Liquid walls and interfaces in arbitrary directions stabilized by vibrations

Gravity shapes liquids and plays a crucial role in their internal balance. Creating new equilibrium configurations irrespective of the presence of a gravitational field is challenging with applications on Earth as well as in zero-gravity environments. Vibrations are known to alter the shape of liquid interfaces and also to change internal dynamics and stability in depth. Here, we show that vibrations can also create an “artificial gravity” in any direction. We demonstrate that a liquid can maintain an inclined interface when shaken in an arbitrary direction. A necessary condition for the equilibrium to occur is the existence of a velocity gradient determined by dynamical boundary conditions. However, the no-slip boundary condition and incompressibility can perturb the required velocity profile, leading to a destabilization of the equilibrium. We show that liquid layers provide a solution, and liquid walls of several centimeters in height can thus be stabilized. We show that the buoyancy equilibrium is not affected by the forcing.

Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

<p style='text-indent:20px;'>The aim of this paper is to give global nonexistence and blow–up results for the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &amp;\text{in $(0,\infty)\times\Omega$,}\\ u = 0 &amp;\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &amp;\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) &amp; \text{in $\overline{\Omega}$,} \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open <inline-formula><tex-math id="M2">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> subset of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Gamma = \partial\Omega $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ (\Gamma_0,\Gamma_1) $\end{document}</tex-math></inline-formula> is a partition of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \Gamma_1\not = \emptyset $\end{document}</tex-math></inline-formula> being relatively open in <inline-formula><tex-math id="M9">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \Delta_\Gamma $\end{document}</tex-math></inline-formula> denotes the Laplace–Beltrami operator on <inline-formula><tex-math id="M11">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> is the outward normal to <inline-formula><tex-math id="M13">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, and the terms <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ Q $\end{document}</tex-math></inline-formula> represent nonlinear damping terms, while <inline-formula><tex-math id="M16">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ g $\end{document}</tex-math></inline-formula> are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.</p>

The Stabilization Of The Problem Of Transmission Of The Wave Equation with dynamical control

In this paper, we address exponential stablization of transmission problem of the wave equation with dynamical boundary conditions. Using the classical energy method and multiplier technique, we prove that the energy of system exponentially decays.

From Steklov to Neumann via homogenisation

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.