Energy Critical Wave Research Articles

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Ground state of the Gross–Pitaevskii equation with a harmonic potential in the energy-critical case

Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin–Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus the supremum norm.

On classification of non-radiative solutions for various energy-critical wave equations

Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region |x|>R+|t| vanishes asymptotically in both time directions. This notion, introduced by [7], has been key in solving the soliton resolution conjecture for these equations in the radial case. In the present paper, we first classify their asymptotic behaviour at infinity, showing that they correspond to a k-parameters family of solutions where k depends on the dimension. This generalises the previous results [8,6] in three and four dimensions. We then establish a unique maximal extension of these solutions.

Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation

We consider the wave maps problem with domain R 2 + 1 \mathbb {R}^{2+1} and target S 2 \mathbb {S}^{2} in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from R 2 \mathbb {R}^{2} to S 2 \mathbb {S}^{2} , with polar angle equal to Q 1 ( r ) = 2 arctan ⁡ ( r ) Q_{1}(r) = 2 \arctan (r) . By applying the scaling symmetry of the equation, Q λ ( r ) = Q 1 ( r λ ) Q_{\lambda }(r) = Q_{1}(r \lambda ) is also a harmonic map, and the family of all such Q λ Q_{\lambda } are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the Q λ Q_{\lambda } family. More precisely, for b > 0 b>0 , and for all λ 0 , 0 , b ∈ C ∞ ( [ 100 , ∞ ) ) \lambda _{0,0,b} \in C^{\infty }([100,\infty )) satisfying, for some C l , C m , k > 0 C_{l}, C_{m,k}>0 , C l log b ⁡ ( t ) ≤ λ 0 , 0 , b ( t ) ≤ C m log b ⁡ ( t ) , | λ 0 , 0 , b ( k ) ( t ) | ≤ C m , k t k log b + 1 ⁡ ( t ) , k ≥ 1 t ≥ 100 \begin{equation*} \frac {C_{l}}{\log ^{b}(t)} \leq \lambda _{0,0,b}(t) \leq \frac {C_{m}}{\log ^{b}(t)}, \quad |\lambda _{0,0,b}^{(k)}(t)| \leq \frac {C_{m,k}}{t^{k} \log ^{b+1}(t) }, k\geq 1 \quad t \geq 100 \end{equation*} there exists a wave map with the following properties. If u b u_{b} denotes the polar angle of the wave map into S 2 \mathbb {S}^{2} , we have u b ( t , r ) = Q 1 λ b ( t ) ( r ) + v 2 ( t , r ) + v e ( t , r ) , t ≥ T 0 \begin{equation*} u_{b}(t,r) = Q_{\frac {1}{\lambda _{b}(t)}}(r) + v_{2}(t,r) + v_{e}(t,r), \quad t \geq T_{0} \end{equation*} where − ∂ t t v 2 + ∂ r r v 2 + 1 r ∂ r v 2 − v 2 r 2 = 0 \begin{equation*} -\partial _{tt}v_{2}+\partial _{rr}v_{2}+\frac {1}{r}\partial _{r}v_{2}-\frac {v_{2}}{r^{2}}=0 \end{equation*} | | ∂ t ( Q 1 λ b ( t ) + v e ) | | L 2 ( r d r ) 2 + | | v e r | | L 2 ( r d r ) 2 + | | ∂ r v e | | L 2 ( r d r ) 2 ≤ C t 2 log 2 b ⁡ ( t ) , t ≥ T 0 \begin{equation*} ||\partial _{t}(Q_{\frac {1}{\lambda _{b}(t)}}+v_{e})||_{L^{2}(r dr)}^{2}+||\frac {v_{e}}{r}||_{L^{2}(r dr)}^{2} + ||\partial _{r}v_{e}||_{L^{2}(r dr)}^{2} \leq \frac {C}{t^{2} \log ^{2b}(t)}, \quad t \geq T_{0} \end{equation*} and λ b ( t ) = λ 0 , 0 , b ( t ) + O ( 1 log b ⁡ ( t ) log ⁡ ( log ⁡ ( t ) ) ) \begin{equation*} \lambda _{b}(t) = \lambda _{0,0,b}(t) + O\left (\frac {1}{\log ^{b}(t) \sqrt {\log (\log (t))}}\right ) \end{equation*}

