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Evolution Equations & Control Theory

June 2020 , Volume 9 , Issue 2

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On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations
Manil T. Mohan
2020, 9(2): 301-339 doi: 10.3934/eect.2020007 +[Abstract](1136) +[HTML](492) +[PDF](485.25KB)
Abstract:

In this work, we consider the three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt fluids in bounded and unbounded domains. We investigate the global solvability results, asymptotic behavior and also address some control problems of such viscoelastic fluid flow equations with "fading memory" and "memory of length \begin{document}$ \tau $\end{document}". A local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique are used to obtain global solvability results. Since we are not using compactness arguments in the proofs, the global solvability results are also valid in unbounded domains like Poincaré domains. We also remark that using an \begin{document}$ m $\end{document}-accretive quantization of the linear and nonlinear operators, one can establish the existence and uniqueness of strong solutions for the Navier-Stokes-Voigt equations and avoid the tedious Galerkin approximation scheme. We examine the asymptotic behavior of the stationary solutions and also establish the exponential stability results. Finally, under suitable assumptions on the Galerkin basis, we consider the controlled Galerkin approximated 3D Kelvin-Voigt fluid flow equations. Using the Hilbert uniqueness method combined with Schauder's fixed point theorem, the exact controllability of the finite dimensional Galerkin approximated system is established.

Regularized solution for a biharmonic equation with discrete data
Tran Ngoc Thach, Nguyen Huy Tuan and Donal O'Regan
2020, 9(2): 341-358 doi: 10.3934/eect.2020008 +[Abstract](750) +[HTML](369) +[PDF](830.24KB)
Abstract:

In this work, we focus on the Cauchy problem for the biharmonic equation associated with random data. In general, the problem is severely ill-posed in the sense of Hadamard, i.e, the solution does not depend continuously on the data. To regularize the instable solution of the problem, we apply a nonparametric regression associated with the Fourier truncation method. Also we will present a convergence result.

Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping
Jun-Ren Luo and Ti-Jun Xiao
2020, 9(2): 359-373 doi: 10.3934/eect.2020009 +[Abstract](745) +[HTML](319) +[PDF](299.02KB)
Abstract:

The paper is concerned with the Cauchy problem for second order hyperbolic evolution equations with nonlinear source in a Hilbert space, under the effect of nonlinear time-dependent damping. With the help of the method of weighted energy integral, we obtain explicit decay rate estimates for the solutions of the equation in terms of the damping coefficient and two nonlinear exponents. Specialized to the case of linear, time-independent damping, we recover the corresponding decay rates originally obtained in [3] via a different way. Moreover, examples are given to show how to apply our abstract results to concrete problems concerning damped wave equations, integro-differential damped equations, as well as damped plate equations.

Stochastic porous media equations with divergence Itô noise
Ioana Ciotir
2020, 9(2): 375-398 doi: 10.3934/eect.2020010 +[Abstract](637) +[HTML](310) +[PDF](365.63KB)
Abstract:

We study the existence and uniqueness of solution to stochastic porous media equations with divergence Itô noise in infinite dimensions. The first result prove existence of a stochastic strong solution and it is essentially based on the non-local character of the noise. The second result proves existence of at least one martingale solution for the critical case corresponding to the Dirac distribution.

Null-controllability properties of a fractional wave equation with a memory term
Umberto Biccari and Mahamadi Warma
2020, 9(2): 399-430 doi: 10.3934/eect.2020011 +[Abstract](749) +[HTML](339) +[PDF](461.26KB)
Abstract:

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset \begin{document}$ \omega(t) $\end{document} which is moving with a constant velocity \begin{document}$ c\in\mathbb{R} $\end{document}, the main result of the paper states that the equation is null controllable in a sufficiently large time \begin{document}$ T $\end{document} and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.

Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation
Prasanta Kumar Barik
2020, 9(2): 431-446 doi: 10.3934/eect.2020012 +[Abstract](610) +[HTML](321) +[PDF](321.76KB)
Abstract:

In this paper we study the continuous coagulation and multiple fragmentation equation for the mean-field description of a system of particles taking into account the combined effect of the coagulation and the fragmentation processes in which a system of particles growing by successive mergers to form a bigger one and a larger particle splits into a finite number of smaller pieces. We demonstrate the global existence of mass-conserving weak solutions for a wide class of coagulation rate, selection rate and breakage function. Here, both the breakage function and the coagulation rate may have algebraic singularity on both the coordinate axes. The proof of the existence result is based on a weak \begin{document}$ L^1 $\end{document} compactness method for two different suitable approximations to the original problem, namely, the conservative and non-conservative approximations. Moreover, the mass-conservation property of solutions is established for both approximations.

