In [16], Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocaldrift variant of the 1D Burgers equation does not have global classical solutions ∂tθ + u ∂xθ = 0, u = Hθ, where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to t...

We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximat...

A nonlinear, nonlocal dielectric continuum model, called the nonlocal modified PoissonBoltzmann equation (NMPBE), has been proposed to reflect the spatial-frequency dependence of dielectric permittivity in the calculation of electrostatics of ionic-solvated biomolecules. However, its analysis is difficult due to its definition involving Dirac delta distributions for modeling point charges, expo...

The recent appearance of nonlocal methods is examined in the light of traditional continuum mechanics. A comparison of nonlocal approaches in the fields of solid and fluid mechanics reveals that no consistent definition of a nonlocal theory has been used. We suggest a definition based on the violation of the principle of local action in continuum mechanics. From the consideration of the implica...