Integrated Local Energy Decay for Damped Magnetic Wave Equations on Stationary Space-Times

In this work, we will establish (integrated) local energy decay for a broad class of damped wave equations on stationary, asymptotically flat space-times, including those which possess magnetic and scalar potential terms. This generalizes the result of [Kof22], which lacked potential terms and assumed that the damping was compactly-supported. The presence of the potentials allow for interaction with the damping and the existence of both complex eigenvalues and real resonances embedded in the continuous spectrum, all of which stand to inhibit local energy decay. We will discuss each of these obstructions carefully. Along the way, we also establish an energy dichotomy when non-zero real resonances are permissible.

To set up the problem which we will study, let $(\mathbb{R}^{4},g)$ be a Lorentzian manifold with coordinates $(t,x)\in\mathbb{R}\times\mathbb{R}^{3}$ and metric signature $(-+++).$ We will consider damped wave operators of the form

where $a,V,$ and the components of $A$ are smooth and complex-valued; here, we are utilizing the notation $D_{\alpha}=-i\partial_{\alpha}$, which should be interpreted as an operator. We will call $A$ the magnetic potential, $V$ the scalar potential, and $a$ the damping. If $a\equiv 0$, then we will simply refer to $P$ as a wave operator. We will require that $P$ is asymptotically flat. More precisely, we first define the family of norms

where $A_{j}=\{\left\langle x\right\rangle\approx 2^{j}\}$ for $j\geq 0$ denote inhomogeneous dyadic annuli, $\partial=(\partial_{t},\nabla_{x})$ denotes the space-time gradient, and $\left\langle x\right\rangle=(1+|x|^{2})^{1/2}$ denotes the Japanese bracket of $x$. The notation $A\lesssim B$ indicates that $A\leq CB$ for some $C>0,$ and the notation $A\approx B$ means that $B\lesssim A\lesssim B$. In the definition of the $A_{j}$’s, we require that these implicit constants are compatible to cover $\mathbb{R}^{3}.$ We may now define the $AF$ norm topology via

The latter conditions are placed to work in standard symbol classes, as opposed to those with limited regularity. They extend the flavor of the estimate present in the definition of asymptotic flatness, but one does not require summability over the differentiation indices.

We will primarily be interested in when $P$ is stationary, which occurs when $(g,A,a,V)$ are independent of $t$. In particular, $\partial_{t}$ is a Killing field for $g$ when $P$ is stationary. When $P$ is a damped wave operator, we will always assume that $a$ is independent of $t$, $a$ is non-negative for $|x|\leq 2R_{0}$, and $a$ is positive on an open subset of $\{|x|\leq R_{0}\}$. For $|x|\leq 2R_{0}$, the damping term in $P$ behaves as typical viscous damping, and we consider it as a general time-independent first-order perturbation term for $|x|>2R_{0}$. Further, we will assume throughout $\partial_{t}$, while $dtt$ the constant time slices are uniformly space-like (i.e. $dt$ is uniformly time-like). The former condition implies an ellipticity condition on the terms in $P$ which are independent of $D_{t}$ (see Section 1.2 for more), while the latter guarantees that $g^{00}\gtrsim-1$, allowing $P$ to be reduced to a normal form of $g^{00}=-1$.

Next, we introduce parameters that quantify various aspects of asymptotic flatness. Namely, we instantiate

• •

  the parameters $M_{0}$, $R_{0}$ and c, which are such that

  $\displaystyle\left\lVert(g-m,A,a,V)\right\rVert_{AF}$

  $\displaystyle\leq M_{0},\qquad\qquad\left\lVert(g-m,A,a,V)\right\rVert_{AF_{>R_{0}}}\leq\textbf{c}\ll 1,$

  where the subscript denotes the restriction of the norm to $\{|x|>R_{0}\}.$ The parameter c should be viewed as being fixed first, after which we find a suitable $R_{0}$ for which the above holds.

• •

  the sequence $(c_{j})_{j\geq\log_{2}R_{0}}$ of positive real numbers satisfying that

  $\displaystyle\left\lVert(g-m,A,a,V)\right\rVert_{AF(A_{j})}\lesssim c_{j},\qquad\qquad\sum\limits_{j}c_{j}\lesssim\textbf{c},$

  where $\left\lVert\cdot\right\rVert_{AF(A_{j})}$ denotes the restriction of the $AF$ norm to the dyadic region $A_{j}.$ We may assume without any loss of generality that the sequence is slowly-varying, i.e.

  $$c_{j}/c_{k}\leq 2^{\delta|k-j|},\qquad\delta\ll 1.$$

  The integers $0\leq j<\log_{2}R_{0}$ index finitely many dyadic regions. Since the union of such regions is compact, we can extend $(c_{j})$ to such indices in an arbitrary manner, although we will assume that the appended terms in the sequence are positive and allow $(c_{j})$ to remain slowly-varying.

