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Person# Joachim Krieger

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Energy

In physics, energy () is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light. Energy is a conser

Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) (mɪŋˈkɔːfski,_-ˈkɒf-) combines inertial space and time manifolds (x,y) with a non-inertial reference frame of space and time (x',t')

Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages

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MATH-301: Ordinary differential equations

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.

MATH-478: Dispersive PDEs

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.

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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 $L≠tan\:L$.

2020Joachim 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 $k=1$.

2022We 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