time-independent potential q = q(x) ∈ L ∞ (Ω). In fact, the well-posedness for a space-time dependent potential q = q(x, t) ∈ L ∞ (Ω T ) has been studied. We refer to [Bre11,Theorem 10.14] for the well-posedness of the abstract wave equations, and to [DFA20] for the well-posedness result for non-local semi-linear integro-differential wave equations which involve both the fractional Laplacian (in space) and the Caputo fractional derivative operator (in time).
2.3. The DN map and its duality. With the well-posedness at hand, one can define the corresponding DN map (1.2) for the fractional wave equation (1.1). Let us define the solution operator where W ⊂ Ω e is a Lipschitz set with W ∩ Ω = ∅, and u is the solution of (1.1). Given any ϕ(x, t) defined in (Ω e ) T , we define and we define the following backward DN-map: Proof. Let u 1 = P q f 1 and u 2 = P q f 2 . Using integration by parts, we have Finally, changing the variable t → T − t, we have which is our desired lemma.
Since Λ * q is self-adjoint, we can derive the following identity immediately. Lemma 2.4 (Integral identity). Let q 1 , q 2 ∈ L ∞ (Ω), and given any . Let u 1 := P q1 f 1 and u 2 := P q2 f 2 , where the operator P q is given in (2.5), for q = q 1 and q = q 2 , respectively. Then Proof. Using (2.6), we have Combining with (2.7), we obtain which is our desired lemma.

Global uniqueness for the fractional wave equation
In this section, let us state and prove a qualitative Runge type approximation for the fractional wave equation, and then prove Theorem 1.1. Before further discussion, let us comment on the speeds of propagation of the local and nonlocal wave equations.
, for x ∈ R n , such that φ ≡ 0 and φ is compactly supported, then for every t > 0, the solution u(·, t) has compact support.
On the other hand, the speed of propagation for the fractional wave equation is infinite due to the nonlocal nature of the fractional Laplacian. To prove this rigorously, let us recall the strong uniqueness property for the fractional Laplacian. Given 0 < s < 1, r ∈ R, if u ∈ H r (R n ) satisfies u = (−∆) s u = 0 in any nonempty open subset of R n , then u ≡ 0 in R n . By this property, we can prove the following lemma.
Proof. Suppose the contrary, that the speed of propagation of (3.1) is finite.
Using the strong uniqueness for the fractional Laplacian, we conclude that u ≡ 0, which implies φ ≡ 0, this is a contradiction.
3.1. Qualitative Runge approximation. The qualitative approximation property is based on the strong uniqueness for the fractional Laplacian ([GSU20, Theorem 1.2]).
Let Ω ⊂ R n be a bounded Lipschitz domain for n ∈ N, and W ⊂ Ω e be an open set with Lipschitz boundary satisfying W ∩ Ω = ∅. For s ∈ (0, 1), let P q be the solution operator given by (2.5), and define Remark 3.2. The Runge approximation plays an essential role in the study of fractional inverse problems, for example, see [GSU20, GLX17, RS20, CLR20] and references therein.
Proof of Theorem 3.1. By using the Hahn-Banach theorem and the duality arguments, it suffices to show that if v ∈ L 2 (Ω T ), which satisfies Similar to the proof of Theorem 2.1, it is easy to see that (3.3) is well-posed.
For f ∈ C ∞ c (W T ), let u and w be the solutions of (1.1) and (3.3), respectively. Note that u − f is only supported in Ω T , then we have where we have used u is the solution of (1.1), u(x, 0) = ∂ t u(x, 0) = 0 and w(x, T ) = ∂ t w(x, T ) = 0 for x ∈ R n in last equality of (3.4) . By using the conditions (3.2) and (3.4), one must have (f, (−∆) s w) L 2 (WT ) = 0, for any f ∈ C ∞ c (W T ), which implies that Fix any fixed t ∈ (0, T ), the strong uniqueness for the fractional Laplacian (see [GSU20, Theorem 1.2]) yields that w(·, t) = 0 in R n × {t}, for all t ∈ (0, T ). Therefore, we derive v = 0 as desired, and the Hahn-Banach theorem infers the density property. This proves the assertion.
Remark 3.3. By using similar arguments, one can also consider the well-posedness (Theorem 2.1) and the Runge approximation (Theorem 3.1) also hold for the case q = q(x, t) ∈ L ∞ (Ω T ). In this work, we are only interested in time-independent potentials q = q(x). Proof of Theorerm 1.1. Given any g ∈ L 2 (Ω T ), using Theorem 3.1, there exists a sequence Combining (1.4) and (2.8), we know that Taking the limit k → ∞, we obtain Finally, by the arbitrariness of g ∈ L 2 (Ω T ), we conclude that q 1 = q 2 in Ω T .

