Approximate Cloaking for The Heat Equation via Transformation Optics

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $\epsilon$ in three dimensions and $|\ln\epsilon|^{-1}$ in two dimensions, where $\epsilon$ is the regularization parameter.


Introduction and statement of the results
Cloaking using transformation optics (changes of variables) was introduced by Pendry, Schurig, and Smith [28] for the Maxwell system and by Leonhardt [14] in the geometric optics setting. These authors used a singular change of variables, which blows up a point into a cloaked region. The same transformation had been used to establish (singular) non-uniqueness in Calderon's problem in [9]. To avoid using the singular structure, various regularized schemes have been proposed. One of them was suggested by Kohn, Shen, Vogelius, and Weinstein [10], where instead of a point, a small ball of radius ε is blown up to the cloaked region. Approximate cloaking for acoustic waves has been studied in the quasistatic regime [10,24], the time harmonic regime [11,17,25,18], and the time regime [26,27], and approximate cloaking for electromagnetic waves has been studied in the time harmonic regime [4,13,22], see also the references therein. Finite energy solutions for the singular scheme have been studied extensively [8,30,31]. There are also other ways to achieve cloaking effects, such as the use of plasmonic coating [2], active exterior sources [29], complementary media [12,20], or via localized resonance [21] (see also [15,19]).
The goal of this paper is to investigate approximate cloaking for the the heat equation using transformation optics. Thermal cloaking via transformation optics was initiated by Guenneau, Amra, and Venante [7]. Craster, Guenneau, Hutridurga, and Pavliotis [6] investigate the approximate cloaking for the heat equation using the approximate scheme in the spirit of [10]. They show that for the time large enough, the largeness depends on ε, the degree of visibility is of the order ε d (d = 2, 3) for sources that are independent of time. Their analysis is first based on the fact that as time goes to infinity, the solutions converge to the stationary states and then uses known results on approximate cloaking in the quasistatic regime [10,24].
In this paper, we show that approximate cloaking is achieved at any positive time and established the degree of invisibility of order ε in three dimensions and | ln ε| −1 in two dimensions. Our results hold for a general source that depends on both time and space variables, and our estimates are independent of the content of the materials inside the cloaked region. The degree of visibility obtained herein is optimal due to the fact that a finite time interval is considered (compare with [6]). We next describe the problem in more detail and state the main result. Our starting point is the regularization scheme [10] in which a transformation blows up a small ball B ε (0 < ε < 1/2) instead of a point into the cloaked region B 1 in R d (d = 2, 3). Here and in what follows, for r > 0, B r denotes the ball centered at the origin and of radius r in R d . Our assumption on the geometry of the cloaked region is mainly to simplify the notations. Concerning the transformation, we consider the map F ε : In what follows, we use the standard notations for the "pushforward" of a symmetric, matrix-valued function A, and a scalar function ρ, by the diffeomorphism F , and I denotes the identity matrix. The cloaking device in the region B 2 \ B 1 constructed from the transformation technique is given by a pair of a matrix-valued function and a function that characterize the material properties in B 2 \ B 1 . Physically, A is the thermal diffusivity and ρ is the mass density of the material.
Let Ω with B 2 ⊂⊂ Ω ⊂ R d (d = 2, 3) 1 be a bounded region for which the heat flow is considered. Suppose that the medium outside B 2 (the cloaking device and the cloaked region) is homogeneous so that it is characterized by the pair (I, 1), and the cloaked region is characterized by a pair (a O , ρ O ) where a O is a matrix-valued function and ρ O is a real function, both defined in B 1 . The medium in the whole space is then given by In what follows, we make the usual assumption that a O is symmetric and uniformly elliptic, i.e., for a.e. x ∈ B 1/2 , for some Λ ≥ 1, and σ is a positive function bounded above and below by positive constants.
Given a function f ∈ L 1 (0, +∞), L 2 (Ω) and an initial condition u 0 ∈ L 2 (Ω), in the medium characterzied by (A c , ρ c ), one obtains a unique weak solution u c ∈ L 2 (0, ∞); H 1 (Ω) ∩C [0, +∞), L 2 (Ω) of the equation and in the homogeneneous medium characterized by (I, 1), one gets a unique weak solution u ∈ L 2 (0, ∞); The approximate cloaking meaning of the scheme (1.12) is given in the following result: Assume that u c and u are the solution of (1.6) and (1.7) respectively. Then, for As a consequence of Theorem One therefore cannot detect the difference between (A c , ρ c ) and (I, 1) as ε → 0 by observation of u c outside B 2 : cloaking is achieved for observers outside B 2 in the limit as ε → 0.
We now briefly describe the idea of the proof. The starting point of the analysis is the invariance of the heat equations under a change of variables which we now state.
a bounded open subset of R d of class C 1 , and let A be an elliptic symmetric matrix-valued function, and ρ be a bounded, measurable function defined on Ω bounded above and below by positive constants. Let F : Ω → Ω be bijective such that F and F −1 are Lipschitz, det ∇F > c for a.e. x ∈ Ω for some c > 0, and F (x) = x near ∂Ω. Let f ∈ L 1 (0, T ); L 2 (Ω) and u 0 ∈ L 2 (Ω). Then u ∈ L 2 (0, T ); Recall that F * is defined in (1.2). In this paper, we use the following standard definition of weak solutions: (1.10) in the distributional sense for all ϕ ∈ H 1 0 (Ω).
We now return to the idea of the proof of Theorem 1.1. Set Then u ε is the unique solution of the system Moreover, In comparing the coefficients of the systems verified by u and u ε , the analysis can be derived from the study of the effect of a small inclusion B ε . The case in which finite isotropic materials contain inside the small inclusion was investigated in [3] (see also [5] for a related context). The analysis in [3] partly involved the polarization tensor information and took the advantage of the fact that the coefficients inside the small inclusion are finite. In the cloaking context, Craster et al. [6] derived an estimate of the order ε d for a time larger than a threshold one. Their analysis is based on long time behavior of solutions to parabolic equations and estimates for the degree of visibility of the conducting problem, see [10,24], hence the threshold time goes to infinity as ε → 0.
In this paper, to overcome the blow up of the coefficients inside the small inclusion and to achieve the cloaking effect at any positive time, we follow the approach of Nguyen and Vogelius in [26]. The idea is to derive appropriate estimates for the effect of small inclusions in the time domain from the ones in the frequency domain using the Fourier transform with respect to time. Due to the dissipative nature of the heat equation, the problem in the frequency for the heat equation is more stable than the one corresponding to the acoustic waves, see, e.g., [25,26], and the analysis is somehow easier to handle in the high frequency regime. After using a standard blow-up argument, a technical point in the analysis is to obtain an estimate for the solutions of the equation ∆v + iωε 2 v = 0 in R d \ B 1 (ω > 0) at the distance 1/ε in which the dependence on ε and ω are explicit (see Lemma 2.2). Due to the blow up of the fundamental solution in two dimensions, the analysis requires some new ideas. We emphasize that even though our setting is in a bounded domain with zero Dirichlet boundary condition, we employs Fourier transform in time instead of eigenmodes decomposition as in [6]. This has the advantage that one can put both systems of u ε and u in the same context.

