Energy Critical Wave Research Articles

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

AbstractWe exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ℝ2 into a smooth compact revolution surface of ℝ3 that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with 𝕊2 target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc.

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.

Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation

We exhibit $\mathcal C^{\infty}$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton $$u(t,x)=\frac{1}{\lambda^{\frac{N-2}{2}}(t)}(Q+\e(t))(\frac{x}{\lambda(t)}) \mbox{with} \|\e(t),\pa_t\e(t)\|_{\dot{H}^1\times L^2}\ll1 $$ where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\to L^{2^*}$, and a blow up speed $$\lambda(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))} \mbox{as} t\to T.$$

Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case

Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of W concentrating at the origin.

Profiles of bounded radial solutions of the focusing, energy-critical wave equation

In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence.

On similarity in the evolution of semilinear wave and Klein-Gordon equations: Numerical surveys

The aim of this paper is threefold. The first and also main purpose is to provide numerical evidence for the conjecture proposed by Bizoń et al. [“Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation,” J. Math. Phys. 52, 103703 (2011)]10.1063/1.3645363 that the blowup evolution of spherically symmetric semilinear Klein-Gordon equations is similar to the evolution of spherically symmetric semilinear wave equations, i.e., the mass term can be neglected when the amplitude of a solution grows. The second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations. The third goal is to present numerical evidence for the fact that the special class of self-similar profiles of semilinear wave equations found by Kycia [“On self-similar solutions of semilinear wave equations in higher space dimensions,” Appl. Math Comput. 217, 9451–9466 (2011)]10.1016/j.amc.2011.04.039 play the same role in the evolution of semilinear wave and Klein-Gordon equations as the previously known ordinary profiles. All the results are presented in spherical symmetry.

Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems

We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

Strichartz estimates on Kerr black hole backgrounds

We study the dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of this paper is to establish Strichartz estimates for solutions of the aforementioned equation. As an application, we then prove global well-posedness and uniqueness for the energy critical semilinear wave equation.

Geometric Renormalization Below the Ground State

The caloric gauge was introduced by Tao with studying large data energy critical wave maps mapping from $\mathbf{R}^{2+1}$ to hyperbolic space $\mathbf{H}^m$ in view. In \cite{BIKT} Bejenaru, Ionescu, Kenig, and Tataru adapted the caloric gauge to the setting of Schr\"odinger maps from $\mathbf{R}^{d + 1}$ to the standard sphere $S^2 \hookrightarrow \mathbf{R}^3$ with initial data small in the critical Sobolev norm. Here we develop the caloric gauge in a bounded geometry setting with a construction valid up to the ground state energy.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions.Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded domain Ω ⊂ R 2 . First, we prove an appropriate Strichartz type estimate using the L q spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser–Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.

Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation

The analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions. In this paper we study the focusing, cubic, nonlinear Klein–Gordon equation in R3 with large radial data in the energy space. This equation admits a unique positive stationary solution Q, called the ground state. In 1975 Payne and Sattinger showed that solutions u(t) with energy E[u,u˙]strictly below that of the ground state are divided into two classes, depending on a suitable functional K(u): If K(u)<0, then one has finite time blow-up, if K(u)⩾0 global existence; moreover, these sets are invariant under the flow. Recently, Ibrahim, Masmoudi and the first author [22] improved this result by establishing scattering to zero for K[u]⩾0 by means of a variant of the Kenig–Merle method (Kenig and Merle, 2006, 2008 [25,26]). In this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case. For example, in a small neighborhood of Q one encounters the following trichotomy: On one side of a center-stable manifold one has finite time blow-up for t⩾0, on the other side scattering to zero, and on the manifold itself one has scattering to Q, both as t→+∞. In total, the class of data with energy at most slightly above that of Q is divided into nine disjoint non-empty sets each displaying different asymptotic behavior as t→±∞, which includes solutions blowing up in one time direction and scattering to zero on the other. The analogue of the solutions found by Duyckaerts and Merle (2009, 2008) [13,14] for the energy critical wave and Schrödinger equations appear here as the unique one-dimensional stable/unstable manifolds approaching ±Q exponentially as t→∞ or t→−∞, respectively. The main technical ingredient in our proof is a “one-pass” theorem which excludes the existence of (almost) homoclinic orbits between Q (as well as −Q) and (almost) heteroclinic orbits connecting Q with −Q. In a companion paper (Nakanishi and Schlag, 2010 [31]) we establish analogous properties for the NLS equation.

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 .

Dynamics of Threshold Solutions for Energy-Critical Wave Equation

We consider the energy-critical nonlinear focusing wave equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [8], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u0, u1) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analogous to [3], which treats the energy-critical nonlinear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.

Scattering for the two-dimensional energy-critical wave equation

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation.

On the equipartition of energy for the critical NLW

We study some qualitative properties of global solutions to the following focusing and defocusing critical NLW: □ u + λ u | u | 2 ∗ − 2 = 0 , λ ∈ R , u ( 0 ) = f ∈ H ˙ 1 ( R n ) , ∂ t u ( 0 ) = g ∈ L 2 ( R n ) on R × R n for n ⩾ 3 , where 2 ∗ ≡ 2 n n − 2 . We will consider the global solutions of the defocusing NLW whose existence and scattering property is shown in [J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. (7) (1994) 303–309 (electronic); H. Bahouri, J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (6) (1998) 783–789] and [H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175], without any restriction on the initial data ( f , g ) ∈ H ˙ 1 ( R n ) × L 2 ( R n ) . As well as the solutions constructed in [H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (2) (1984) 261–270] to the focusing NLW for small initial data and to the ones obtained in [C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, preprint], where a sharp condition on the smallness of the initial data is given. We prove that the solution u ( t , x ) satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see [C. Morawetz, Time decay for the nonlinear Klein–Gordon equation, Proc. Roy. Soc. London Ser. A 306 (1968) 291–296]). As a by-product we deduce that any global solution to critical NLW belonging to a natural functional space satisfies: lim R → ∞ 1 R ∫ R ∫ | x | < R | ∇ x u ( t , x ) | 2 d x d t = lim R → ∞ 1 2 R ∫ R ∫ | x | < R ( | ∇ t , x u ( t , x ) | 2 + 2 λ 2 ∗ | u ( t , x ) | 2 ∗ ) d x d t = ∫ R n ( | ∇ t , x u ( 0 , x ) | 2 + 2 λ 2 ∗ | u ( 0 , x ) | 2 ∗ ) d x .

Classification of the radial solutions of the focusing, energy-critical wave equation

Classification of the radial solutions of the focusing, energy-critical wave equation