arXiv subjects

Joachim Krieger

Publications and source records attributed to Joachim Krieger.

At least 19 recordsLinked to original sources

arXiv2026

Long finite time bubble trees for two co-rotational wave maps

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ into $\mathbb{S}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in\mathbb{N}$, we construct an $n$-bubble solution concentrating at scales $λ_1(t)\gg λ_2(t)\gg \ldots\gg λ_n(t)$, where $λ_n(t)=t^{-1}\vert \log t\vert^β$, and $λ_j(t)\gtrsim \exp( \int_t^{t_0} λ_{j+1}(s)ds)$, for any $j \tfrac32$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs.

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arXiv2025

Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy

We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory.

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arXiv2025

The cubic NLS on the line with an inverse square potential

We establish modified scattering for solutions of the cubic NLS on the line with a repulsive inverse square potential and small localized data. The method is based on a comparison between the free and distorted Galilei vector fields and a wave packet transform.

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arXiv2025

Concentric bubbles concentrating in finite time for the energy critical wave maps equation

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation.

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arXiv2024

Finite time blow up for the energy critical Zakharov system I: approximate solutions

We construct approximate solutions $ (ψ_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (λe^{i θ}W, λ^2 W^2)$ . To be precise for $ N \in \mathbb{Z}_+,\;N \gg1 $ we obtain a magnetic envelope/ion density pair of the form $$ ψ_*(t, x)= e^{iα(t)}λ(t) W(λ(t)x) + η(t, x), \;n_*(t,x) = λ^2(t) W^2(λ(t) x) + χ(t,x), $$ where $ W(x) = (1 + \frac{|x|^2}{8})^{-1}$, $α(t) = α_0 \log(t)$, $λ(t)= t^{-\frac{1}{2}-ν}$ with large $ν> 1 $ and further $$ i \partial_t ψ_* + Δψ_* + n_* ψ_* = \mathcal{O}(t^N),\; \Box n_* - Δ(|ψ_*|^2) = \mathcal{O}(t^N),\;\;η(t) \to η_0, χ(t) \to χ_0, $$ as $ t \to 0^+$ in a suitable sense. The method of construction is inspired by matched asymptotic regions and approximation procedures in the context of blow up solutions introduced by the first author jointly with W. Schlag and D. Tataru, as well as the subsequently developed methods in the Schrödinger context by G. Perelman et al.

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arXiv2024

Finite time blow up for the energy critical Zakharov system II: exact solutions

Based on our companion paper [Krieger-Schmid, 2024], we show that the 4D energy critical Zakharov system admits finite time type II blow up solutions. The main new difficulty this work deals with is the appearance of a term in the linearization around the approximate solution, which is non-local with respect to both space and time. In particular this cannot be handled by straightforward adaptation of the methods developed in [Krieger-Schlag-Tataru, 2008/09]. The key new ingredients we use are a type of approximate modulation theory, taking advantage of frequency localisations, and the exploitation of an inhomogeneous wave equation with both a non-local, as well as a local potential term. These terms arise for the main non-perturbative component of the ion density $n$ and can be solved via inversion of a certain Fredholm type operator, as well as by using distorted Fourier methods. Our result relies on a number of numerical non-degeneracy assumptions.

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arXiv2024

A stability theory beyond the co-rotational setting for critical Wave Maps blow up

We exhibit non-equivariant perturbations of the blowup solutions constructed in \cite{KST} for energy critical wave maps into $\mathbb{S}^2$. Our admissible class of perturbations is an open set in some sufficiently smooth topology and vanishes near the light cone. We show that the blowup solutions from \cite{KST} are rigid under such perturbations, including the space-time location of blowup. As blowup is approached, the dynamics agree with the classification obtained in \cite{DJKM}, and all six symmetry parameters converge to limiting values. Compared to the previous work \cite{KMiao} in which the rigidity of the blowup solutions from \cite{KST} under equivariant perturbations was proved, the class of perturbations considered in the present work does not impose any symmetry restrictions. Separation of variables and decomposing into angular Fourier modes leads to an infinite system of coupled nonlinear equations, which we solve for small admissible data. The nonlinear analysis is based on the distorted Fourier transform, associated with an infinite family of Bessel type Schrödinger operators on the half-line indexed by the angular momentum~$n$. A semi-classical WKB-type spectral analysis relative to the parameter $\hbar=\frac{1}{n+1}$ for large $|n|$ allows us to effectively determine the distorted Fourier basis for the entire infinite family. Our linear analysis is based on the global Liouville-Green transform as in the earlier works \cite{CSST, CDST}.