Strichartz estimates and blowup stability for energy critical nonlinear wave equations

We prove Strichartz estimates in similarity coordinates for the radial wave equation with a self-similar potential in dimensions d ≥ 4 d\geq 4 . As an application of these, we establish the asymptotic stability of the ODE blowup profile of the energy critical radial nonlinear wave equation for 4 ≤ d ≤ 6 4\leq d\leq 6 .

Global, Non-Scattering Solutions to the Energy Critical Wave Maps Equation

We consider the 1-equivariant energy critical wave maps problem with two-sphere target. Using a method based on matched asymptotic expansions, we construct infinite time relaxation, blow-up, and intermediate types of solutions that have topological degree one. More precisely, for a symbol class of admissible, time-dependent length scales, we construct solutions which can be decomposed as a ground state harmonic map (soliton) re-scaled by an admissible length scale, plus radiation, and small corrections which vanish (in a suitable sense) as time approaches infinity. Our class of admissible length scales includes positive and negative powers of t, with exponents sufficiently small in absolute value. In addition, we obtain solutions with soliton length scale undergoing damped or undamped oscillations in a bounded set, or undergoing unbounded oscillations, for all sufficiently large t.

On Channels of Energy for the Radial Linearised Energy Critical Wave Equation in the Degenerate Case

Abstract 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 three dimensions in [8], then for the general case in [10]. We consider even dimensions, for which such estimates are known to fail [2]. We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us in [1] to prove the soliton resolution conjecture in six dimensions.

Inverse scattering for critical semilinear wave equations

We show that the scattering operator for defocusing energy critical semilinear wave equations \square u+f(u)=0, with f C-infinity and f ~ u^5, in three space dimensions, determines f.

Dynamics of bubbling wave maps with prescribed radiation

We study energy critical one-equivariant wave maps taking values in the two-sphere. It is known that any finite energy wave map that develops a singularity does so by concentrating the energy of (possibly) several copies of the ground state harmonic map at the origin. If only a single bubble of energy is concentrated, the solution decomposes into a dynamically rescaled harmonic map plus a term that accounts for the energy that radiates away from the singularity. In this paper, we construct blow up solutions by prescribing the radiative component of the map. In addition, we give a sharp classification of the dynamical blow up rate for every solution with this prescribed radiation.

An asymptotic expansion of two-bubble wave maps in high equivariance classes

This is the first part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps equation on 1+2 dimensional Minkowski space, with equivariance class k>3. It is known that every topologically trivial wave map with energy less than twice that of the unique k-equivariant harmonic map Q scatters in both time directions. We study maps with precisely the threshold energy, i.e., twice the energy of Q. In this paper, we give a refined construction of a wave map with threshold energy that converges to a superposition of two harmonic maps (bubbles), asymptotically decoupling in scale. We show that this two-bubble solution possesses H^2 regularity. We give a precise dynamical description of the modulation parameters as well as an expansion of the map into profiles. In the next paper in the series, we show that this solution is unique (up to the natural invariances of the equation) relying crucially on the detailed properties of the solution constructed here. Combined with our earlier work, we can now give an exact description of every threshold wave map.

Uniqueness of Two‐Bubble Wave Maps in High Equivariance Classes

AbstractThis is the second part of a two‐paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.Consider the ‐valued equivariant energy critical wave maps equation on , with equivariance class . It is known that every topologically trivial wave map with energy less than twice that of the unique k‐equivariant harmonic map scatters in both time directions. We study maps with precisely the threshold energy .In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two‐bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Corrigendum to “Multi-travelling waves for the nonlinear Klein-Gordon equation”

We provide a correction to the proof of Proposition 3 by Côte and Martel [Trans. Amer. Math. Soc. 370 (2018), pp. 7461–7487]. We refer to Yuan [Nonlinearity 32 (2019), pp. 5017–5048] for a self-contained proof in the analogue context of the energy critical wave equation. We also refer to Chen and Jendrej [Trans. Amer. Math. Soc. 372 (2019), pp. 7461–7496] for an alternate proof.

