# Remark on the energy channel property for the radial linear wave equation

Charles Collot, Thomas Duyckaerts, Carlos Kenig, Frank Merle

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Charles Collot, Thomas Duyckaerts, Carlos Kenig, Frank Merle. Remark on the energy channel property for the radial linear wave equation. Comptes Rendus. Mathématique, 2025, 363 (13), pp.1457-1460. 10.5802/crmath.803. hal-05528795

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
REMARK ON THE ENERGY CHANNEL PROPERTY FOR THE
RADIAL LINEAR WAVE EQUATION
IN THE MEMORY OF HAİM BREZIS

CHARLES COLLOT<fnref n="1" />, THOMAS DUYCKAERTS<fnref n="2" />, CARLOS KENIG<fnref n="3" />,
AND FRANK MERLE<fnref n="4" />

ABSTRACT. This note concerns an exterior energy bound (channel of energy property) for the linear radial wave equations, which is crucial in the proof of the soliton resolution for the energy-critical wave equations. We give a short and synthetic proof that this property in general odd space dimension follows from the case of space dimension 3, which is elementary. This gives a simple proof of the channel of energy property in all odd space dimensions. We also show by the same proof that the analogous bound in even space dimensions follows from the case of space dimension 4.

# 1. INTRODUCTION AND STATEMENT OF THE RESULT

Let $N \ge 3$, and denote by $\dot{H}_N^1(R)$ and $L_N^2(R)$ the vector space of radial functions in $\{x \in \mathbb{R}^N, |x| > R\}$, with the norms:

$$ \|f\|_{\dot{H}_N^1}^2 = \int_R^\infty |f'(r)|^2 r^{N-1} dr, \quad \|g\|_{L_N^2}^2 = \int_R^\infty |g(r)|^2 r^{N-1} dr. $$

When there is no ambiguity on the dimension, we will drop the index $N$. We will consider the radial wave equation in space dimension $N$:

$$ (1) \qquad \partial_t^2 u - \partial_r^2 u - \frac{N-1}{r} \partial_r u = 0, \quad r > R + |t|, $$

with initial data

$$ (2) \qquad (u, \partial_t u)|_{t=0} = (u_0, u_1) \in \dot{H}^1(R) \times L^2(R). $$

Extending the initial data $(u_0, u_1)$ to an element of $\dot{H}^1 \times L^2(\mathbb{R}^N)$, we see that the solution $u$ of (1), (2) is defined without ambiguity on $\{r > R + |t|\}$. Indeed, by finite speed of propagation the restriction of $u$ to this set is independent of the choice of the extension of $(u_0, u_1)$.

<footnote n="1">¹AGM (UMR 8088), Université de Cergy-Pontoise. Supported by the CY Initiative of Excellence Grant “Investissements d’Avenir” ANR-16-IDEX-0008 via Labex MME-DII, the Grant “Chaire Professeur Junior” ANR-22-CPJ2-0018-01 of the French National Research Agence and the Grant BOURGEONS ANR-23-CE40-0014-01 of the French National Research Agence.</footnote>

<footnote n="2">²LAGA (UMR 7539), Université Sorbonne Paris Nord, and Département de mathématiques et applications (UMR 8553), École Normale Supérieure.</footnote>

<footnote n="3">³University of Chicago. Partially supported by NSF Grants DMS-2153794 and DMS-2052710.</footnote>

<footnote n="4">⁴AGM (UMR 8088), Université de Cergy-Pontoise, and Institut des Hautes Études Scientifiques.</footnote>
We define the following subspace of $\dot{H}_N^1(R)$ (respectively $L_N^2(R)$):

$$P_N(R) = \operatorname{span} \left\{ \frac{1}{r^{N-2\ell}} \mid \ell = 1, \dots, \left\lfloor \frac{N+1}{4} \right\rfloor \right\}.$$

$$Q_N(R) = \operatorname{span} \left\{ \frac{1}{r^{N-2\ell}} \mid \ell = 1, \dots, \left\lfloor \frac{N-1}{4} \right\rfloor \right\},$$

and note that $P_N(R)$ (respectively $Q_N(R)$) is exactly the intersection of the generalized kernel of $\Delta$ with $\dot{H}_N^1(R)$ (respectively $L_N^2(R)$).