Next, we introduce definitions regarding the skewness of $P-iaD_{t}$ and the size of the damping relative to the magnetic potential. We will generically refer to these as symmetry conditions.

We note that $\operatorname{Im}A$ is already guaranteed to be small for $|x|>R_{0}$ by asymptotic flatness, so the $\varepsilon$-weakly magnetic condition is more so a condition for $|x|\leq R_{0}$. This is a weakening of the more standard $\varepsilon$-almost symmetry condition, which will arise briefly in Section 2.3. The $\varepsilon$-damping dominant conditions are, to our knowledge, more original and seemingly more natural; we will only discuss the strong version for now since it is simpler to parse and easier to verify (as well as stronger). It stipulates that the damping must be the dominant skew-adjoint term (at the principal level) for $|x|\leq 2R_{0}.$ Thus, the magnetic potential is permitted to have large imaginary components for $|x|\leq 2R_{0},$ so long as the damping is more significant, in the sense stated above. Moreover, the $\varepsilon$ in the strongly $\varepsilon$-damping dominant condition must satisfy that $\varepsilon<(2\left\lVert g-m\right\rVert_{L^{\infty}}+2)^{-1}$ for our results to hold in this work (in which case the weakly $\varepsilon$-damping dominant condition holds with a different $\varepsilon$, as discussed in the previous remark). This indicates that if the metric perturbation is sufficiently large within $\{|x|\leq R_{0}\}$, then $|\operatorname{Im}A|$ must be sufficiently smaller than $a$. If the perturbation is small, then $|\operatorname{Im}A|$ and $a$ are permitted to be close for $|x|\leq 2R_{0}$.

Finally, we remark that neither the $\varepsilon$-weakly magnetic nor the strongly $\varepsilon$-damping dominant condition is stronger than the other, although working with the conditions is somewhat similar in the high frequency analysis; the same is true for the $\varepsilon$-weakly magnetic condition and the weakly $\varepsilon$-damping dominant condition, for similar reasons. One should view the damping condition as allowing $|\operatorname{Im}A|$ to be potentially quite large for $|x|\leq 2R_{0}$, so long as $a$ is sufficiently larger (although, of course, both are inherently limited by the $M_{0}$ parameter, as well). However, let us say that $a$ is zero outside of a ball of radius $\delta\ll 1$ and is small within the ball. Then, the strongly $\varepsilon$-damping dominant condition would require that $\operatorname{Im}A$ be zero for $\delta\leq|x|\leq 2R_{0}$ and small everywhere. The $\varepsilon$-weakly magnetic condition allows for $|\operatorname{Im}A_{\alpha}|>0$ for $|x|\leq 2R_{0},$ so long as we control the $AF$ size uniformly.

The study of localized energy estimates dates back to the work of [Mora66, Mora68, Mora75], [MRS77] on Minkowski space-time. Local energy estimates constitute a powerful measure of dispersion, implying Strichartz estimates ([BT07, BT08], [JSS90, JSS91], [MMT08], [MMTT10], [MT09, MT12], [Tat08], [Toh12]) and pointwise decay estimates (e.g. [MTT12], [Tat13], [Morg20], [MW21], [Looi21], [Hi23]). The latter has been used to tackle various generalizations of Price’s law which, in its simplest form, conjectured a $t^{-3}$ pointwise decay rate of waves on non-rotating black hole space-times; see the listed references on pointwise decay and [Hi22] (which settles the conjecture affirmatively in full generality) for more. The particular local energy estimate of interest in this work, integrated local energy decay, is a powerful quantitative statement given in the form of an inequality which may be qualitatively interpreted as expressing that the energy of a wave must disperse quickly enough to be time-integrable within any compact region of space.

In order to define the relevant energy inequalities explicitly, we will first define the local energy norms

The predual norm to the $LE$ norm is the $LE^{*}$ norm, which is defined as

If we wish for the time interval to be e.g. $[0,T]$ in the above norms, then we will use the notation e.g. $\left\lVert u\right\rVert_{LE[0,T]}$. A subscript of $c$ on any of these spaces denotes compact spatial support, whereas a subscript of $0$ denotes the closure of $C_{c}^{\infty}$ in the relevant norm.

