arXiv subjects

Wilhelm Schlag

Publications and source records attributed to Wilhelm Schlag.

At least 55 records · Page 3Linked to original sources

arXiv2014

Channels of energy for the linear radial wave equation

Exterior channel of energy estimates for the radial wave equation were first considered in three dimensions by Duyckaerts, the first author, and Merle, and recently for the 5-dimensional case by the first, second, and fourth authors. In this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation. This will be used in a companion paper to establish soliton resolution for equivariant wave maps in 3 dimensions exterior to the ball B(0,1), and in all equivariance classes.

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arXiv2014

Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}

For the radial energy-supercritical nonlinear wave equation $$\Box u = -u_{tt} + \triangle u = \pm u^7$$ on $\R^{3+1}$, we prove the existence of a class of global in forward time $C^\infty$-smooth solutions with infinite critical Sobolev norm $\dot{H}^{\frac{7}{6}}(\R^3)\times \dot{H}^{\f16}(\R^3)$. Moreover, these solutions are stable under suitably small perturbations . We also show that for the {\em defocussing} energy supercritical wave equation, we can construct such solutions which moreover satisfy the size condition $\|u(0, \cdot)\|_{L_x^\infty(|x|\geq 1)}>M$ for arbitrarily prescribed $M>0$. These solutions are stable under suitably small perturbations. Our method proceeds by regularization of self-similar solutions which are smooth away from the light-cone but singular on the light-cone. The argument crucially depends on the supercritical nature of the equation. Our approach should be seen as part of the program initiated in \cite{KrSchTat1}, \cite{KrSchTat2}, \cite{DoKri}.

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arXiv2014

Profiles for the radial focusing 4d energy-critical wave equation

Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.

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arXiv2013

Relaxation of wave maps exterior to a ball to harmonic maps for all data

In this paper we study 1-equivariant wave maps of finite energy from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the unit sphere in R^3 gets mapped to the north pole. Finite energy implies that spacial infinity gets mapped to either the north or south pole. In particular, each such equivariant wave map has a well-defined topological degree which is an integer. We establish relaxation of such a map of arbitrary energy and degree to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizon, Chmaj, Maliborski who observed this asymptotic behavior numerically.

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arXiv2013

Center-stable manifold of the ground state in the energy space for the critical wave equation

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.

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arXiv2013

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 the natural energy space 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 time approaches infinity. In our previous paper arxiv:1010.3799 we treated the radial case, and this paper provides the natural nonradial extension of these results. However, the present paper is self-contained and in fact develops a somewhat different formalism in order to handle the more complex nonradial situation.

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arXiv2012

Exotic blowup solutions for the u^5 focusing wave equation in R^3

For the energy critical focusing wave equation in three dimensions we establish the existence of blowup solutions via an "exotic" dynamical rescaling of the ground state solution W(x). The scaling law is not of the pure-power type, but rather eternally oscillates between such classes. Moreover, there is a continuum of these scaling laws.

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arXiv2012

Full range of blow up exponents for the quintic wave equation in three dimensions

For the critical focusing wave equation \Box u = u^5 on R^{3+1} in the radial case, we prove the existence of type II blow up solutions with scaling parameter λ(t) = t^{-1-ν} for all ν>0. This extends the previous work by the authors and Tataru where the condition ν> 1/2 had been imposed, and gives the optimal range of polynomial blow up rates in light of recent work by Duyckaerts, Kenig and Merle.

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arXiv2012

Scattering for wave maps exterior to a ball

We consider 1-equivariant wave maps from \R \times (\R^3 \setminus B) to S^3 where B is a ball centered at 0, and the boundary of B gets mapped to a fixed point on S^3. We show that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.

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arXiv2012

Energy partition for the linear radial wave equation

We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy relative to a light cone, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts, Merle and the second author obtained in odd dimensions, fails in even dimensions. Positive results for restricted classes of data are obtained. This is a companion paper to our two nonlinear papers with Andrew Lawrie.

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arXiv2012

Threshold phenomenon for the quintic wave equation in three dimensions

For the critical focusing wave equation $\Box u = u^5$ on $\R^{3+1}$ in the radial case, we establish the role of the "center stable" manifold $Σ$ constructed in \cite{KS} near the ground state $(W,0)$ as a threshold between type I blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń, Chmaj, Tabor. The underlying topology is stronger than the energy norm.

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arXiv2011

Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials

We consider semiclassical Schrödinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $f_\pm(\cdot,E;\hbar)$ with error terms that are uniformly controlled for small $E$ and $\hbar$, and construct the scattering matrix as well as the semiclassical spectral measure associated to $H(\hbar)$. This is crucial in order to obtain decay bounds for the corresponding wave and Schrödinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta $\ell$ where the role of the small parameter $\hbar$ is played by $\ell^{-1}$. It follows from the results in this paper and \cite{DSS2}, that the decay bounds obtained in \cite{DSS1}, \cite{DS} for individual angular momenta $\ell$ can be summed to yield the sharp $t^{-3}$ decay for data without symmetry assumptions.

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arXiv2011

On pointwise decay of linear waves on a Schwarzschild black hole background

We prove sharp pointwise $t^{-3}$ decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates $t^{-4}$, and $t^{-6}$, respectively. We proceed by decomposition into angular momentum $\ell$ and summation of the decay estimates on the Regge-Wheeler equation for fixed $\ell$. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in $\ell$ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

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arXiv2011

On the spectral properties of L_{+-} in three dimensions

This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L_{+-}, satisfy the gap property, at least over the radial functions. This means that the interval (0,1] does not contain any eigenvalues of L_{+-} and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the center-stable manifold associated with these states. This paper therefore provides the final installment in the proof of this scattering property for the cubic Klein-Gordon and Schrodinger equations in the radial case, see the recent theory of Nakanishi and the third author, as well as the earlier work of the third author and Beceanu on NLS. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations.

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arXiv2011

Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation

We present some numerical findings concerning the nature of the blowup vs. global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial data. The context of this study is provided by the classical paper by Payne, Sattinger from 1975, as well as the recent work by K. Nakanishi, and the second author arXiv:1005.4894. Specifically, we numerically investigate the boundary of the forward scattering region. At this point we do not have sufficient numerical evidence that might indicate whether or not the boundary remains a smooth manifold for general energies. In this updated version we include some fine-scale computations that reveal more complicated structures than one might expect.

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arXiv2011

Global dynamics above the ground state energy for the cubic NLS equation in 3D

We extend our previous result on the nonlinear Klein-Gordon equation to the nonlinear Schrodinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by Duyckaerts and Roudenko appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with the Klein-Gordon case, the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a "one-pass" theorem which precludes "almost homoclinic orbits", i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein-Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference from the Klein-Gordon case is the need to control two modulation parameters.

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arXiv2011

Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing energy sub-critical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph transform (or Hadamard) approach, which requires less spectral information on the linearized operator, and less decay of the nonlinearity, than the Lyapunov-Perron method employed previously in this context. The only assumption on the stationary solution is that the kernel of the linearized operator is spanned by its spatial derivatives, which is known to hold for the ground states. The main novelty of this paper lies with the fact that the graph transform method is carried out in the presence of modulation parameters corresponding to the symmetries.

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arXiv2010

Global dynamics above the ground state energy for the one-dimensional NLKG equation

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case arXiv:1005.4894, the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein-Gordon equation, similar to those of Mizumachi arXiv:math/0605031. The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.

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