Voigt Equations Research Articles

Existence for fractional evolutionary inclusions involving nonlinear weakly continuous operators with applications

In this paper, our primary purpose is to settle the existence for a type of fractional-order evolutionary inclusions. By utilizing the Rothe method and the surjective result of weakly continuous operators, we show that the solution of the evolutionary equation exists under several kinds hypotheses on the data. Finally, the obtained results are applied to verify the existence of solutions for a time-fractional evolutionary hemivariational inequality, a time-fractional Navier–Stokes–Voigt equation and a quasistatic friction contact problem.

Decay characterization of solutions to incompressible Navier–Stokes–Voigt equations

Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r ∗ of the initial data in H 1 ( R 3 ). Motivated by this work, we focus on characterizing the large-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting methods. In particular, for the case − n 2 < r ∗ ⩽ 1, we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case.

Stabilization estimates for the Brinkman–Forchheimer–Kelvin–Voigt equation backward in time

The final value value problem for the Brinkman–Forchheimer–Kelvin–Voigt equations is analysed for quadratic and cubic types of Forchheimer nonlinearity. The main term in the Forchheimer equations is allowed to be fully anisotropic. It is shown that the solution depends continuously on the final data provided the solution satisfies an a priori bound in L^3. The technique employed avoids the use of a specialist method for an improperly posed problem such as logarithmic convexity.

Time Optimal Feedback Control for 3D Navier–Stokes-Voigt Equations

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study the existence result of a time optimal control for the feedback control systems. Lastly, asymmetrical Clarke’s subdifferential inclusions and asymmetrical 3D Navier–Stokes–Voigt differential variational inequalities are given to explain our main results.

Scattering of Love wave from an interface crack in a viscoelastic waveguide layer bonded to a piezoelectric substrate: an analytical estimate and numerical validation

This study uses an analytical method to examine the scattering of a Love wave from an interfacial crack in a thin lossy media (viscoelastic material) that serves as a waveguide layer attached to an unbounded loss-less media (piezoelectric material). The viscoelastic material is modeled using the Kelvin–Voigt equation. Phase velocity and attenuation have been calculated by equating the real and imaginary parts of the dispersion relation to zero, respectively, due to the complex wave number of viscoelastic material. Using the integral transform approach and the dislocation density function, the scattering field close to the crack tip is determined, resulting in a Cauchy singular integral equation of the first type. It has been transformed into a system of linear algebraic equations using Gauss-Chebyshev numerical integration. It has been investigated how frequency, waveguide layer thickness, and viscoelastic material loss factor affect phase velocity and Love wave attenuation. The mode-III dynamic stress intensity factor at the left and right crack-tip is investigated for various thicknesses of the waveguide layer and loss factor of viscoelastic material. The outcomes were verified using numerical data from the COMSOL Multiphysics 5.6 FEA software. The findings are fundamental and could be used in the design of particular sensors.

Existence for a class of time-fractional evolutionary equations with applications involving weakly continuous operator

The aim of this paper is to deal with a new class of fractional evolutionary equations involving the time-fractional order integral and nonlinear weakly continuous operators. Exploiting the Rothe method and using a surjectivity result for weakly continuous operators, the solvability for the problem is established. The result is applied to prove the existence of solutions to time-fractional nonstationary incompressible Navier–Stokes equation and Navier–Stokes–Voigt equation.

The Navier–Stokes–Voigt equations with position-dependent slip boundary conditions

The Navier–Stokes–Voigt equations with position-dependent slip boundary conditions

Inverse problem for integro-differential Kelvin–Voigt equations

Abstract In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind.

A phase separation model for binary fluids with hereditary viscosity

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier–Stokes–Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn–Hilliard equation with Flory–Huggins potential. The resulting system is subject to no‐slip condition for the (volume averaged) fluid velocity and to no‐flux boundary conditions for the order parameter and the chemical potential . We first establish the well‐posedness of the initial and boundary value problem. Then, taking advantage of an Ekman‐type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two.