Analogous to the local energy spaces, we define the spatial local energy spaces $\mathcal{LE},\mathcal{LE}^{1},\mathcal{LE}^{*}$ when the time variable is fixed (and there is no time derivative nor integral involved in the norms, either). We will also require spaces which allow us to track dependence on a spectral parameter $\omega$, namely

These spaces are equipped with the norms

respectively. They will become relevant in this work when we introduce the resolvent formalism in Section 3. Now, we will define local energy decay.

The next estimate is a weaker variant of local energy decay which does not see complex eigenvalues nor non-zero real resonances (to be defined in Section 3).

The last estimate is a standard uniform energy bound, which embodies the behavior that one observes when the energy of the system is non-increasing.

Notice that the uniform energy bounds provide a link between the two-point local energy decay estimate and the local energy decay estimate. In order to discuss the uniform estimate further, let $Pu=f$, and assume that $P$ is stationary, asymptotically flat, and of symmetric wave type. Consider the sesquilinear map $\boldsymbol{E}$ on the energy space $\mathcal{E}:=\dot{H}^{1}\times L^{2}$ defined by

This induces a quadratic energy functional, i.e. an energy form

When $\partial_{t}$ is a uniformly time-like vector field (which we will assume throughout), $P_{0}$ is guaranteed to be elliptic (in the principal sense). Direct integration by parts gives that

If $a$ is real-valued and non-negative, then it follows that

If $A,V\equiv 0$, then $P_{0}$ is uniformly elliptic (in the sense of the full symbol), which implies that the energy is coercive, i.e.

Energy coercivity applied to the above now provides the uniform energy bound (1.3).

When $A$ and $V$ are non-zero, one is not guaranteed energy coercivity (and thus not guaranteed uniform energy bounds), even if $a$ is non-negative. However, one does get an almost-coercive energy statement from the ellipticity of $P_{0}$ and the asymptotic flatness assumption:

When $P$ is not of symmetric wave type, then one redefines the energy by replacing $P_{0}$ with its symmetric part (see Section 2.3); the estimate (1.4) still holds for the energy associated to $P_{0}$ by ellipticity regardless of the symmetry. The uniform bound (1.3) still need not hold when $P_{0}$ is replaced by its symmetric part. Thus, it is not straightforward to transition from (1.2) to (1.1) when $A,V\not\equiv 0$, even for well-signed dampings; this is true even in the symmetric wave type case.

In [MST20], the authors proved (amongst other results) that local energy decay for stationary $AF$ wave operators is equivalent to an absence of geodesic trapping, negative eigenfunctions, and real resonances. We will describe trapping in Section 2.1 and the spectral objects in Section 3. In short, trapping occurs when there exist bicharacteristic rays which live within a compact set for all time. The spectral obstructions correspond to singular behavior of the resolvent - negative eigenfunctions live in $L^{2}$ and have corresponding eigenvalue in the lower half-plane, whereas real resonances lie on the real line and have a corresponding resonant state which lives in a local energy space (one must distinguish between zero and non-zero resonances). In [MST20], the authors also did not necessarily possess a coercive energy; they proved local energy decay by establishing (1.2) using local energy estimates in different frequency regimes, then they utilized resolvent estimates.

The work [BR14] utilized dissipative Mourre commutator methods to establish that if the space-time is stationary and asymptotically Euclidean (i.e. $(\mathbb{R}^{4},g)$ is a product manifold and hence possesses no non-trivial metric cross terms $dt\otimes dx^{j}$), then one has local energy decay for $AF$ stationary damped wave operators with $A,V\equiv 0$ and $a$ being a non-negative short-range potential, provided that $a$ satisfied a dynamical hypothesis called geometric control. This condition requires that all trapped null bicharacteristic rays intersect where $a>0$ (see Definition 2.1 for a precise definition), although the authors of [BR14] only required this for trapped geodesics due to the product manifold structure. Geometric control dates back to [RT74], which utilized it to obtain exponential energy decay (i.e. uniform stabilization) for dissipative problems on compact manifolds. In [Kof22], we generalized the work of [BR14] to the asymptotically flat case (i.e. allowed the metric to possess cross terms). To be precise, we will record this result, which is Theorem 1.9 in [Kof22] (adding in the missing assumption of uniformly space-like time slices).

In [Kof22], uniform energy bounds held as a result of the conditions on the damping and lack of potentials, which made it sufficient to prove the two-point bound in order to establish local energy decay. In order to establish the former estimate, [Kof22] followed the strategy set forth in [MST20]:

1. (1)

   Establish local energy estimates that imply local energy decay for Schwartz functions, whose corresponding function space we denote as $\mathcal{S}$, which are cutoff to high, medium, and low frequency regimes.