**Theorem 1.** There exists a constant $C > 0$, independent of $N$, with the following property. Assume that $N \ge 3$ is an integer, and $R > 0$. Let $u$ be a solution of (1), (2).

* Assume that $N$ is odd, and that $(u_0, u_1)$ is in the orthogonal of $P_N(R) \times Q_N(R)$ in $\dot{H}^1(R) \times L^2(R)$. Then:

$$(3) \quad \sum_{\pm} \lim_{t \to \pm\infty} \int_{R+|t|}^{\infty} (\partial_{t,r} u(t,r))^2 r^{N-1} dr \ge \|(u_0, u_1)\|_{\dot{H}^1(R) \times L^2(R)}^2.$$

* Assume that $N = 4k$, $k \in \mathbb{N}$, and that $u_0$ is orthogonal to $P_N(R)$ in $\dot{H}^1(R)$. Then

$$(4) \quad \sum_{\pm} \lim_{t \to \pm\infty} \int_{R+|t|}^{\infty} (\partial_{t,r} u(t,r))^2 r^{N-1} dr \ge \frac{1}{C} \|u_0\|_{\dot{H}^1(R)}^2.$$

* Assume that $N = 4k + 2$, $k \in \mathbb{N} \setminus \{0\}$, and that $u_1$ is orthogonal to $Q_N(R)$ in $L^2$. Then

$$(5) \quad \sum_{\pm} \lim_{t \to \pm\infty} \int_{R+|t|}^{\infty} (\partial_{t,r} u(t,r))^2 r^{N-1} dr \ge \frac{1}{C} \|u_1\|_{L^2(R)}^2.$$

The estimates of Theorem 1 are crucial for the proof of the soliton resolution for the energy critical semi-linear wave equation: see [7], [8], [5] and [1].

Theorem 1 was proved in space dimension 3 in [6] and generalized to higher odd space dimensions in [9]. The difficulty of the proof in [9] increases with the dimensions.

Channels of energy in even space dimensions were first studied in [2], where the limit of the outer energy (the left-hand side of (4) and (5)) is computed in the case $R = 0$. Theorem 1 was proved for $N = 4$ and $N = 6$ in [5], using the formula in [2] and explicit computation. The general case of Theorem 1 in even space dimensions was obtained in [10]. The proof in [10] relies on explicit formulas for the solution of the wave equation whose complexity increases with the dimension. We refer to [4, 3] for other recent works on channels of energy.

In this note, we remark, using a short argument, that it is sufficient to prove Theorem 1 in space dimensions 3 and 4. More precisely, the estimate (3) in all odd space dimensions follows from the same estimate in space dimension 3, and the estimate (4) and (5) in all even space dimension follows from the estimate (4) in space dimension 4.

Theorem 1 in the case $N = 3$ (proved in [6]) is elementary: it follows immediately from the explicit formula for radial solutions of (1), (2) for $N = 3$:

$$(6) \quad u(t,r) = \frac{1}{r}(\varphi(r+t) - \varphi(t-r)),$$
where

$$ (7) \qquad \varphi(\eta) = \frac{1}{2}\eta u_0(|\eta|) + \frac{1}{2} \int_0^\eta \sigma u_1(|\sigma|) d\sigma. $$

We thus have a very simple proof of the odd dimension channels of energy estimate (3) in every odd dimension.

The fourth author is particularly pleased to honour Haïm with a short and elegant proof, which was one of the pleasures he has cherished in mathematics.

