Asymptotic Analysis - Volume 118, issue 3

Authors: Moreno-Mérida, Lourdes | Porzio, Maria Michaela

Article Type: Research Article

Abstract: We study a parabolic equation with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity we prove that there exists a solution which becomes immediately bounded. Moreover, we study the asymptotic behavior of this solution in the autonomous case showing that the constructed solution tends to the associate stationary solution.

Keywords: Regularizing effect, asymptotic behavior, regularity of solutions, linear parabolic equations

DOI: 10.3233/ASY-191558

Citation: Asymptotic Analysis, vol. 118, no. 3, pp. 143-159, 2020

Authors: Medjo, T. Tachim | Tone, C. | Tone, F.

Article Type: Research Article

Abstract: In this article we study a globally modified Allen–Cahn–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed in (Adv. Nonlinear Stud. 6 (2006 ) 411–436) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We discretize these equations in time using the implicit Euler scheme and we prove that the approximate solution is uniformly bounded. We also show that the sequence of the approximate solutions of the globally modified Allen–Cahn–Navier–Stokes system converges, as the parameter N goes to infinity, to the solution of the corresponding discrete …two-phase flow system. Using the uniform stability of the scheme and the theory of the multi-valued attractors, we then prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero. Show more

Keywords: Allen–Cahn–Navier–Stokes, globally modified, global attractor, time-discretization, convergence

DOI: 10.3233/ASY-191559

Citation: Asymptotic Analysis, vol. 118, no. 3, pp. 161-208, 2020

Authors: De Maio, Umberto | Kogut, Peter I. | Manzo, Rosanna

Article Type: Research Article

Abstract: We study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction u acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable class of the weak solutions, we provide …asymptotic analysis of the above mentioned optimal control problem posed in a family of perturbed domains and give the characterization of the limiting behavior of its optimal solutions. Show more

Keywords: Nonlinear elliptic equation, boundary control, boundary oscillation, asymptotic analysis

DOI: 10.3233/ASY-191570

Citation: Asymptotic Analysis, vol. 118, no. 3, pp. 209-234, 2020