2. (2)

   Utilize a time frequency partition of unity to prove the estimate

   $$\left\lVert u\right\rVert_{LE^{1}}\lesssim\left\lVert Pu\right\rVert_{LE^{*}},\qquad u\in\mathcal{S}.$$

3. (3)

   Apply an extension procedure to add back in the energy at times $0$ and $T$.

We will generalize the work of [Kof22] to include the lower-order magnetic and scalar potentials, along with more general damping functions. Our first result is an extension of the high frequency estimate present in [MST20] and [Kof22] to our setting.

This theorem is the one of the primary results whose proof must be adapted from [Kof22] to account for the lower-order terms. It is also where one requires the most substantial deviation from [MST20], since trapping is high frequency. We utilize the symmetry-based assumptions in the theorem so that we may deal with the additional lower-order terms. In particular, this is where the conditions on the interaction between the damping and magnetic potentials come into play. In order to leverage the sign of the damping to mitigate the harmful effects of the trapping, we must limit the magnetic potential appropriately.

The medium and low frequencies are not affected by the damping (which may simply be viewed as a general sub-principal $AF$ term, as opposed to being leveraged like in the high frequency setting) and follow directly from the work in [MST20]. This allows us to establish the two-point local energy estimate under the hypothesis that zero is not a resonance, which is needed in the low frequency regime.

We will define the zero non-resonance condition in Section 2.2 and its relation to zero resonant states in Section 3. If we additionally impose that $P$ is of $\varepsilon$-almost symmetric wave type or that $P$ satisfies the $\varepsilon$-damping dominant as well as the analogous estimate with $A$ replaced by $V$, then we obtain a straightforward energy dichotomy (just as in [MST20], where it is Theorem 2.16) as a consequence of the two-point local energy estimate and a uniform energy relation. The given symmetry conditions fulfill a similar role to a condition on the absence of non-zero embedded resonances, and it is needed to obtain the aforementioned uniform energy relation (i.e. an almost-conserved energy property if $Pu=0$) which will appear in the proof.

At least at a heuristic level, the solutions which exhibit case (1) behavior stem from eigenvalues in the lower half-plane of the corresponding stationary problem (see Section 3), which represent poles of the resolvent. The resolvent has meromorphic continuation to the entire lower half-plane, and the poles must occur within a relatively compact subset of frequencies. In particular, there are only finitely many such eigenvalues, and each generalized eigenspace has finite dimension by Fredholm theory. We also remark that we do not obtain improvements from the corresponding result in [MST20] here, nor do we obtain versions of their non-stationary results since our high frequency work exploits the stationarity. While more refined energy space decompositions are likely (and are also present in [MST20] within the non-damped setting), the proper statements and results in the context of damped waves are not clear to us at this time.

Finally, we have local energy decay. Here, we must further assume that $P$ satisfies various spectral hypotheses, which are defined in Section 3.

In order to prove this result, we cannot necessarily rely on uniform energy bounds to pass from the two-point local energy estimate to local energy decay like in [Kof22] and instead rely on spectral theory as in [MST20]. The structure of proving Theorem 1.14, and hence of the overall paper, is as follows and is motivated by [MST20]:

1. (1)

   Section 2.1: Extend the high frequency analysis from [Kof22] to the present context. One must take care to consider how the damping interacts with the remaining principal skew-adjoint terms; this is the purpose of the $\varepsilon$-weakly magnetic and $\varepsilon$-damping dominant conditions. We will explain the results that we borrow and why they apply here.

2. (2)

   Section 2.2: Provide an overview of the low and medium frequency analysis; these do not require change from [MST20], since the damping is simply treated as a first-order $AF$ perturbation term. From here, Theorem 1.11 follows readily from the work in [MST20], [Kof22]. As an immediate consequence, we will establish Theorem 1.12 in Section 2.3, although this has no bearing on the proof of Theorem 1.14.

3. (3)

   Section 3: Summarize the necessary resolvent theory in [MST20] required to prove Theorem 1.14. This will require Theorem 1.10 and Theorem 1.11. Once one is armed with the relevant frequency estimates, the work in [MST20] applies rather directly. We will summarize and/or provide many (but not all) of their arguments for the required results, especially where we believe that further elucidation would be beneficial for the sake of exposition. We require little deviation from their theory in our present work.

We will recall the relevant framework and results from [Kof22] needed to prove the high frequency estimate. When deviation occurs, we will proceed carefully and explicitly. The need for the $\varepsilon$-damping dominant and $\varepsilon$-weakly magnetic conditions only come up in one place, in the proof of Lemma 2.3. Throughout this section, we will assume that $P$ is stationary. This work is motivated by [Kof22], [MST20], and [BR14].

