Voigt Equations Research Articles

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.

Bounds and averages of seismic quality factor Q

An elastic two-phase composite, with no restriction on the shape of the two phases, has stiffness bounds given by the Reuss and Voigt equations, and a narrower range determined by the Hashin-Shtrikman bounds. Averages are given by the Voigt-Reuss-Hill, Hashin-Shtrikman, Gassmann, Backus and Wyllie equations. To obtain stiffness bounds and averages, we invoke the correspondence principle to compute the solution of the viscoelastic problem from the corresponding elastic solution. Then, seismic velocities and attenuation are established for the above — physical and heuristic — models which account for general geometrical shapes, unlike the Backus average. The approach is relevant to the seismic characterization of solid composites such as hydrocarbon source rocks.

Upper bounds on the number of determining nodes for 3D Navier–Stokes–Voigt equations

We consider the 3D Navier–Stokes–Voigt equations with periodic boundary conditions. We give upper bounds on the number of determining nodes for instationary, stationary and periodic solutions to the equations. Here the number of determining nodes is estim

Internal stabilization of stochastic 3D Navier–Stokes–Voigt equations with linearly multiplicative Gaussian noise

Abstract We show that the 3D stochastic Navier–Stokes–Voigt equations with linearly multiplicative Gaussian noise can be stabilized in probability by linear internal feedback controllers with support large enough.

Kelvin–Voigt equations perturbed by anisotropic relaxation, diffusion and damping

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.

Stabilization of 3D Navier–Stokes–Voigt equations

Abstract We consider 3D Navier–Stokes–Voigt equations in smooth bounded domains with homogeneous Dirichlet boundary conditions. First, we study the existence and exponential stability of strong stationary solutions to the problem. Then we show that any unstable steady state can be exponentially stabilized by using either an internal feedback control with a support large enough or a multiplicative Itô noise of sufficient intensity.

On the existence and long-time behavior of solutions to stochastic three-dimensional Navier–Stokes–Voigt equations

ABSTRACTWe consider the 3D stochastic Navier–Stokes–Voigt equations in bounded domains with the homogeneous Dirichlet boundary condition and infinite-dimensional Wiener process. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions.

Regularity of uniform attractor for 3D non-autonomous Navier–Stokes–Voigt equation

In this paper, we study the large time behavior for 3D viscoelastic incompressible fluid flow subject to Kelvin–Voigt damping and a time varying external force. The evolution of the dynamic is governed by a 3D non-autonomous Navier–Stokes–Voigt (NSV) equation. We assume that the external force is in the space of translation bounded functions (or its sub-spaces) that requires less compactness than the space of translation compact functions. We formulate the system in the framework of skew product flow. By defining an appropriate energy space that incorporates the Kelvin–Voigt damping, we established the existence of regular strong uniform attractor for the NSV equation in energy space W˜ and provide a description of the general structure of the uniform attractor. Our result improves the H01 regularity of global attractor of the system under study.

Influence of mode conversions in the skull on transcranial focused ultrasound and temperature fields utilizing the wave field separation method: A numerical study**Project supported by the National Natural Science Foundation of China (Grant Nos. 81527901, 11604361, and 91630309).

Transcranial focused ultrasound is a booming noninvasive therapy for brain stimuli. The Kelvin–Voigt equations are employed to calculate the sound field created by focusing a 256-element planar phased array through a monkey skull with the time-reversal method. Mode conversions between compressional and shear waves exist in the skull. Therefore, the wave field separation method is introduced to calculate the contributions of the two waves to the acoustic intensity and the heat source, respectively. The Pennes equation is used to depict the temperature field induced by ultrasound. Five computational models with the same incident angle of 0° and different distances from the focus for the skull and three computational models at different incident angles and the same distance from the focus for the skull are studied. Numerical results indicate that for all computational models, the acoustic intensity at the focus with mode conversions is 12.05% less than that without mode conversions on average. For the temperature rise, this percentage is 12.02%. Besides, an underestimation of both the acoustic intensity and the temperature rise in the skull tends to occur if mode conversions are ignored. However, if the incident angle exceeds 30°, the rules of the over- and under-estimation may be reversed. Moreover, shear waves contribute 20.54% of the acoustic intensity and 20.74% of the temperature rise in the skull on average for all computational models. The percentage of the temperature rise in the skull from shear waves declines with the increase of the duration of the ultrasound.

