Formation of singularities for a transport equation with nonlocal velocity

Abstract

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

Wellposedness and regularity of solutions of an aggregation equation

“…Aggregation equations and other equations similar to (1.1) with fractional diffusion have been studied in the literature (see [7], [10], [9] and [24]). While the case γ = 2 corresponds to the usual diffusion, the regime 0 < γ < 2 corresponds to the so-called anomalous diffusion which in probabilistic terms has a connection with stochastic equations driven by Lévy α-stable flights 1 .…”

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

“…While the singularity formation for the SQG equation remains open, it was shown in Córdoba et al (2005) that the 1d model can develop a singularity from smooth initial data in finite time.…”

“…Important modifications of the original model of Constantin et al (1985) were proposed by De Gregorio (1990Gregorio ( , 1996 and somewhat differently motivated work of Córdoba et al (2005). All these works include modeling of two important features of incompressible flow: (i) the vorticity transport (either as a scalar, as in 2d Euler, or as a vector field, as in 3d Euler, with the vector field transport also covering the vortex stretching) and (ii) the Biot-Savart law, which expresses the velocity field which transports the vorticity in terms of the vorticity itself.…”

“…finite-time blowup when a < 0, Castro and Córdoba (2010) Córdoba et al (2005), SQG analogy u = Hω ω t + uω x = 0 finite-time blow-up from smooth data HL-model, Hou and Luo (2013), 2d Boussinesq/3d axi-symmetric Euler analogy u x = Hω ω t + uω x = θ x θ t + uθ x = 0 finite time blow-up from smooth data, the main new result of this paper CKY-model, Choi et al (2015), simplified HL-model…”

One-dimensional model equations for hyperbolic fluid flow

“…From the regularity standpoint, the α-patch model is between 1D Euler u x = Hω and the Córdoba-Córdoba-Fontelos model u = Hω (see [5,11]), which is an analogue of the SQG equation. These two models differ however from a geometric perspective, since the symmetry properties of the Biot-Savart laws are different.…”