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Concentration close to the cone for linear waves

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Abstract

We are concerned with solutions to the linear wave equation. Our main result concerns the computation of the asymptotic exterior energy outside of the cone |x| \ge |t|+R for R>0 and odd dimension. This proves, in the general case, the results of Kenig–Lawrie–Liu–Schlag (2015) (which were restricted to radial data). Also, along the proof, we derive further expressions of the exterior energy (outside a shifted light cone), valid in all dimension and for non-radial data. In particular, we generalize the formulas of Côte–Kenig–Schlag (2014) obtained in the radial setting.

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... We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at 0 is given by non radiative solutions to the linear equation (described in [2]). We also construct nonlinear solutions with an arbitrary prescribed radiation field. ...
... We now define the linear and nonlinear flows: if (u 0 , u 1 ) ∈ H, then u L (t) = S L (t)(u 0 , u 1 ) is the solution of the linear wave equation (2) u L = 0, u L (0) = (u 0 , u 1 ). ...
... We described in [2], for odd dimensions, the linear space P (R) of initial datum (v 0 , v 1 ) ∈ H that give rise to a solution v = S L (v 0 , v 1 ) to the linear wave equation such that E ext,R ( v) = 0, in terms of the the Radon transform of the initial data (v 0 , v 1 ) and according to its decomposition in spherical harmonics: for the convenience of the reader, we give further details in the Appendix A, see in particular (20). This was first done for radial data in odd dimension by [9], and in even dimension in [11] (see also [1]), and it was extended to non radial data for odd dimensions in [2] and later in even dimension in [10]. ...
Preprint
Non radiative solutions of the energy critical non linear wave equation are global solutions u that furthermore have vanishing asymptotic energy outside the lightcone at both t±t \to \pm \infty: limt±t,xu(t)L2(xt+R)=0, \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, for some R>0R \gt 0. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein. We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at 0 is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.
... If there exists a radius r α > max{R α , c 1 |α| 2 } for each α = 0, so that the non-radiative solution u α satisfies u α Ḣ1 ({x:|x|>rα}) = A, then any radial solution to (CP1) with a maximal lifespan (−T − , T + ) and lim sup t→T+ (u(·, t), u t (·, t)) Ḣ1 ×L 2 (R 3 ) < A, must satisfy T + = +∞ and scatter in the whole space in the positive time direction. 1 In the case (a), we understand 0 > Rα = 0 − . Remark 1.6. ...
... The formula between G ± is relatively simpler in odd dimensions than even dimensions. In this work we only need to use the 3-dimensional case (please see [1,13] for all dimensions, for example) ...
... We first fix a positive constant η = η(γ) ≪ 1 so that C 1 γη 4 < 1/256. Here C 1 ≥ 1 is the constant in the Strichartz estimate (1). We then split the region Ω R,2 N into N 1 = N 1 (γ, c 1 , M ) pieces Φ k accordingly, with ...
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In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions u to the non-linear wave equation is similar to that of a linear free wave vLv_L in an exterior region {x:x>R+t}\{x: |x|>R+|t|\}, i.e. limt±x>R+t((uvL)2+uttvL2)dx=0. \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 + |u_t-\partial_t v_L|^2) dx = 0. We classify all such solutions for a given linear free wave vLv_L in this work. We also give some applications of our theory on the global behaviours of radial solutions to this kind of equations. In particular we show the scattering of all finite-energy radial solutions to the defocusing energy critical wave equations.
... We call G ± the radiation profile of u, or equivalently, its initial data (u 0 , u 1 ). In the 5-dimensional case, the radiation profiles G ± satisfies G + (s, θ) = G − (−s, −θ), as shown in Côte-Laurent [2] and Li-Shen-Wei [16]. Therefore a linear free wave is R-weakly non-radiative if and only if its radiation profiles are compactly supported in the region [−R, R] × S 4 . ...
... An explicit formula in term of Fourier transforms can also be found in a recent work Côte-Laurent [2]. In this work, if we mention the radiation profile of a free wave, or equivalently its initial data, then we mean the radiation profile in the negative time direction unless specified otherwise. ...
