The structure of singularities in nonlocal transport equations (original) (raw)

Finite Time Singularities in Transport Equations with Nonlocal Velocities and Fluxes

2007

Navier-Stokes and Euler equations, when written in terms of vorticity, contain nonlinear convective terms involving singular integral (nonlocal) operators of the vorticity itself. This fact suggests the analysis of the role played by nonlocal velocities and fluxes in the formation of singularities. We consider the following one-dimensional analogs of Euler equations, namely: 1) θt + ((Hθ)θ)x = 0 2) θt − (Hθ)θx = 0 with H being the Hilbert transform of, and their viscous versions obtained by adding a dissipative term at the right hand side of the equations. We prove that the inviscid equations do develop singularities in finite time while the solutions of the viscous versions do exist for all time. We also discuss connections of these problems with finite time singularities in Birkhoff-Rott equation.

On Singularity Formation of a Nonlinear Nonlocal System

Archive for Rational Mechanics and Analysis, 2011

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.

Global existence, singularities and ill-posedness for a nonlocal flux

Advances in Mathematics, 2008

In this paper we study a one-dimensional model equation with a nonlocal flux given by the Hilbert transform that is related with the complex inviscid Burgers equation. This equation arises in different contexts to characterize nonlocal and nonlinear behaviors. We show global existence, local existence, blow-up in finite time and ill-posedness depending on the sign of the initial data for classical solutions.

The regularity of the boundary of vortex patches for some non-linear transport equations

2021

We prove the persistence of boundary smoothness of vortex patches for nonlinear transport equations with velocity fields given by convolution of the density with a kernel of the form L(∇N), where N is the fundamental solution of the laplacian in Rn and L a linear mapping of Rn into itself. This allows the velocity field to have non-trivial divergence. The quasi-geostrophic equation in R3 and the Cauchy kernel in the plane are examples. AMS 2010 Mathematics Subject Classification: 31A15 (primary); 49K20 (secondary).

Periodic solutions for a 1D-model with nonlocal velocity via mass transport

This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing, via numerics and rigorous analysis, that this model presents singular behavior of solutions. For instance, they can blow up by forming mass-concentration. We develop a global well-posedness theory for periodic measure initial data that allows, in particular, to analyze how the model evolves from those singularities. Our results are based on periodic mass transport theory and the abstract gradient flow theory in metric spaces developed by Ambrosio et al. (2005). A viscous version of the model is also analyzed and inviscid limit properties are obtained.

Finite time singularities in a 1D model of the quasi-geostrophic equation

Advances in Mathematics, 2005

In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data.

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

2011

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction

Finite time singularities in a class of hydrodynamic models

Physical review, 2001

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L ∼ k α |v k | 2 d 3 k in 3D Fourier representation, where α is a constant, 0 < α < 1. Unlike the case α = 0 (the usual Eulerian hydrodynamics), a finite value of α results in a finite energy for a singular, frozenin vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t * − t) 1/(2−α) , where t * is the singularity time.

On the large time behavior of the solutions of a nonlocal ordinary di?erential equation with mass conservation

2015

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.