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2163-2480

## Evolution Equations & Control Theory

December 2012 , Volume 1 , Issue 2

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2012, 1(2): 235-250
doi: 10.3934/eect.2012.1.235

*+*[Abstract](2685)*+*[PDF](362.9KB)**Abstract:**

In this paper we consider a class of partially observed semilinear stochastic evolution equations on infinite dimensional Hilbert spaces subject to measurement uncertainty. We prove the existence of optimal feedback control law from a class of operator valued functions furnished with the Tychonoff product topology. This is an extension of our previous results for uncertain systems governed by deterministic differential equations on Banach spaces. Also we present a result on existence of optimal feedback control law for a class of uncertain stochastic systems modeled by differential inclusions.

2012, 1(2): 251-270
doi: 10.3934/eect.2012.1.251

*+*[Abstract](2114)*+*[PDF](509.4KB)**Abstract:**

We analyze an abstract version of the evolution system ruling the dynamics of a memory relaxation of a type III thermoelastic extensible beam or Berger plate occupying a volume $\Omega$ \begin{equation} \begin{cases} u_{tt}-ωΔ u_{tt}+Δ^2 u-[b +||\nabla u\|^2_{L^2(\Omega)}]\Delta u+Δ α_t=g\\ α_{tt}-Δ α-∫_0^\infty u(s)Δ[α(t)-α(t-s)]d s-Δ u_t=0 \end{cases} \end{equation} subject to hinged boundary conditions for $u$ and to the Dirichlet boundary condition for $\alpha$, where the dissipation is entirely contributed by the convolution term in the second equation. The study of the asymptotic properties of the related solution semigroup is addressed.

2012, 1(2): 271-296
doi: 10.3934/eect.2012.1.271

*+*[Abstract](2486)*+*[PDF](499.5KB)**Abstract:**

A Carleman estimate for some first-order elliptic systems is established. This estimate is extended to elliptic boundary value problems provided the boundary condition satisfies a Lopatinskii-type requirement. Based on these estimates conservative hyperbolic systems of first order can be stabilized with a logarithmic decay rate by introducing a localized interior dissipation. The support of the dissipative term does not need to satisfy a geometric condition.

2012, 1(2): 297-314
doi: 10.3934/eect.2012.1.297

*+*[Abstract](2356)*+*[PDF](404.9KB)**Abstract:**

We provide a new simple proof of local-in-time existence of regular solutions to the Euler equation on a domain with a free moving boundary and without surface tension in 2 space dimensions. We prove the existence under the condition that the initial velocity belongs to the Sobolev space $H^{2.5+δ}$ where $\delta>0$ is arbitrary.

2012, 1(2): 315-336
doi: 10.3934/eect.2012.1.315

*+*[Abstract](1998)*+*[PDF](469.3KB)**Abstract:**

This paper is motivated by a piezoelectric/piezomagnetic phenomenon in the presence of thermal effects. The evolution system we consider is linear and coupled between one hyperbolic , two elliptic and one parabolic equation. We show the equivalence between ``the exponential decay of the total energy of our system" and an ``observability inequality for an anisotropic elastic wave system" assuming that a geometric condition is satisfied. This geometric condition ensures that the elliptic operator associated with the mechanical part of our system has no eigenfunctions $ \Psi $ such that the divergence

*div*(Λ $ \Psi $ ) = 0 in $\Omega$ where $ Λ $ denotes the thermal expansion tensor.

2012, 1(2): 337-354
doi: 10.3934/eect.2012.1.337

*+*[Abstract](3247)*+*[PDF](462.6KB)**Abstract:**

So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [21].

2012, 1(2): 355-392
doi: 10.3934/eect.2012.1.355

*+*[Abstract](3976)*+*[PDF](583.6KB)**Abstract:**

In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Itô-Lévy noise in bounded and unbounded domains in $ \mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.

2012, 1(2): 393-429
doi: 10.3934/eect.2012.1.393

*+*[Abstract](2141)*+*[PDF](579.6KB)**Abstract:**

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin [9]. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.

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