Moving and oblique observations of beams and plates
Philippe Jaming and Vilmos Komornik
2020, 9(2): 447-468 doi: 10.3934/eect.2020013 +[Abstract](538) +[HTML](311) +[PDF](371.59KB)
Abstract:

We study the observability of the one-dimensional Schrödinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at the end of the paper.

Robust attractors for a Kirchhoff-Boussinesq type equation
Zhijian Yang, Na Feng and Yanan Li
2020, 9(2): 469-486 doi: 10.3934/eect.2020020 +[Abstract](430) +[HTML](219) +[PDF](436.96KB)
Abstract:

The paper studies the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoff-Boussinesq type equation: \begin{document}$ u_{tt}-\Delta u_{t}+\Delta^{2} u = div\Big\{\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Big\}+\Delta g(u)+f(x,t) $\end{document}. We show that when the growth exponent \begin{document}$ p $\end{document} of the nonlinearity \begin{document}$ g(u) $\end{document} is up to the critical range: \begin{document}$ 1\leq p\leq p^*\equiv\frac{N+2}{(N-2)^{+}} $\end{document}, (ⅰ) the IBVP of the equation is well-posed, and its solution has additionally global regularity when \begin{document}$ t>\tau $\end{document}; (ⅱ) the related dynamical process \begin{document}$ \{U_f(t,\tau)\} $\end{document} has a pullback attractor; (ⅲ) in particular, when \begin{document}$ 1\leq p< p^* $\end{document}, the process \begin{document}$ \{U_f(t,\tau)\} $\end{document} has a family of pullback exponential attractors, which is stable with respect to the perturbation \begin{document}$ f\in \Sigma $\end{document} (the sign space).

On a Kirchhoff wave model with nonlocal nonlinear damping
Vando Narciso
2020, 9(2): 487-508 doi: 10.3934/eect.2020021 +[Abstract](457) +[HTML](227) +[PDF](403.23KB)
Abstract:

This paper is concerned with the well-posedness as well as the asymptotic behavior of solutions for a quasi-linear Kirchhoff wave model with nonlocal nonlinear damping term \begin{document}$ \sigma\left(\int_{\Omega}|\nabla u|^2\,dx\right )g(u_t), $\end{document} where \begin{document}$ \sigma $\end{document} and \begin{document}$ g $\end{document} are nonlinear functions under proper conditions. The analysis of such a damping term is presented for this kind of Kirchhoff models and consists the main novelty in the present work.

Uniform stabilization of a wave equation with partial Dirichlet delayed control
Xiaorui Wang and Genqi Xu
2020, 9(2): 509-533 doi: 10.3934/eect.2020022 +[Abstract](480) +[HTML](254) +[PDF](424.45KB)
Abstract:

In this paper, we consider the uniform stabilization of some high-dimensional wave equations with partial Dirichlet delayed control. Herein we design a parameterization feedback controller to stabilize the system. This is a new approach of controller design which overcomes the difficulty in stability analysis of the closed-loop system. The detailed procedure is as follows: At first we rewrite the system with partial Dirichlet delayed control into an equivalence cascaded system of a transport equation and a wave equation, and then we construct an exponentially stable target system; Further, we give the form of the parameterization feedback controller. To stabilize the system under consideration, we choose some appropriate kernel functions and define a bounded inverse linear transformation such that the closed-loop system is equivalent to the target system. Finally, we obtain the stability of closed-loop system by the stability of target system.

Null controllability for a heat equation with dynamic boundary conditions and drift terms
Abdelaziz Khoutaibi and Lahcen Maniar
2020, 9(2): 535-559 doi: 10.3934/eect.2020023 +[Abstract](550) +[HTML](235) +[PDF](454.46KB)
Abstract:

We consider the heat equation in a bounded domain of \begin{document}$ \mathbb{R}^N $\end{document} with distributed control (supported on a small open subset) subject to dynamic boundary conditions of surface diffusion type and involving drift terms on the bulk and on the boundary. We prove that the system is null controllable at any time. The result is based on new Carleman estimates for this type of boundary conditions.

On a backward problem for two-dimensional time fractional wave equation with discrete random data
Nguyen Huy Tuan, Tran Ngoc Thach and Yong Zhou
2020, 9(2): 561-579 doi: 10.3934/eect.2020024 +[Abstract](500) +[HTML](209) +[PDF](420.19KB)
Abstract:

This paper is concerned with a backward problem for a two- dimensional time fractional wave equation with discrete noise. In general, this problem is ill-posed, therefore the trigonometric method in nonparametric regression associated with Fourier truncation method is proposed to solve the problem. We also give some error estimates and convergence rates between the regularized solution and the sought solution under some assumptions.

2018  Impact Factor: 1.048

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