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We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential, generalizing work of Soffer and Weinstein, and of Tsai and Yau. The novelty is that we allow generic spectra associated to the potential. This is a new application of the idea of interpreting the nonlinear Fermi golden rule as a consequence of the Hamiltonian structure.
We study existence and nonexistence results for generalized transition wave solutions of space-time heterogeneous Fisher–KPP equations. When the coefficients of the equation are periodic in space but otherwise depend in a fairly general fashion on time, we prove that such waves exist as soon as their speed is sufficiently large in a sense. When this speed is too small, transition waves do not exist anymore; this result holds without assuming periodicity in space. These necessary and sufficient conditions are proved to be optimal when the coefficients are periodic both in space and time. Our method is quite robust and extends to general nonperiodic space-time heterogeneous coefficients, showing that transition wave solutions of the nonlinear equation exist as soon as one can construct appropriate solutions of a given linearized equation.
It is known that, using the Gaussian beam approximation, one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long, time. We show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular, we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner–Nordström black hole necessarily leads to a “loss of derivative” in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long, time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of subextremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity.
We prove a height estimate (distance from the tangent hyperplane) for -minimizers of the perimeter in the sub-Riemannian Heisenberg group. The estimate is in terms of a power of the excess (-mean oscillation of the normal) and its proof is based on a new coarea formula for rectifiable sets in the Heisenberg group.
We propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly nonresonant dispersive equations. As an example, we obtain unconditional well-posedness of the Cauchy problem in the energy space for a large class of one-dimensional dispersive equations with a dispersion that is greater than the one of the Benjamin–Ono equation. At the level of dispersion of the Benjamin–Ono, we also prove the well-posedness in the energy space but without unconditional uniqueness. Since we do not use a gauge transform, this enables us in all cases to prove strong convergence results in the energy space for solutions of viscous versions of these equations towards the purely dispersive solutions. Finally, it is worth noting that our method of proof works on the torus as well as on the real line.
We establish an algebraic error estimate for the stochastic homogenization of fully nonlinear, uniformly parabolic equations in stationary ergodic spatiotemporal media. The approach is similar to that of Armstrong and Smart in the study of quantitative stochastic homogenization of uniformly elliptic equations.