Jonas Lührmann
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View article: Stability of the catenoid for the hyperbolic vanishing mean curvature equation outside symmetry
Stability of the catenoid for the hyperbolic vanishing mean curvature equation outside symmetry Open
We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski …
View article: Asymptotic stability of the sine-Gordon kink
Asymptotic stability of the sine-Gordon kink Open
We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to cap…
View article: Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation under even perturbations
Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation under even perturbations Open
We establish the full asymptotic stability of solitary waves for the focusing cubic Schrödinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances appro…
View article: On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation
On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation Open
We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold …
View article: On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation
On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation Open
We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold …
View article: Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry
Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry Open
We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski …
View article: Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry
Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry Open
We establish the conditional asymptotic stability in a local energy norm of the unstable soliton for the one-dimensional quadratic Klein-Gordon equation under even perturbations. A key feature of the problem is the positive gap eigenvalue …
View article: On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials
On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials Open
We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to …
View article: The wave maps equation and Brownian paths
The wave maps equation and Brownian paths Open
We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ as initial data. Our main th…
View article: Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution
Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution Open
We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich ne…
View article: Asymptotic stability of the sine-Gordon kink under odd perturbations
Asymptotic stability of the sine-Gordon kink under odd perturbations Open
We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon m…
View article: On modified scattering for 1D quadratic Klein-Gordon equations with\n non-generic potentials
On modified scattering for 1D quadratic Klein-Gordon equations with\n non-generic potentials Open
We consider the asymptotic behavior of small global-in-time solutions to a 1D\nKlein-Gordon equation with a spatially localized, variable coefficient\nquadratic nonlinearity and a non-generic linear potential. The purpose of this\nwork is …
View article: Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation
Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation Open
We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency pr…
View article: Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data
Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data Open
We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{…
View article: Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities
Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities Open
We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to…
View article: Local smoothing estimates for Schr\\"odinger equations on hyperbolic\n space
Local smoothing estimates for Schr\\"odinger equations on hyperbolic\n space Open
We establish global-in-time frequency localized local smoothing estimates for\nSchr\\"odinger equations on hyperbolic space $\\mathbb{H}^d$. In the presence of\nsymmetric first and zeroth order potentials, which are possibly time-dependent…
View article: Concentration Compactness for Critical Radial Wave Maps
Concentration Compactness for Critical Radial Wave Maps Open
We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere $\mathbb{S}^m$, $m \geq 1$, and prove global regularity and scattering for classical smooth data of finite energy. In a…
View article: On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $\mathbb{R}^3$
On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $\mathbb{R}^3$ Open
We consider energy sub-critical defocusing nonlinear wave equations on $\mathbb{R}^3$ and establish the existence of unique global solutions almost surely with respect to a unit-scale randomization of the initial data on Euclidean space. I…
View article: Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation
Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation Open
We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1+4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gerard profile …