Related papers: Quantitative Mode Stability for the Wave Equation …
We study an inverse resonance problem for the scalar wave equation on the Kerr-de Sitter family. In a compact subextremal slow-rotation regime and at a fixed overtone index, high-frequency quasinormal modes admit semiclassical quantization…
In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form…
We study dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of thispaper is to establish uniform energy bounds and local energy decay for such backgrounds.
In this article we prove optimal pointwise bounds for solutions to the semilinear wave equation with integer powers $p\geq 3$ on Kerr backgrounds with small angular momentum and small initial data.
Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for…
We prove a stability threshold theorem for 2D Navier-Stokes on three unbounded domains: the whole plane $\mathbb{R} \times \mathbb{R}$, the half plane $\mathbb{R} \times [0,\infty)$ with Navier boundary conditions, and the infinite channel…
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…
This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_g{\psi} = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general {\psi} or (II2) |a| < M, {\psi}…
We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary…
This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $\Phi$ defined from a fixed…
In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability…
We review the latest progress and properties of the families of bright and dark one-dimensional periodic waves propagating in saturable Kerr-type and quadratic nonlinear media. We show how saturation of the nonlinear response results in…
We study scalar waves on subextremal Kerr-de Sitter spacetimes in a compact slow-rotation regime and at a fixed overtone index. Working initially at a fixed cosmological constant $\Lambda>0$ and uniformly for $(M,a)$ in a compact…
We show the stability of Kerr-de Sitter black holes, in the full subextremal range, as solutions of the vacuum Einstein equation with a positive cosmological constant under the assumption that mode stability holds for these spacetimes. The…
We present a fully discrete finite element method for the interior null controllability problem subject to the wave equation. For the numerical scheme, piece-wise affine continuous elements in space and finite differences in time are…
We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_g{\psi} = 0 on Kerr exterior backgrounds. We begin…
The analytic proof of mode stability of the Kerr black hole was provided by Whiting. In his proof, the construction of a conserved quantity for unstable mode was crucial. We extend the method of the analysis for the Kerr-de Sitter geometry.…
This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e $|a|/m\ll 1$, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide…
We consider the Cauchy problem for the scalar wave equation in the Kerr geometry for smooth initial data supported outside the event horizon. We prove that the solutions decay in time in L^\infty_loc. The proof is based on a representation…
We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class…