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The null controllability of the heat equation is known for decades [19,23,30]. The finite time stabilizability of the one dimensional heat equation was proved by Coron--Nguy\^en [13], while the same question for high dimensional spaces…
Numerical simulations of quantum field theories on lattices serve as a fundamental tool for studying the non-perturbative regime of the theories, where analytic tools often fall short. Challenges arise when one takes the continuum limit or…
These notes are a written version of lectures given in the 2024 Les Houches Summer School on {\it Large deviations and applications}. They are are based on a series of works published over the last 25 years on steady properties of…
In the present Letter we use the Direct Numerical Simulation (DNS) of the Navier-Stokes equation for a two-phase flow (water and air) to study the dynamics of the modulational instability of free surface waves and its contribution to the…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
This paper surveys a few aspects of the global theory of wave equations. This material is structured around the contents of a minicourse given by the second author during the CMI/ETH Summer School on evolution equations during the Summer of…
This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter…
In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target…
In this paper we consider some stabilization problems for the wave equation with switching. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula and some…
The dynamics of a viscous accretion disc subject to a slowly varying warp of large amplitude is considered. Attention is restricted to discs in which self-gravitation is negligible, and to the generic case in which the resonant wave…
We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combine the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow…
A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the…
As described in the classic works of Lee--Stewart and Short--Stewart, the numerical evaluation of linear stability of planar detonation waves is a computationally intensive problem of considerable interest in applications. Reexamining this…
The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical…
The goal of these lecture notes is to introduce the developing research area of gravitational-wave phenomenology. In more concrete terms, they are meant to provide an overview of gravitational-wave sources and an introduction to the…
This paper is devoted to the study of the approximate controllability for a one-dimensional wave equation in domains with moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving.…
We prove global well-posedness for the defocusing cubic wave equation with data in $H^{s} \times H^{s-1}$, $1>s>{13/18}$. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We…
The purpose of this note is to offer a birds-eye view on the history and the state-of-the-art in the research surrounding the Millenium Prize problem for the Navier-Stokes equations, the general problem of singularities in fluid dynamics…
Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in…
This paper focuses on adaptive control of the discrete-time linear quadratic regulator (adaptive LQR). Recent literature has made significant contributions in proving non-asymptotic convergence rates, but existing approaches have a few…