Construction of excited multi-solitons for the focusing 4D cubic wave equation

Consider the focusing 4D cubic wave equation∂ttu−Δu−u3=0,on(t,x)∈[0,∞)×R4. The main result states the existence in energy space H˙1×L2 of multi-solitary waves where each traveling wave is generated by Lorentz transform from a specific excited state, with different but collinear Lorentz speeds. The specific excited state is deduced from the non-degenerate sign-changing state constructed in Musso-Wei [34]. The proof is inspired by the techniques developed for the 5D energy-critical wave equation and the nonlinear Klein-Gordon equation in a similar context by Martel-Merle [30] and Côte-Martel [6].The main difficulty originates from the strong interactions between solutions in the 4D case compared to other dispersive and wave-type models. To overcome the difficulty, a sharp understanding of the asymptotic behavior of the excited states involved and of the kernel of their linearized operator is needed.

Unconditional well-posedness for the energy-critical nonlinear damped wave equation

Unconditional well-posedness for the energy-critical nonlinear damped wave equation

Construction of excited multi-solitons for the 5D energy-critical wave equation

For the 5D energy-critical wave equation, we construct excited [Formula: see text]-solitons with collinear speeds, i.e. solutions [Formula: see text] of the equation such that [Formula: see text] where for [Formula: see text], [Formula: see text] is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel–Merle [Construction of multi-solitons for the energy-critical wave equation in dimension 5, Arch. Ration. Mech. Anal. 222(3) (2016) 1113–1160] and Côte–Martel [Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] developed for the energy-critical wave and nonlinear Klein–Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.

Almost sure scattering for the energy critical nonlinear wave equation

We study the defocusing energy critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random perturbations of the initial data. The random perturbation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the perturbation.

Asymptotic simplification for solutions of the energy critical nonlinear wave equation

In this article, we survey recent results (with Duyckaerts and Merle) on the long-time behavior of radial solutions of the energy critical nonlinear wave equation in odd dimensions.

Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations

We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.

Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

For the critical generalized KdV equation partial _t u + partial _x (partial _x^2 u + u^5)=0 on {mathbb {R}}, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q''+Q^5=Q. For any nu in (0,frac{1}{3}), there exist global (for tge 0) solutions of the equation with the asymptotic behavior u(t,x)=t-ν2Qt-ν(x-x(t))+w(t,x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} u(t,x)= t^{-\\frac{\\nu }{2}} Q\\left( t^{-\\nu } (x-x(t))\\right) +w(t,x) \\end{aligned}$$\\end{document}where, for some c>0, x(t)∼ct1-2νand‖w(t)‖H1(x>12x(t))→0ast→+∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x(t)\\sim c t^{1-2\\nu } \\quad \\text{ and }\\quad \\Vert w(t)\\Vert _{H^1(x>\\frac{1}{2} x(t))} \\rightarrow 0\\quad \\text{ as } t\\rightarrow +\\infty . \\end{aligned}$$\\end{document}Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

Almost-sure scattering for the radial energy-critical nonlinear wave equation in three dimensions

We study the Cauchy problem for the radial energy critical nonlinear wave equation in three dimensions. Our main result proves almost sure scattering for radial initial data below the energy space. In order to preserve the spherical symmetry of the initial data, we construct a radial randomization that is based on annular Fourier multipliers. We then use a refined radial Strichartz estimate to prove probabilistic Strichartz estimates for the random linear evolution. The main new ingredient in the analysis of the nonlinear evolution is an interaction flux estimate between the linear and nonlinear components of the solution. We then control the energy of the nonlinear component by a triple bootstrap argument involving the energy, the Morawetz term, and the interaction flux estimate.