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Joachim Krieger

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https://people.epfl.ch/joachim.krieger?lang=fr

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Le cours donne une introduction à la théorie des EDO, y compris existence de solutions locales/globales, comportement asymptotique, étude de la stabilité de points stationnaires et applications, en particulier aux systèmes dynamiques et en biologie.

This course will give an introduction to some aspects of nonlinear dispersive partial differential equations. These are time evolution problems that arise in many contexts in physics, such as quantum mechanics, electrodynamics, fluid motion and relativity.

Cours enseignés par cette personne

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Publications associées (54)

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Publications associées (54)

Semi-global controllability of a geometric wave equation

Joachim Krieger, Shengquan Xiang

We prove the semi-global controllability and stabilization of the

(1+1)-

dimensional wave maps equation with spatial domain 𝕊1 and target

𝕊k

. 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

2022

Chargement

A stability theory beyond the co-rotational setting for critical wave maps blow up

Joachim Krieger, Shuang Miao

We exhibit non-equivariant perturbations of the blowup solutions constructed in [18] 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 [18] are rigid under such perturbations, including the space-time location of blowup. As blowup is approached, the dynamics agrees with the classification obtained in [7], and all six symmetry parameters converge to limiting values. Compared to the previous work [16] in which the rigidity of the blowup solutions from [18] 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 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 work [4, 5].

2020

Chargement

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in

\mathbb{R}^3

. First we show that the system is linearly unstable near the static solution

u\equiv1

for any dissipative boundary condition

u_t+au_ν=0,a∈(0,1)

. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially with rate less than

\frac{\sqrt 2}{2L} log \frac{1+a}{1-a}

, which is sharp, provided that the radius of the ball

L

satisfies

L≠tan\:L

2020

Chargement

Afficher plus

Semi-global controllability of a geometric wave equation

Joachim Krieger, Shengquan Xiang

We prove the semi-global controllability and stabilization of the

(1+1)-

dimensional wave maps equation with spatial domain 𝕊1 and target

𝕊k

. First we show that damping stabilizes the system when the energy is strictly below the threshold

2π

k=1

2022

A stability theory beyond the co-rotational setting for critical wave maps blow up

Joachim Krieger, Shuang Miao

We exhibit non-equivariant perturbations of the blowup solutions constructed in [18] for energy critical wave maps into

\mathbb{S}^2

n

. A semi-classical WKB-type spectral analysis relative 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 work [4, 5].

2020

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in

\mathbb{R}^3

. First we show that the system is linearly unstable near the static solution

u\equiv1

for any dissipative boundary condition

u_t+au_ν=0,a∈(0,1)

. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially with rate less than

\frac{\sqrt 2}{2L} log \frac{1+a}{1-a}

, which is sharp, provided that the radius of the ball

L

satisfies

L≠tan\:L

2020