Browsing by Author "Antontsev, S. N."
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- Cauchy problem for the Navier–Stokes–Voigt model governing nonhomogeneous flowsPublication . Antontsev, S. N.; Oliveira, H. B. deThe Navier-Stokes-Voigt model that governs flows with non-constant density of incompressible fluids with elastic properties is considered in the whole space domain R-d and in the entire time interval. If d is an element of{2,3,4}, we prove the existence of weak solutions (velocity, density and pressure) to the associated Cauchy problem. We also analyse some issues of regularity of the weak solutions to the considered problem and the large time behavior in special unbounded domains.
- Evolution problems of Navier-Stokes type with anisotropic diffusionPublication . Antontsev, S. N.; de Oliveira, H. B.In this work, we consider the evolutive problem for the incompressible Navier-Stokes equations with a general diffusion which can be fully anisotropic. The existence of weak solutions is proved for the associated initial problem supplemented with no-slip boundary conditions. We prove also the properties of extinction in a finite time, exponential time decay and power time decay. With this respect, we consider the important case of a forces fields with possible different behavior in distinct directions. Perturbations of the asymptotically stable equilibrium are established as well.
- Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and dampingPublication . Antontsev, S. N.; de Oliveira, H. B.; Khompysh, Kh.In this work, we consider the nonlinear initial-boundary value problem posed by the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with fully anisotropic diffusion, relaxation and damping. Moreover, we assume that the momentum equation is perturbed by a damping term which, depending on whether its signal is positive or negative, may account for the presence of a source or a sink within the system. In the particular case of considering this problem with a linear and isotropic relaxation term, we prove the existence of global and local weak solutions for the associated initial-boundary value problem supplemented with no-slip boundary conditions. When the damping term describes a sink, we establish the conditions for the polynomial time decay or for the exponential time decay of these solutions.
- Kelvin-Voigt equations perturbed by anisotropic relaxation, diffusion and dampingPublication . Antontsev, S. N.; de Oliveira, H.B.; Khompysh, Kh.The purpose of this work is the analysis of the existence and uniqueness of weak solutions to a Kelvin-Voigt problem wherein the viscous and relaxation parts of the stress tensor are given by distinct power-laws. We assume that the viscous and relaxation terms may be fully anisotropic and that the momentum equation is perturbed by a damping term which may also be fully anisotropic. In the particular case of considering this problem with a linear and isotropic relaxation term, we prove the existence of global and local weak solutions. The uniqueness of weak solutions is established in this case as well. For the full anisotropic problem, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients must interact in order to be established global and local in time a priori estimates. (C) 2019 Elsevier Inc. All rights reserved.
- The classical Kelvin-Voigt problem for incompressible fluids with unknown non-constant density: existence, uniqueness and regularityPublication . Antontsev, S. N.; Oliveira, H. B. de; Khompysh, KhThe classical Kelvin-Voigt equations for incompressible fluids with non-constant density are investigated in this work. To the associated initial-value problem endowed with zero Dirichlet conditions on the assumed Lipschitz-continuous boundary, we prove the existence of weak solutions: velocity and density. We also prove the existence of a unique pressure. These results are valid for d is an element of {2, 3, 4}. In particular, if d is an element of {2, 3}, the regularity of the velocity and density is improved so that their uniqueness can be shown. In particular, the dependence of the regularity of the solutions on the smoothness of the given data of the problem is established.
- The Oberbeck-Boussinesq problem modified by a thermo-absorption termPublication . Antontsev, S. N.; de Oliveira, H. B.We consider the Oberbeck-Boussinesq problem with an extra coupling, establishing a suitable relation between the velocity and the temperature. Our model involves a system of equations given by the transient Navier-Stokes equations modified by introducing the thermo-absorption term. The model involves also the transient temperature equation with nonlinear diffusion. For the obtained problem, we prove the existence of weak solutions for any N >= 2 and its uniqueness if N = 2. Then, considering a low range of temperature, but upper than the phase changing one, we study several properties related with vanishing in time of the velocity component of the weak solutions. First, assuming the buoyancy forces field extinct after a finite time, we prove the velocity component will extinct in a later finite time, provided the thermo-absorption term is sublinear. In this case, considering a suitable buoyancy forces field which vanishes at some instant of time, we prove the velocity component extinct at the same instant. We prove also that for non-zero buoyancy forces, but decaying at a power time rate, the velocity component decay at analogous power time rates, provided the thermo-absorption term is superlinear. At last, we prove that for a general non-zero bounded buoyancy force, the velocity component exponentially decay in time whether the thermo-absorption term is sub or superlinear. (C) 2011 Elsevier Inc. All rights reserved.