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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.
- 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.
- Generalized Navier-Stokes equations with nonlinear anisotropic viscosityPublication . de Oliveira, H. B.The purpose of this work is to study the generalized Navier-Stokes equations with nonlinear viscosity that, in addition, can be fully anisotropic. Existence of very weak solutions is proved for the associated initial and boundary-value problem, supplemented with no-slip boundary conditions. We show that our existence result is optimal in some directions provided there is some compensation in the remaining directions. A particular simplification of the problem studied here, reduces to the Navier-Stokes equations with (linear) anisotropic viscosity used to model either the turbulence or the Ekman layer in atmospheric and oceanic fluid flows.
- 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.
- On a one-equation turbulent model with feedbacksPublication . de Oliveira, H. B.; Paiva, A.; Pinelas, S.; Došlá, Z.; Došlý, O.; Kloeden, P. E.A one-equation turbulent model is derived in this work on the basis of the approach used for the k-epsilon model. The novelty of the model consists in the consideration of a general feedback forces field in the momentum equation and a rather general turbulent dissipation function in the equation for the turbulent kinetic energy. For the steady-state associated boundary value problem, we prove the uniqueness of weak solutions under monotonous conditions on the feedbacks and smallness conditions on the solutions to the problem. We also discuss the existence of weak solutions and issues related with the higher integrability of the solutions gradients.
- Existence and large time behavior for generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluidsPublication . Antontsev, S N; de Oliveira, H. B.; Khompysh, KhGeneralized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluids are considered in this work. We assume that, in the momentum equation, the diffusion and relaxation terms are described by two distinct power-laws. Moreover, we assume that the momentum equation is perturbed by an extra 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. For the associated initial-boundary value problem, we study the existence of weak solutions as well as the large time behavior of the solutions.
- Parabolic reaction-diffusion systems with nonlocal coupled diffusivity termsPublication . Ferreira, Jorge; de Oliveira, Hermenegildo BorgesIn this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several bow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.
- Generalized Kelvin-Voigt equations for nonhomogeneous andincompressible fluidsPublication . Antontsev, Stanislav N.; de Oliveira, H.B.; Khompysh, KhonatbekIn this work, we consider the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with the diffusion and relaxation terms described by two distinct power-laws. Moreover, we assume that the momentum equation is perturbed by an extra 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. For the associated initial-boundary value problem, we study the existence of weak solutions as well as the large-time behavior of the solutions. In the case the extra term is a sink, we prove the global existence of weak solutions and we establish the conditions for the polynomial time decay and for the exponential time decay of these solutions. If the extra term is a source, we show how the exponents of nonlinearity must interact to ensure the local existence of weak solutions.
- Existence of weak solutions for the generalized Navier-Stokes equations with dampingPublication . de Oliveira, H. B.In this work we consider the generalized Navier-Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any and any sigma > 1, where q is the exponent of the diffusion term and sigma is the exponent which characterizes the damping term.
- Kelvin-Voigt equations with anisotropic diffusion, relaxation and damping: Blow-up and large time behaviorPublication . Antontsev, S.; Oliveira, H. B. de; Khompysh, KhA nonlinear initial and boundary-value problem for the Kelvin-Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473(2) (2019) 1122-1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.