We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decreas...

The paper is concerned with linear thermoelastic plate equations in a domain Ω: utt +∆u+∆θ = 0 and θt −∆θ −∆ut = 0 in Ω× (0,∞), subject to Dirichlet boundary condition: u|Γ = Dνu|Γ = θ|Γ = 0 and initial condition: (u, ut, θ)|t=0 = (u0, v0, θ0) ∈W 2 p,D(Ω)×Lp×Lp. Here, Ω is a bounded or exterior domain in R (n ≥ 2). We assume that the boundary Γ of Ω is a C hypersurface and we define W 2 p,D by ...

A map from the initial conditions to the values of the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with n interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed-fo...

This paper contains a description of a general class of city formation models. Individual economic agents have preferences for locations that depend upon the population distribution. A location's attractiveness depends upon some combination of its population and its average distance to other agents. Economic variables enter indirectly. Taking this broad perspective leads to a deeper understandi...

In this paper, we consider analytic initial conditions with nite energy, whose complex spatial continuation is a superposition of a smooth background ow and a singular eld. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in nite time. This research was partly supported by the ...

In this paper we investigate the epochs in which the Universe started accelerating and when it began to become dark energy–dominated (i.e., the dynamics of the expansion of the Universe dominated by the dark energy). We provide analytic expressions to calculate the redshifts of these epochs as a function of density parameters. Moreover, we review and discuss cosmological models with a dark ener...

We consider the homogenization of the wave equation with high frequency initial conditions propagating in a medium with highly oscillatory random coefficients. By appropriate mixing assumptions on the random medium, we obtain an error estimate between the exact wave solution and the homogenized wave solution in the energy norm. This allows us to consider the limiting behavior of the energy dens...

have global solutions. Results of Saari [9] and Sperling [II] imply, however, that the basic problem in proving this classical result would be the proof that, for almost every initial condition, x(t)” = XL, xj(t>” cannot become infinite in finite time. Interestingly enough, while Kato’s result “solves” the dynamical existence question in the quantum case, it says nothing about the question of x...

We prove that the set of initial conditions so that gradient descent converges to strict saddle points has (Lebesgue) measure zero, even for non-isolated critical points, answering an open question in [1].

In this paper we revisit the problem of the solution to Balitsky-Kovchegov equation deeply in the saturation domain. We find that solution has the form given in ref. [23] but it depends on variable z̄ = ln(xQs)+Const and the value of Const is calculated in this paper. We propose the solution for full BFKL kernel at large z̄ in the entire kinematic region that satisfies the McLerran-Venugopalan-ty...