# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations & Control Theory

March 2018 , Volume 7 , Issue 1

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2018, 7(1): 1-31 doi: 10.3934/eect.2018001 +[Abstract](1099) +[HTML](393) +[PDF](612.96KB)
Abstract:

In this paper, the controllability for a thermoelastic plate problem with a rotational inertia parameter is considered under two scenarios. In the first case, we prove the exact and approximate controllability when the controls act in the whole domain. In the second case, we prove the interior approximate controllability when the controls act only on a subset of the domain. The distributed controls are determined explicitly by the physical constants of the plate in the first case, while this is no longer possible in the second case as the relation (79) is no longer valid. In this case, we propose an approximation of the control function with an error that tends to zero. By means of a powerful and systematic approach based on spectral analysis, we improve some already existing results on the optimal rate of the exponential decay and on the analyticity of the associated semigroup.

2018, 7(1): 33-52 doi: 10.3934/eect.2018002 +[Abstract](722) +[HTML](369) +[PDF](463.43KB)
Abstract:

We present a continuous data assimilation algorithm for three-dimensional viscous simplified Bardina turbulence model, based on the fact that dissipative dynamical systems possess finite degrees of freedom. We construct an approximating solution of simplified Barbina model through an interpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the finite-dimensional spatial resolution of collected measurements sufficient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.

2018, 7(1): 53-60 doi: 10.3934/eect.2018003 +[Abstract](885) +[HTML](343) +[PDF](323.9KB)
Abstract:

Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space \begin{document} $C([0, T]; H^{s}(\Bbb{R}))$ \end{document} \begin{document} $(s<0)$ \end{document} to the system (1) which is \begin{document} $L^2(\Bbb{R})$ \end{document} scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.

2018, 7(1): 61-77 doi: 10.3934/eect.2018004 +[Abstract](718) +[HTML](328) +[PDF](435.73KB)
Abstract:

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.

2018, 7(1): 79-93 doi: 10.3934/eect.2018005 +[Abstract](1219) +[HTML](373) +[PDF](421.72KB)
Abstract:

The paper deals with the age-dependent model which is a generalization of the classical Lotka-Volterra model. Age structure of both species, predators and preys is concerned. The model is based on the system of partial differential and integro-differential equations. We study the existence and uniqueness of the solution for the considered population problem. The stability problem for trivial stationary solution of the model is also proved.

2018, 7(1): 95-116 doi: 10.3934/eect.2018006 +[Abstract](813) +[HTML](309) +[PDF](499.0KB)
Abstract:

In this paper we study a distributed control problem for a phase-field system of conserved type with a possibly singular potential. We mainly handle two cases: the case of a viscous Cahn-Hilliard type dynamics for the phase variable in case of a logarithmic-type potential with bounded domain and the case of a standard Cahn-Hilliard equation in case of a regular potential with unbounded domain, like the classical double-well potential, for example. Necessary first order conditions of optimality are derived under natural assumptions on the data.

2018, 7(1): 117-152 doi: 10.3934/eect.2018007 +[Abstract](724) +[HTML](305) +[PDF](583.24KB)
Abstract:

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the \begin{document}$L_p$\end{document} in time \begin{document}$L_q$\end{document} in space maximal regularity class, (\begin{document}$2 < p < ∞$\end{document}, \begin{document}$N < q < ∞$\end{document}, and \begin{document}$2/p + N/q < 1$\end{document}), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the \begin{document}$L_p$\end{document}-\begin{document}$L_q$\end{document} framework for the linearized equations with the help of maximal \begin{document}$L_p$\end{document}-\begin{document}$L_q$\end{document} regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

2018, 7(1): 153-182 doi: 10.3934/eect.2018008 +[Abstract](838) +[HTML](398) +[PDF](626.99KB)
Abstract:

We consider a heat-plate interaction model where the 2-dimensional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each components evolves. In this paper, we apply "low", physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point \begin{document}$\lambda = -1/ρ$\end{document} in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.

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