Stability for the fractional wave equation
In order to understand the stability estimate for the fractional wave equation, let us recall the famous Caffarelli-Silvestre extension [CS07] for the fractional Laplacian. For each x ′ ∈ R n and x n+1 ∈ R n+1 where a n,s : 4.1. Logarithmic stability of the Caffarelli-Silvestre extension. We now defineΩ We now prove a lemma, which concerns the propagation of smallness for the Caffarelli-Silvestre extension. By using similar ideas as in [RS20, Section 5], we can derive the following boundary logarithmic stability estimate.
Suppose that there exist constants C 1 > 1 and E > 0 such that η(t) ≤ E and for some constants C > 1 and µ > 0, both depending only on n, s, C 1 , Ω, W . Moreover, given any γ > 0, we have for some constants C > 1 and µ > 0, both depending only on n, s, C 1 , Ω, W , as well as γ.
Replacing [RS20, (5.67) of Theorem 5.1] by the following inequality: we can prove (4.4) using the similar argument as in the proof of [RS20, (5.5), Theorem 5.1], with a slight modification as indicated above.
for some constant C 0 > 0 independent of v F and F . Choosing E = C 0 F L 2 (ΩT ) , the condition (4.12) satisfies, and then we can employ Corollary 4.
Lemma 4.4. Suppose that q ∈ L ∞ (Ω). Let P q be the Poisson operator given in (2.5). Then (4.21) P q − Id : L 2 (0, T ; H 2s W ) → L 2 (Ω T ) is a compact injective linear operator. Moreover, for each F ∈ L 2 (Ω T ), the adjoint operator of P q − Id is given by where v F is the solution of (4.13).
Proof of Lemma 4.4. It is worth pointing out that the function (P q − Id)f is equal to P q f | ΩT , for any f ∈ L 2 (0, T ; H 2s W ). We split the proof into several steps.
Therefore, we see that the operator (4.21) is compact.
Suppose that f ∈ ker(P q − Id), then P q f = 0 in Ω T . From the definition of the Poisson operator, u = P q f satisfies (4.23) ∂ 2 t + (−∆) s + q u = 0 in Ω T .
Since u = 0 in Ω T , from (4.23), we have that (−∆) s u = 0 in Ω T . Therefore, using the strong uniqueness for the fractional Laplacian again, we know that u ≡ 0 throughout R n , and hence f ≡ 0, which concludes that P q is injective.
Step 3. Computing the adjoint operator.

4.4.
Proof of Theorem 1.2. Finally, we can prove our logarithmic stability estimate of the inverse problem for the fractional wave equation.
Recalling the definition of the function space Z −s (Ω, T ) in Definition 1.2, we finally prove the assertion.

Exponential instability of the inverse problem
In the last section of this paper, we demonstrate that the logarithmic stability in Theorem 1.2 is optimal, by showing the exponential instability phenomenon for the fractional wave equation. The ideas of the construction of the instability are motivated by Mandache's pioneer work [Man01].

Matrix representation via an orthonormal basis.
For r > 0, let B r be the ball of radius r > 0 with center at 0. First of all, we introduce a set of basis of L 2 (B 3 \ B 2 ). The following proposition can be found in [RS18, Lemma 2.1 and Remark 2.2]: Proposition 5.1. Let n ≥ 2. Given any m ≥ 0, we define There exists an orthonormal basis {Y mkℓ : for some constant C ′ n,s and C ′′ n,s (both depending only on n and s), whereỸ mkℓ ∈ H s (R n ) is the unique solution to Remark 5.2. For n = 1, the "sphere" ∂B 1 ⊂ R consists only two end points {−1, 1}, which is no longer a sphere. Therefore, we need to find another basis for the one-dimensional case. We shall discuss the case of n = 1 later.

Given any bounded linear operator
Let a m2k2ℓ2 m1k1ℓ1 be the tensor with entries a m2k2ℓ2 m1k1ℓ1 , and consider the following Banach space: see e.g. [RS18, Definition 2.7]. The following lemma can be found in [RS18,(20)], which plays an essential role in our work.
Lemma 5.3. If n ≥ 2, then Thanks to Lemma 5.3, we can regard the tensor a m2k2ℓ2 m1k1ℓ1 (q) as the matrix representation of the bounded linear operator A.