A note on the Euler–Voigt system in a 3D bounded domain: Propagation of singularities and absence of the boundary layer

We consider the Euler–Voigt equations in a smooth bounded domain as an approximation for the 3D Euler equations. We show that the solutions of the Voigt equations are global, do not smooth out the data, and converge to the solutions of the Euler equations. For these reasons they represent a good model, also for computations of turbulent flows.

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

ABSTRACTAn optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.

Navier–Stokes–Voigt Equations with Memory in 3D Lacking Instantaneous Kinematic Viscosity

We consider a Navier-Stokes-Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. The dissipative character of our model is weaker than the one where hereditary and instantaneous viscosity coexist, previously studied by Gal and Tachim-Medjo. Nevertheless, we prove the existence of a regular exponential attractor of finite fractal dimension under rather sharp assumptions on the memory kernel.

Time Optimal Control of the Unsteady 3D Navier–Stokes–Voigt Equations

We consider an optimal control problem for the 3D Navier–Stokes–Voigt equations in bounded domains, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions, and then establish the first-order necessary optimality conditions and the second-order sufficient optimality conditions.

Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms

This study considers the blow-up in finite time and large-time behavioral properties of solutions to the initial-boundary value problem for generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms:v→t+(v→⋅∇)v→+∇P=div(ϰ|D(v→)|q−2D(v→)t+ν|D(v→)|p−2D(v→))+γ|v→|m−2v→,divv→(x,t)=0,(x,t)∈QT, where D(v→)=12(∇v→+∇v→T) is the rate of the strain tensor, v→(x,t) is the velocity field, P(x,t) is the pressure, ν is the viscosity kinematic coefficient, and ϰ is the viscosity relaxation coefficient. The coefficient γ and the exponents p, m are given constants.Two different cases are considered. In the case where γ<0 (with an absorption term), we prove that the solutions of the associated problem exhibit exponential and power decay, and, in the case where γ>0 (with nonlinear source term), the solutions blow up in a finite time under suitable assumptions on the exponents p, q, m and the coefficients ν, ϰ, γ, with certain initial data.

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $$\alpha >0$$ . Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

Noise Effect on Ice Surface Softening During Friction

The ice surface softening by friction is investigated considering the additive non-correlated fluctuations of the shear strain and stress, and the temperature. The premelting is construed by the Kelvin–Voigt equation for shear strain and by the relaxation equations of Landau–Khalatnikov type for shear stress and temperature. Taking into account the noises in these equations, the Langevin and Fokker–Planck equations are derived. Their analysis is based on the investigation of extrema of the distribution function, i.e., steady-state values of the shear strain using the Stratonovich interpretation. The phase diagrams are constructed, where the noises intensities and thermostat temperature determine the regions of ice, softened ice and their mixture (stick–slip rubbing). We present that domain of ice friction is bounded by relatively small background sliding block temperatures and fluctuation intensities of the stress and temperature. The ice film softens with growth of the stress noise intensity even at small thermostat temperatures. The friction force time series for all rubbing modes are calculated and compared with experimentally observed ones.

Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin–Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed.

On the regularity and convergence of solutions to the 3D Navier–Stokes–Voigt equations

In this paper we consider the 3D Navier–Stokes–Voigt equations with periodic boundary conditions. We first prove the higher-order global regularity, including both Sobolev and Gevrey regularity, of solutions to the Navier–Stokes–Voigt equations. Then we show the convergence of solutions of the 3D Navier–Stokes–Voigt equations to the corresponding strong solution of the limit 3D Navier–Stokes equations on the interval of existence of the latter as the parameter tends to zero.

Stabilizing the long-time behavior of the Navier–Stokes–Voigt equations by fast oscillating-in-time forces

Stabilizing the long-time behavior of the Navier–Stokes–Voigt equations by fast oscillating-in-time forces

Decay rate of solutions to 3D Navier–Stokes–Voigt equations in Hm spaces

In this paper, we first prove the regularity in H m ( R 3 ) of weak solutions to the Navier–Stokes–Voigt equations with initial data in H K ( R 3 ) for all m ≤ K . Then we compute the upper bound of decay rate for these solutions, specifically, we prove that ‖ ∇ m u ( t ) ‖ 2 + ‖ ∇ m + 1 u ( t ) ‖ 2 ≤ c ( 1 + t ) − 3 / 2 − m , for large t , when u 0 ∈ H σ m + 1 ( R 3 ) ∩ L 1 ( R 3 ) , m ∈ N .