The high frequency analysis is rooted in the behavior of the bicharacteristic flow generated by the principal symbol of $P$. First, we make a minor simplification. Since the constant time slices are assumed uniformly space-like, it follows that $g^{00}\lesssim-1$. Dividing through by $-g^{00}$ preserves the assumptions on the operator coefficients (see e.g. [MT12]); hence, we may assume that $g^{00}=-1$.

After these modifications, the principal symbol of $P$ is the smooth function

Notice that since $P$ is stationary, $p$ is independent of $t$. This symbol generates a bicharacteristic flow on $\mathbb{R}\times T^{*}\mathbb{R}^{4}$ given by $\varphi_{s}(w)=\left(t_{s}(w),x_{s}(w),\tau_{s}(w),\xi_{s}(w)\right)$ which solves

with initial data $w\in T^{*}\mathbb{R}^{4}.$ Since $g$ is smooth and asymptotically flat, and $\partial_{t}$ is uniformly time-like, we have a unique, smooth, globally-defined flow with smooth dependence on the data. We will have particular interest in null bicharacteristics, i.e. those with initial data lying in the zero set of $p$, denoted $\operatorname{Char}(P)$. We remark that there is no distinguishing between $\pm p$ on $\operatorname{Char}(P)$, hence the minus sign in front of $p$ is somewhat inconsequential on it.

Using the flow $\varphi_{s}$, we define the forward and backward trapped and non-trapped sets with respect to $\varphi_{s}$, respectively, as

The trapped and non-trapped sets are defined as

respectively. The flow is said to be non-trapping if $\Omega^{p}_{tr}{\,=\,}\emptyset$. Otherwise, the flow is said to possess trapping. Now, we may precisely state the geometric control condition.

In order to leverage the sign of the damping, [Kof22] utilized a scaling argument which we will also exploit. Namely, if $u$ solves $Pu=f$, we consider

For this discussion, a tilde over a function will denote dilation by $\gamma$ in each coordinate, as done to define $\tilde{u}$. If we call

then

The benefit of this scaling is that we obtain an arbitrarily large constant in front of the damping function. However, we underscore that such a large constant is also inherited by the magnetic potential.

Analogous Hamiltonian systems and trapped/non-trapped sets exist for the principal symbol $\tilde{p}$ of $\tilde{P}$, which amounts to simply dilating the coordinates of $g$. If we assume that geometric control holds for the flow generated by $p$, then it is proven in [Kof22] that it holds for the flow generated by $\tilde{p}$. Since the proof is straightforward, we will omit it here and only record the result, which is Proposition 2.6 in [Kof22].

Since the lower-order terms $A$ and $V$ are not at the principal level, they do not affect $p$ nor $\tilde{p}$. Henceforth, we will fix a large $\gamma>0$ and study the problem from the scaled perspective while reverting back to our original notation (e.g. no tildes). It is readily seen that it is equivalent to prove Theorem 1.10 for the scaled problem, where we now have a large constant in front of the damping term.

The proof of the version of Theorem 1.10 present in [Kof22] (i.e. $A,V\equiv 0,\ \operatorname{supp}a\subset\{|x|\leq R_{0}\}$) is a positive commutator argument. At the symbolic level, this requires the construction of an escape function and a lower-order correction term. Let $p$ and $s_{skew}$ represent the principal symbols of the self and skew-adjoint parts of $P$, respectively. Namely,

The multiplication by $\gamma$ in $s_{skew}$ will prove advantageous for a bootstrapping argument, which is precisely why we implement the $\gamma$-scaling. However, the imaginary part of $A$ has interaction with the damping and also features multiplication by $\gamma$. The $\varepsilon$-damping dominant and $\varepsilon$-weakly magnetic conditions are applied in order to retain the positivity effects of the damping.

The precise escape function construction is as follows.

Here, $H_{p}$ is the Hamiltonian vector field induced by $p$, and $S^{m}(T^{*}\mathbb{R}^{3})$ denotes the standard Kohn-Nirenberg symbol class of order $m$. To each symbol $b\in S^{m}(T^{*}\mathbb{R}^{3})$, we have the associated Weyl quantization of $b$, denoted $b^{\operatorname{w}}$, which is a pseudodifferential operator of order $m$ defined by the action

We will use $\Psi^{m}(\mathbb{R}^{3})$ to denote the space of pseudodifferential operators on $\mathbb{R}^{3}$ of order $m$, and write