Special weak solutions.
In view of Proposition 5.1, we need to introduce some special solutions. We begin with the following lemma.
Based on Lemma 5.4, we can define the DN map where u is given in (5.2). In view of (5.3), we know that is a bounded operator. However, the regularity given in (5.11) is insufficient for our purpose. In the following lemma, we improve (5.11) by modifying the ideas in [RS18, Remark 2.5].
Proof. We first note that it suffices to take C ′ R,T,n,s ≥ 1 and c ′ n,s > 0 described in Lemma 5.7.
Step 2. Construction of sets.
Our goal is to construct an appropriate b m2k2ℓ2 m1k1ℓ1 (t) ∈ Y (t) that is an approximation of Γ m2k2ℓ2 m1k1ℓ1 (q)(t) .
Step 4. Calculating the cardinality of Y (t).
5.6. Proof of Theorem 1.3. With Lemma 5.12 and Lemma 5.12 at hand, we can prove the exponential instability of the inverse problem for the fractional wave equation.
Proof of Theorem 1.3. Let µ and c R,T,n,s be the constants given in Lemma 5.12 and in (5.24), respectively. For each 0 < ǫ < min {c R,T,n,s , R, µβ}, we can construct an ǫ-discrete set Z as described in Lemma 5.12. Let δ be the constant chosen in (5.25). Next, for each t ∈ (0, T ), we construct a δ-net Y (t) as in Lemma 5.9 and (5.26) holds. Clearly, Y (t) is also a δ-net for Γ m2k2ℓ2 m1k1ℓ1 (Z)(t), · X . We now choose β = β(R, n, α) sufficiently large (which is independent of ǫ) such that µβ ≥ R and Therefore, by the pigeonhole principle, for each t ∈ (0, T ), there exist y m2k2ℓ2 In view of Lemma 5.3, we have sup t∈(0,T ) The arbitrariness of 0 ≡ χ ∈ C ∞ c ((0, T )) leads to the estimate (1.6), while the estimate (1.5) immediately follows form the definition of Z. Moreover, since ǫ < R, q i L ∞ ≤ R, for i = 1, 2. The proof is now completed.
We next prove the exponential instability in the case of n = 1, Theorem 1.4. The proof of Theorem 1.4 is very similar to that of Theorem 1.3. The main difference is that when n = 1, the boundary ∂B 1 of the interval B 1 = (−1, 1) consists only two points {−1, 1}. Therefore, we need to modify the proof of Proposition 5.1.
We first construct an orthonormal basis {Y k } of L 2 ((2, 3)) such that the solutioñ Y k of (5.27) for all x ∈ (−1, 1). If we choose Y k = e 2πik(x−2) to be the usual orthonormal basis of L 2 ((2, 3)), it will be difficult to obtain an exponential decay bound forỸ k . Therefore, we would like to find another orthonormal basis for L 2 ((2, 3)) to meet our goal.
Proof. In view of (5.28), we want to find real-valued Y k of the form Plugging the ansatz (5.29) into (5.28), we obtain that for x ∈ (−1, 1) ((2, 3)). In order to derive the desired decaying properties, for each k ≥ 1, we will choose g k such that (5.31a) 3 2 r −j g k (r) dr = 0 for all 0 ≤ j ≤ k − 1.

We observe that
for all k > j. Combining this estimate with (5.31a), we have Plugging (5.33) into (5.30), we obtain that which is our desired result.
Similar to Lemma 5.3, we can prove the following lemma.
Similar to preceding discussions, let us consider the one spatial dimensional case.
• Special weak solutions.
• Construction of a family of δ-net.
Proof of Theorem 1.4. Finally, we can prove Theorem 1.4 by following the lines in the proof of Theorem 1.3.

Appendix A. Proofs related to the forward problem
In the end of this work, we prove Theorem 2.1 in details for the self-containedness. The proof of Theorem 2.1 is similar to the proof of the case s = 1, i.e., the wellposedness of the classical wave equation. The main difference is that the estimates and results hold in the fractional Sobolev space. Here we will utilize similar ideas shown in [Eva98, Chapter 7]: The Galerkin approximation.
We now set up the Galerkin approximation for the fractional wave equation. To this end, let us consider an eigenbasis {w k } k∈N associated with the Dirichlet fractional Laplacian in a bounded domain Ω, that is, in Ω e .
Moreover, we can normalize these eigenfunctions so that Proof. Due to the orthonormality property (A.2) of {w k } k∈N ⊂ L 2 (Ω), we have In addition, we have where e kℓ := B[w ℓ , w k ] for k, ℓ = 1, . . . , m. Let us write F k (t) := ( F (t), w k ) for k = 1, . . . , m. As a result, by using (A.5) becomes the linear system of ordinary differential equation (ODE) Our next goal is to take m → ∞, whenever we have a suitable energy estimate, uniform in m ∈ N.
Lemma A.2 (Energy estimate). Under the assumptions of Theorem 2.1, there exists a constant C > 0, independent of m ∈ N, such that for all m ∈ N.
Proof. We divide the proof into several steps.
Multiplying the equation (A.5) by d k m ′ (t), and summing over k = 1, . . . , m, with the condition (A.2) at hand, we have for a.e. 0 ≤ t ≤ T . Note that the first term of (A.8) can be written as On the other hand, we can express for some constant C > 0.
This proves the assertion. Now, we are ready to prove Theorem 2.1.
Proof of Theorem 2.1. Our goal is to pass the limits in the previous Galerkin approximations.
Step 1. Existence of weak solution.
Passing the limit m → ∞, the estimate (2.3) follows directly from above.