... In addition, the integrals • The characteristic numbers α, β can be uniquely determined by the asymptotic behaviour of initial data. In fact, the pair (α, β) satisfying (2) and (3) is unique since we have ...
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In this work we classify all radial non-radiative solutions to the 5D nonlinear wave equations with a wide range of energy critical nonlinearity. We show that such a solution always comes with two characteristic numbers. These characteristic numbers can be determined by either the radiation profile of the initial data or the asymptotic behaviour of the solution. In addition, two radial weakly non-radiative solutions with the same characteristic numbers must coincide with each other in the overlap part of their exterior regions. Finally we give a few applications of our theory on the global behaviours of solutions to the nonlinear wave equations.
... Extensions to domains of the form {|x| > R + |t|} for R > 0, obtained in [KLLS15,DKMM22,LSW21], will be used in the present paper. Other recent results on the asymptotic behaviour of linear waves can be found in [Del21,CL21,LSWW22]. For the linearised wave equation (1.1), two natural counter examples to estimates like (1.5) and (1.6) are ΛW and tΛW (for N ≥ 5), as they are non-radiative i.e. ...
Preprint
Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation they have been obtained in the radial case in odd dimensions, first in 3 dimensions by Duyckaerts, Kenig and Merle (Camb. J. Math., 2013), then for general odd dimensions by the same authors (Comm. Math. Phys., 2020). We consider even dimensions, for which such estimates are known to fail (C\^ote, Kenig and Schlag, Math. Ann., 2014). We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us to prove the soliton resolution conjecture in six dimensions (Collot, Duyckaerts, Kenig and Merle, arXiv preprint 2201.01848, 2022 versions 1 and 2).
... • After completion of this work, formula (1. 1.10) has been also obtained by Côte and Laurent in [5]. ...
... The radiation fields G ± = T ± (u 0 , u 1 ) can be given in term of the initial data (see Cöte-Laurent [2], please note that our notations are slightly different from their original work) ...
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Radiation field and channel of energy method have become important tools in the study of nonlinear wave equations in recent years. In this work we give basic theory of radiation fields of free waves in the energy sub-critical case. We also show that the asymptotic behaviours of non-radiative solutions to a wide range of non-linear wave equations resemble those of non-radiative free waves. Our theory is completely given in the critical Sobolev spaces of the corresponding nonlinear wave equation and avoids any assumption on the energy of the solutions.
... We start by the odd dimensional case and then deal with the even dimensional case. Please note that a similar result for odd dimensions has been proved in Côte-Laurent [1] by the Radon transform. The novelty of our result includes ...
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In this work we consider weakly non-radiative solutions to both linear and non-linear wave equations. We first characterize all weakly non-radiative free waves, without the radial assumption. Then in dimension 3 we show that the initial data of non-radiative solutions to a wide range of nonlinear wave equations are similar to those of non-radiative free waves in term of asymptotic behaviour.
... A similar formula has been known for many years, see Friedlander [10]. One may also refer to Li-Shen-Wei [14] for an explicit formula for all dimensions d ≥ 2. This map between initial data and radiation profiles can also be given in term of their Fourier transforms, as given in a recent work Côte-Laurent [1]. We may also give a formula of free waves in term of the radiation fields G − via a time translation ...
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In this work we consider the operator (TG)(x)=S2G(xω,ω)dω,xR3,  GL2(R×S2). (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot \omega, \omega) d\omega, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). This is the adjoint operator of the Radon transform. We manage to give an optimal L6L^6 decay estimate of TG\mathbf{T} G near the infinity by a geometric method, if the function G is compactly supported. As an application we give decay estimate of non-radiative solutions to the 3D linear wave equation in the exterior region {(x,t)R3×R:x>R+t}\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}. This kind of decay estimate is useful in the channel of energy method for wave equations
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