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We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the…
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the…
In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various…
In this paper we study the system of two falling balls in continuous time. We modell the system by a suspension flow over a two dimensional, hyperbolic base map. By detailed analysis of the geometry of the system we identify special…
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially…
This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power…
We study a one dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated non-linear…
Recent studies on viscous streaming flows in two dimensions have elucidated the impact of body curvature variations on resulting flow topology and dynamics, with opportunities for microfluidic applications. Following that, we present here a…
This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively.…
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on…
The question on expansion of moving volume inside of a smooth flow of the compressible liquid is under consideration. We find a condition on initial data such that if it holds, then within a finite time either the boundary of the moving…
In this paper, we investigate the volume-prserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space. We prove that the tubeness is preserved along…
We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…
We consider a volume preserving curvature evolution of surfaces in an asymptotically Euclidean initial data set with positive ADM-energy. The speed is given by a nonlinear function of the mean curvature which generalizes the spacetime mean…
We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous…
We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat $3$-manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and…
We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption…
We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing…
We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in…
We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with…