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arXiv2023

Global controllability and stabilization of the wave maps equation from a circle to a sphere

Continuing the investigations started in the recent work [Krieger-Xiang, 2022] on semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$, where {\it semi-global} refers to the $2π$-energy bound, we prove global exact controllability of the same system for $k>1$ and show that the $2π$-energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [Krieger-Xiang, 2022] is sharp. Lastly, the global exact controllability for $\mathbb{S}^1$-target within minimum time is discussed.

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arXiv2022

Boundary stabilization of focusing NLKG near unstable equilibria: radial case

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball of radius L in $\mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $u\equiv 1$ for any dissipative boundary condition $u_t+ au_ν=0, a\in (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $\sqrt{2}L\neq \tan \sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ \frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.

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arXiv2022

Semi-global controllability of a geometric wave equation

We prove the semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2π$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k=1$.

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arXiv2020

Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation

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 procedure and a modified linear-nonlinear decomposition furnished by a delicate "probabilistic" parametrix construction. This is the first global existence result for a geometric wave equation for random initial data at scaling super-critical regularity.

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arXiv2019

Cost for a controlled linear KdV equation

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.

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arXiv2019

Small data global regularity for half-wave maps in $n = 4$ dimensions

We prove that the half-wave maps problem on $\mathbb{R}^{4+1}$ with target $S^2$ is globally well-posed for smooth initial data which are small in the critical $l^1$ based Besov space. This is a formal analogue of the result for wave maps by Tataru.

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arXiv2019

Randomization improved Strichartz estimates and global well-posedness for supercritical data

We introduce a novel data randomisation for the free wave equation which leads to the same range of Strichartz estimates as for radial data, albeit in a non-radial context. We then use these estimates to establish global well-posedness for a wave maps type nonlinear wave equation for certain supercritical data, provided the data are suitably small and randomised.

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arXiv2018

Type II blow up solutions with optimal stability properties for the critical focussing nonlinear wave equation on R^{3+1}

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\mathbb R^{3+1}$ constructed by Krieger-Schlag-Tataru are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $λ(t) = t^{-1-ν}$ is sufficiently close to the self-similar rate, i. e. $ν>0$ is sufficiently small. This result is qualitatively optimal in light of a result by Krieger-Nakanishi-Schlag. The paper builds on the analysis in an earlier paper by the second author.

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arXiv2018

On stability of blow up solutions for the critical co-rotational Wave Maps problem

We show that the finite time blow up solutions for the co-rotational Wave Maps problem constructed in [7,15] are stable under suitably small perturbations within the co-rotational class, provided the scaling parameter $λ(t) = t^{-1-ν}$ is sufficiently close to $t^{-1}$, i. e. the constant $ν$ is sufficiently small and positive. The method of proof is inspired by [3,12], but takes advantage of geometric structures of the Wave Maps problem already used in [1,21] to simplify the analysis. In particular, we heavily exploit that the resonance at zero satisfies a natural first order differential equation.

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arXiv2018

Concentration Compactness for Critical Radial Wave Maps

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 addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig-Merle [6, 7] and a "twisted" Bahouri-Gerard type profile decomposition [1], following the implementation of this strategy by the second author and Schlag [17] for energy critical wave maps into the hyperbolic plane as well as by the last two authors [16] for the energy critical Maxwell-Klein-Gordon equation.

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arXiv2017

On stability of type II blow up for the critical NLW on \R^{3+1}

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\R^{3+1}$ constructed in earlier work by Krieger-Schlag-Tataru are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $λ(t) = t^{-1-ν}$ is sufficiently close to the self-similar rate, i. e. $ν>0$ is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -\partial_t^2 + \partial_r^2 + \frac2r\partial_r +V(λ(t)r) \] for suitable monotone scaling parameters $λ(t)$ and potentials $V(r)$ with a resonance at zero.

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