Related papers: Singularity formation on a fluid interface during …
The relativistic membrane equation can be rewritten as a first order hyperbolic system. Making use of the characteristic decomposition method, a new blow-up theorem is established. As an application, it demonstrates the formation of…
We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable…
The dynamics of small, yet heavy, identical particles in turbulence exhibits singularities, called caustics, that lead to large fluctuations in the spatial particle-number density, and in collision velocities. For large particle, inertia…
We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak…
The interface formation model is applied to describe the initial stages of the coalescence of two liquid drops in the presence of a viscous ambient fluid whose dynamics is fully accounted for. Our focus is on understanding (a) how this…
Many dynamic pipe flow simulator tools are capable of predicting the onset of hydrodynamic flow instability through detailed simulation. These instabilities provide a natural mechanism for flow regime transition. The quality and reliability…
A recent laboratory experiment (Blaauwgeers et al. 2003) suggests that a Kelvin-Helmholtz (KH) instability at the interface between two superfluids, one rotating and anisotropic, the other stationary and isotropic, may trigger sudden…
An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal…
An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts…
Ambient interstellar material may become entrained in outflows from massive stars as a result of shear flow instabilities. We study the linear theory of the Kelvin - Helmholtz instability, the simplest example of shear flow instability, in…
The potential flow of two-dimensional ideal incompressible fluid with a free surface is studied. Using the theory of conformal mappings and Hamiltonian formalism allows us to derive exact equations of surface evolution. Simple form of the…
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…
At the superfluid-solid 4He interface there exist crystallization waves having much in common with gravitational-capillary waves at the interface between two normal fluids. The Rayleigh-Taylor instability is an instability of the interface…
We study the formation of singularities for the Euler-Alignment system with influence function $\psi=\frac{k_\alpha}{|x|^\alpha}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of…
The minimalist approach in the study of perturbations in fluid dynamics and magnetohydrodynamics involves describing their evolution in the linear regime using a single first-order ordinary differential equation, dubbed principal equation.…
We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain…
The translational motion of a solid sphere near a deformable fluid interface is studied in the low Reynolds number regime. In this problem, the fluid flow driven by the sphere is dynamically coupled the instantaneous conformation of the…
For the water waves equations, the existence of splat singularities has been shown in [3], i.e., the interface self-intersects along an arc in finite time. The aim of this paper is to show the absence of splat singularities for the…
We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a…
The {\it two-fold singularity} has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that…