Energy Critical Nonlinear Wave Equation 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.

Spacetime integral bounds for the energy-critical nonlinear wave equation

In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the L t ∞ L x 2 L_{t}^{\infty } L_{x}^{2} and L t ∞ H ˙ 2 L_{t}^{\infty } \dot {H}^{2} norms of the solution to the quintic problem. The main motivation for this paper is the use of an interaction Morawetz estimate for the nonlinear wave equation.

Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

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 .

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.

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.

Sharp criteria of blow-up for the energy-critical nonlinear wave equation with a damping term

In terms of the best constant of the critical Sobolev inequality, by establishing two invariant flows, we find the sharp criteria of blow-up and global existence of solutions for the energy-critical nonlinear wave equation with a damping term.

On multi-solitons for the energy-critical wave equation in dimension 5

We construct K-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions u of the equation such that where and for any , Wk is the Lorentz transform of the explicit standing soliton , with any speed , satisfying for , and an explicit smallness condition. The proof extends the refined method of construction of asymptotic multi-solitons from Martel and Merle (2016 Arch. Ration. Mech. Anal. 222 1113–60; Martel and Merle 2018 Inventiones Math. 214 1267–363).

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Abstract We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

ASYMPTOTICS OF THE CRITICAL NONLINEAR WAVE EQUATION FOR A CLASS OF NON-STAR-SHAPED OBSTACLES

Scattering for the energy critical nonlinear wave equation for domains exterior to non-trapping obstacles in 3+1 dimension is known for the star-shaped case. In this paper, we extend the scattering for a class of non-star-shaped obstacles called illuminated from exterior. The main tool we use is the method of multipliers with weights that generalize the Morawetz multiplier to suit the geometry of the obstacle.

Global dynamics away from the ground state for the energy-critical nonlinear wave equation

We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground states and gets away from them in the energy space, we can classify its behavior into four cases, according to whether it blows up in finite time or scatters to zero, in forward or backward time direction. We prove that initial data for each case constitute a non-empty open set in the energy space. This is an extension of the recent results by the latter two authors on the subcritical nonlinear Klein- Gordon and Schr¨odinger equations, except for the part of the center manifolds. The key step is to prove the “one-pass” theorem, which states that the transition from the scattering region to the blow-up region can take place at most once along each trajectory. The main new ingredients are the control of the scaling parameter and the blow-up characterization by Duyckaerts, Kenig, and Merle.

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].

Blow-up for the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential

Blow-up for the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential

Global dynamics of the nonradial energy-critical wave equation above the ground state energy

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in H(over dot) x L-2 with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as t -> +/-infinity.

Dynamics for the energy critical nonlinear wave equation in high dimensions

T. Duyckaerts and F. Merle (2008) studied the variational structure near the ground state solution W W of the energy critical wave equation and classified the solutions with the threshold energy E ( W , 0 ) E(W,0) in dimensions d = 3 , 4 , 5 d=3,4,5 . In this paper, we extend the results to all dimensions d ≥ 6 d\ge 6 . The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W W .

Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions

Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime bound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finite quantity M(E(u)) depending only on the energy E(u) of u. We reprove this result, and show that this quantity obeys a bound of at most exponential type in the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for some absolute constant C > 0. The argument combines the quantitative local potential energy decay estimates of these previous papers with arguments used by Bourgain and the author for the analogous nonlinear Schr\odinger equation.