Related papers: Singularity formation for Prandtl's equations
We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.
This paper proposes a new general methodology for finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. The first problem considered is the two-phase Euler vortex sheets…
For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by…
Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical…
We propose a class of spectral singularities that are sensitive to the direction of excitation and are arising in nonlinear systems with broken parity symmetry. These spectral singularities are sensitive to the direction of the incident…
By using Dirichlet form techniques we construct the dynamics of a tagged particle in an infinite particle environment of interacting particles for a large class of interaction potentials. In particular, we can treat interaction potentials…
In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}\o z_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp.…
In this paper, we explore the conformal structure of singularities arising from varying fundamental constants using the method of Penrose diagrams. We employ a specific type of bimetric model featuring two different metrics. One metric…
We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength…
In this paper we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas. In particular, under suitable assumptions we construct a physical solution with a new type of singularities called…
Hypothesis: The impact of droplets is prevalent in numerous applications, and jetting during droplet impact is a critical process controlling the dispersal and transport of liquid. New jetting dynamics are expected in different conditions…
In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new…
We have proposed a method for the dynamic simulation of a collection of self-propelled particles in a viscous Newtonian fluid. We restrict attention to particles whose size and velocity are small enough that the fluid motion is in the…
Actively propelled particles undergoing dissipative collisions are known to develop a state of spatially distributed coherently moving clusters. For densities larger than a characteristic value clusters grow in time and form a stationary…
In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching…
The structure of a finite particle cluster is typically determined by total energy minimization. Here we consider the case where a cluster of soft sphere dipoles becomes active, i.e. when the individual particles exhibit an additional…
We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…
Using the initial peculiar velocity field, we analytically study the hierarchical formation of gravitationally bound objects. The field is smoothed over a scale that corresponds to the mass of a given class of objects. Through the…
We develop the contact singularity theory for singularly-perturbed (or `slow-fast') vector fields of the general form $z' = H(z,\varepsilon)$, $z\in\mathbb{R}^n$ and $\varepsilon\ll 1$. Our main result is the derivation of computable,…
We investigate dynamics of deformable self-propelled particles with a repulsive interaction whose magnitude depends on the relative direction of elongation of a pair of particles. A collective motion of the particles appears in two…