Proof of the main result
To implement the analysis in the frequency domain, let us introduce the Fourier transform with respect to time t: for ϕ ∈ L 2 ((−∞, +∞), L 2 (R d )). Extending u, u c , u ρ , and f by 0 for t < 0, and considering the Fourier with respect to time at the frequency ω > 0, we obtain The main ingredient in the proof of Theorem 1.1 is the following: Proposition 2.1. Let ω > 0, 0 < ε < 1/2, and let g ∈ L 2 (Ω) with supp g ⊂ Ω \ B 2 . Assume that v, v ε ∈ H 1 (Ω) are respectively the unique solution of the systems v ε = 0 on ∂Ω. We have, for 4ε < r < 2, for some positive constant C = C r independent of ε, ω, and g. Here The first result is the following simple one: We have, for R > 2, for some positive constants C R independent of k and v.
Proof. Multiplying the equation byv (the conjugate of v) and integrating by parts, we have This implies Here and in what follows, C denotes a positive constant independent of v and k. By the trace theory, we have Combining (2.6) and (2.7) yields The conclusion follows when k ≥ 1.
The conclusion now follows from (2.20).
Remark 2.1. The estimate in Proposition 2.1 is independent of the coefficients inside B ε and is optimal. In fact, one can choose the coefficients in B ε such that v ε on ∂B ε is as small as one wants.
The conclusion follows.