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We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density \rho_i(t), embedded in a background of uniform exterior density \rho_e(t). In both regions, the fluid is…
A stationary stable solution of the Stokes equations for three identical heavy solid spheres falling in a vertical plane is found. It has no analog in the point-particle approximation. Three spheres aligned horizontally at equal distances…
This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of nonzero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear…
The fluid ball conjecture states that a static perfect fluid space-time is spherically symmetric. In this paper we construct a Robinson's divergence formula for the static perfect fluid space-time. Inspired by this conjecture, a rigidity…
We study the general properties of fluid spheres satisfying the heuristic assumption that their areas and proper radius are equal (the Euclidean condition). Dissipative and non-dissipative models are considered. In the latter case, all…
In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state…
The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the…
We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both…
We study the following open problem, suggested by Barker and Larman. Let $K$ and $L$ be convex bodies in $\mathbb R^n$ ($n\ge 2$) that contain a Euclidean ball $B$ in their interiors. If $\mathrm{vol}_{n-1}(K\cap H) =…
It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow…
We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume…
In this paper we establish the uniformity property of a simplified Ericksen-Leslie system modelling the hydrodynamics of nematic liquid crystals on the two dimensional unit sphere $S^2$, namely the uniform convergence in $L^2$ to a steady…
A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a diffuse bounded Radon measure with distinguished sign. In this paper we are interested in…
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the…
Arnold pointed out that the Euler equation of incompressible ideal hydrodynamics describes geodesics on the group of volume-preserving diffeomorphisms. A simple analogue is the Euler equation for a rigid body, which is the geodesic equation…
We study some properties of static spherically symmetric elastic bodies in general relativity using both analytical and numerical tools. The materials considered belong to the class of John elastic materials and reduce to perfect fluids…
Exact solutions of a classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite strip bounded by the free surface (from above) and (from the…
In this paper we consider the equiform motion of a sphere in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant.…
Let $K$ be a convex body in $\mathbb{R}^{3}$. We denote the volume of $K$ by $Vol(K)$ and the diameter of $K$ by $Diam(K).$ In this paper we prove that there exists a linear bijection $T:\mathbb{R}^{3}\to \mathbb{R}^{3}$ such that…
We prove that if $K$ is a symmetric and isotropic convex body in $\mathbb{R}^n$, then $$\int_K\langle x,u\rangle^2\,dx\int_{K^\circ}\langle x,u\rangle^2\,dx\leq \left(\int_{B_2^n}\langle x,u\rangle^2\,dx\right)^2,\qquad\forall…