Models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study nonuniqueness of steady states in annular domains in , d ≥ 2.
The parabolic equations driven by linearly multiplicative Gaussian noise are stabilizable in probability by linear feedback controllers with support in a suitably chosen open subset of the domain. This procedure extends to Navier − Stokes equations with multiplicative noise. The exact controllability is also discussed.
We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our...
In this paper we study a linear population dynamics model. In this model, the birth process is described by a nonlocal term and the initial distribution is unknown. The aim of this paper is to use a controllability result of the adjoint system for the computation of the density of individuals at some time .
The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator () in the rectangle or with the Kolmogorov-type operator () in the rectangle , under an additive control supported in an open subset of the space domain.We prove that the Grushin-type equation is null controllable in any positive time for and that there is no time for which it is null controllable for ....
We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.
In this paper we introduce a coupled systems of kinetic equations for the linearized Carleman model. We then study the existence theory and the asymptotic behaviour of the resulting coupled problem. In order to solve the coupled problem we propose to use the time marching algorithm. We then develop a convergence theory for the resulting algorithm. Numerical results confirming the theory are then presented.
The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays. First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model.5 (2001) 537–572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue. The classical multi-fluid method consists then in...
We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing...
In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...
In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...