## 2. PROOF

The proof, by induction, is based on a transformation of the radial linear wave equation, that was introduced in [9], and also used in [5] to deduce the case $N = 6$ from the case $N = 4$. More precisely, we will introduce a transformation that maps a solution $u$ of (1) to a solution $v$ of:

$$ (8) \qquad \partial_t^2 v - \partial_r^2 v - \frac{N-3}{r} \partial_r v = 0, \quad r > R + |t|. $$

We prove Theorem 1 when $N$ is even. The proof when $N$ is odd is quasi identical.

We thus fix $N \ge 6$ even and assume that the conclusion of the theorem holds with $N-2$ instead of $N$. We consider a solution $u$ of (1). We let

$$ (9) \qquad v(t,r) = \int_r^\infty \rho \partial_t u(t, \rho) d\rho, \quad v_0(r) = v(0, r), \quad v_1(r) = \partial_t v(0, r). $$

Then by (1),

$$ (10) \qquad \partial_r v = -r \partial_t u, \quad \partial_t v = -(N-2)u(t, r) - r \partial_r u(t, r), $$

so that $v$ satisfies (8). Also,

$$ (11) \qquad \int_{R+|t|}^{\infty} (\partial_r v)^2 r^{N-3} dr = \int_{R+|t|}^{\infty} (\partial_t u)^2 r^{N-1} dr $$

and (by a straightforward integration by parts)

$$ (12) \qquad \int_{R+|t|}^{\infty} (\partial_t v)^2 r^{N-3} dr = \int_{R+|t|}^{\infty} (\partial_r u)^2 r^{N-1} dr - (N-2)(R+|t|)^{N-2} u^2(t, R+|t|). $$

and thus

$$ (13) \qquad \lim_{t \to \pm\infty} \int_{R+|t|}^{\infty} (\partial_t v)^2 r^{N-3} dr \le \lim_{t \to \pm\infty} \int_{R+|t|}^{\infty} (\partial_r u)^2 r^{N-1} dr $$

(indeed it is easy to show the equality between these two limits but the inequality (13) is sufficient for our purpose).

**Case 1.** We assume $N = 4k$, $k \in \mathbb{N}$, $u_1 = 0$ and $u_0 \in P_N(R)$. By (10), $v_0 = 0$. We claim

$$ (14) \qquad v_1 \in Q_{N-2}(R), $$

i.e. that $v_1$ is orthogonal to $1/r^{2j}$, $j \in \{k, \dots, 2k-2\}$ in $L_{N-2}^2(R)$. For such $j$, we compute, using (10),

$$ \int_R^\infty v_1(r) \frac{1}{r^{2j}} r^{N-3} dr = - \int_R^\infty ((N-2)u_0(r) + r \partial_r u_0(r)) \frac{1}{r^{2j}} r^{N-3} dr. $$
We have

$$ \int_R^\infty r \frac{du_0}{dr} \frac{1}{r^{2j}} r^{N-3} dr = \int_R^\infty \frac{du_0}{dr} \frac{1}{r^{2j+1}} r^{N-1} dr = 0, $$

since $u_0 \in P_N(R)$. On the other hand, integrating by parts, we obtain

$$ \int_R^\infty u_0(r) r^{N-3-2j} dr = - \frac{1}{N-2-2j} \int_R^\infty \frac{du_0}{dr} \frac{1}{r^{1+2j}} r^{N-1} dr - \frac{1}{N-2-2j} R^{N-2-2j} u_0(R). $$

Furthermore

$$ u_0(R) = - \int_R^\infty \frac{du_0}{dr} dr = - \int_R^\infty \frac{du_0}{dr} \frac{1}{r^{4k-1}} r^{N-1} dr = 0, $$

where the last equality follows again from the fact that $u_0 \in P_N(R)$, and thus that $u_0$ is orthogonal to $1/r^{4k-2}$ in $\dot{H}_N^1(R)$. This concludes the proof of (14).