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

In this paper, we consider an inverse problem of finding a coefficient of right hand side of the following system of Kelvin–Voigt equations perturbed by an isotropic diffusion and damping terms The damping term γ|v|m − 2v in the momentum equation realizes an absorbtion (sink) if γ ≤ 0, and a source if γ > 0. We show how the exponents p, m, the coefficients ν, ϰ, γ, the dimension of the space d, and data of the problem should interact each other for the existence of weak solutions to the problem. We also establish the conditions for uniqueness of the solutions to this problem.

An inverse problem for generalized Kelvin–Voigt equation with p-Laplacian and damping term

In this paper, we consider the nonlinear inverse problem for generalized Kelvin–Voigt equations with the p-Laplace diffusion and damping term, describing the motion of incompressible viscous fluids. We assume that the damping term in the momentum equation depends on whether its signal is positive or negative, which may realizes the presence of a source or a sink within the system. The investigated inverse problem consists of finding a coefficient f(t) of the right-hand side of the momentum equation, a vector of velocity field v, and a pressure π. An additional information on a solution of the inverse problem is given as integral overdetermination condition. Under several assumptions on the exponents p, m, the coefficients μ, κ, γ, the dimension of the space d, and specified initial data, we prove the existence and uniqueness of the weak solution of the problem.

The classical Kelvin–Voigt problem for incompressible fluids with unknown non-constant density: existence, uniqueness and regularity

The 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 ∈ {2, 3, 4}. In particular, if d ∈ {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.

Continuous dependence for the Brinkman–Darcy–Kelvin–Voigt equations backward in time

We show that the solution to the Brinkman–Darcy–Kelvin–Voigt equations backward in time depends Hölder continuously upon the final data. A logarithmic convexity technique is employed, and uniqueness of the solution is simultaneously achieved.

Nonlinear Dynamic Analysis of the Cutting Process of a Nonextensible Composite Boring Bar

A nonlinear dynamic analysis of the cutting process of a nonextensible composite cutting bar is presented. The cutting bar is simplified as a cantilever with plane bending. The nonlinearity is mainly originated from the nonextensible assumption, and the material of cutting bar is assumed to be viscoelastic composite, which is described by the Kelvin–Voigt equation. The motion equation of nonlinear chatter of the cutting system is derived based on the Hamilton principle. The partial differential equation of motion is discretized using the Galerkin method to obtain a 1-dof nonlinear ordinary differential equation in a generalized coordinate system. The steady forced response of the cutting system under periodically varying cutting force is approximately solved by the multiscale method. Meanwhile, the effects of parameters such as the geometry of the cutting bar (including length and diameter), damping, the cutting coefficient, the cutting depth, the number of the cutting teeth, the amplitude of the cutting force, and the ply angle on nonlinear lobes and primary resonance curves during the cutting process are investigated using numerical calculations. The results demonstrate that the critical cutting depth is inversely proportional to the aspect ratio of the cutting bar and the cutting force coefficient. Meanwhile, the chatter stability in the milling process can be significantly enhanced by increasing the structural damping. The peak of the primary resonance curve is bent toward the right side. Due to the cubic nonlinearity in the cutting system, primary resonance curves show the characteristics of typical Duffing’s vibrator with hard spring, and jump and multivalue regions appear.

Decay characterization of the solutions to the Navier–Stokes–Voigt equations with damping

The Navier–Stokes–Voigt equations are a regularization of the Navier–Stokes equations. Sharing some asymptotic and statistical properties, they have been used in direct numerical simulations of the latter. In this article, we characterize the decay rate of the solutions to the Navier–Stokes–Voigt equations with damping. Applying the Fourier splitting method, we prove the H1 decay of weak solutions for α > 0, β > 2, and μ > ν2.

Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier–Stokes–Voigt equations

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier–Stokes–Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order $$O(\sqrt{\tau }+h)$$ , where $$\tau $$ and h are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.

Feedback control for non-stationary 3D Navier–Stokes–Voigt equations

The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.

Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

Kelvin–Voigt equations with anisotropic diffusion, relaxation and damping: Blow-up and large time behavior

A 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.