By (13), (14) and the induction hypothesis,

$$ (15) \qquad \sum_{\pm} \lim_{t \to \pm\infty} \int_R^\infty |\partial_{t,r} u(t,r)|^2 r^{N-1} dr \ge \frac{1}{C} \|v_1\|_{L_{N-2}^2(R)}^2. $$

By (12)

$$ \|v_1\|_{L_{N-2}^2(R)}^2 = \|u_0\|_{\dot{H}_N^1(R)}^2 - (N-2)R^{N-2}u_0^2(R). $$

Since we have proved that $u_0(R) = 0$, this implies the desired inequality (4).

Case 2. We assume $N = 4k + 2$, $u_0 = 0$ and $u_1 \in Q_N(R)$. We claim

$$ (16) \qquad v_0 \in P_{N-2}(R), $$

that is $v_0$ is orthogonal to $\frac{1}{r^{2j}}$, $k \le j \le 2k - 1$. Indeed, by (10), if $k \le j \le 2k - 1$,

$$ \int_R^\infty \frac{dv_0}{dr} \frac{1}{r^{2j+1}} r^{N-3} dr = - \int_R^\infty u_1 \frac{1}{r^{2j+2}} r^{N-1} dr = 0, $$

by the assumption that $u_0 \in Q_N(R)$. This proves (16). By (13), (16), and the induction hypothesis,

$$ (17) \qquad \sum_{\pm} \lim_{t \to \pm\infty} \int_{R+|t|}^\infty (\partial_{t,r} u(t,r))^2 r^{N-1} dr \ge \frac{1}{C} \|v_0\|_{\dot{H}_{N-2}^1(R)}^2. $$

By (11), $\|v_0\|_{\dot{H}_{N-2}^1(R)}^2 = \|u_1\|_{L_N^2(R)}^2$, which yields (5) and concludes the proof. $\square$

REFERENCES

[1] COLLOT, C., DUYCKAERTS, T., KENIG, C., AND MERLE, F. Soliton resolution for the radial quadratic wave equation in space dimension 6. *Vietnam J. Math.* **52**, 3 (2024), 735–773.

[2] CÔTE, R., KENIG, C. E., AND SCHLAG, W. Energy partition for the linear radial wave equation. *Math. Ann.* **358**, 3-4 (2014), 573–607.

[3] CÔTE, R., AND LAURENT, C. Concentration close to the cone for linear waves. *Rev. Mat. Iberoam.* **40**, 1 (2024), 201–250.

[4] DELORT, J.-M. Microlocal partition of energy for linear wave or Schrödinger equations. *Tunis. J. Math.* **4**, 2 (2022), 329–385.

[5] DUYCKAERTS, T., KENIG, C., MARTEL, Y., AND MERLE, F. Soliton resolution for critical co-rotational wave maps and radial cubic wave equation. *Commun. Math. Phys.* **391**, 2 (2022), 779–871.
[6] DUYCKAERTS, T., KENIG, C., AND MERLE, F. Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. *J. Eur. Math. Soc. (JEMS)* **13**, 3 (2011), 533–599.

[7] DUYCKAERTS, T., KENIG, C., AND MERLE, F. Classification of radial solutions of the focusing, energy-critical wave equation. *Camb. J. Math.* **1**, 1 (2013), 75–144.

[8] DUYCKAERTS, T., KENIG, C., AND MERLE, F. Soliton resolution for the radial critical wave equation in all odd space dimensions. *Acta Math.* **230**, 1 (2023), 1–92.

[9] KENIG, C., LAWRIE, A., LIU, B., AND SCHLAG, W. Channels of energy for the linear radial wave equation. *Adv. Math.* **285**, 1 (2015), 877–936.

[10] LI, L., SHEN, R., AND WEI, L. Explicit formula of radiation fields of free waves with applications on channel of energy. *Anal. PDE* **17**, 2 (2024), 723–748.