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2163-2480

## Evolution Equations & Control Theory

December 2014 , Volume 3 , Issue 4

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2014, 3(4): 557-578
doi: 10.3934/eect.2014.3.557

*+*[Abstract](1155)*+*[PDF](1465.6KB)**Abstract:**

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ coupled to a fourth order plate equation, possibly with rotational inertia parameter $\rho >0$. This plate PDE evolves on a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on $\Omega$ is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on $\Omega$. We note that as the Stokes fluid velocity does not vanish on $\Omega$, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational (``inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.

2014, 3(4): 579-594
doi: 10.3934/eect.2014.3.579

*+*[Abstract](1503)*+*[PDF](418.2KB)**Abstract:**

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. The designed feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers. This observation is of a particular interest since it implies that our approach has far more reaching applications, in particular, in data assimilation. Moreover, we emphasize that our scheme treats all kinds of the determining projections, as well as, the various dissipative equations with one unified approach. However, for the sake of simplicity we demonstrate our approach in this paper to a one-dimensional reaction-diffusion equation paradigm.

2014, 3(4): 595-626
doi: 10.3934/eect.2014.3.595

*+*[Abstract](947)*+*[PDF](567.2KB)**Abstract:**

In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.

2014, 3(4): 627-644
doi: 10.3934/eect.2014.3.627

*+*[Abstract](960)*+*[PDF](455.9KB)**Abstract:**

We study a linear problem that arises in the study of dynamic boundaries, in particular in free boundary problems in connection with fluid dynamics. The equations are also very natural and of interest on their own.

2014, 3(4): 645-670
doi: 10.3934/eect.2014.3.645

*+*[Abstract](1141)*+*[PDF](561.2KB)**Abstract:**

The paper is devoted to studying the stochastic nonlinear wave (NLW) equation $$ \partial_t^2 u + \gamma \partial_t u - \triangle u + f(u)=h(x)+\eta(t,x) $$ in a bounded domain $D\subset\mathbb{R}^3$. The equation is supplemented with the Dirichlet boundary condition. Here $f$ is a nonlinear term, $h(x)$ is a function in $H^1_0(D)$ and $\eta(t,x)$ is a non-degenerate white noise. We show that the Markov process associated with the flow $\xi_u(t)=[u(t),\dot u (t)]$ has a unique stationary measure $\mu$, and the law of any solution converges to $\mu$ with exponential rate in the dual-Lipschitz norm.

2014, 3(4): 671-680
doi: 10.3934/eect.2014.3.671

*+*[Abstract](1009)*+*[PDF](370.8KB)**Abstract:**

We prove that, for $-\infty <\alpha\leq 2$, $1 < p <\infty$, the operator $L = (1+|x|^2)^\frac{\alpha}{2}\sum_{i,j=1}^N a_{ij}(x)D_{ij}$ generates an analytic semigroup in $L^p(\mathbb{R}^N)$ when the diffusion coefficients $a_{ij}$ admit a limit at infinity.

2014, 3(4): 681-698
doi: 10.3934/eect.2014.3.681

*+*[Abstract](1013)*+*[PDF](359.8KB)**Abstract:**

Motivated by the fundamental model of a collisionless plasma, the Vlasov-Maxwell (VM) system, we consider a related, nonlinear system of partial differential equations in one space and one momentum dimension. As little is known regarding the regularity properties of solutions to the non-relativistic version of the (VM) equations, we study a simplified system which also lacks relativistic velocity corrections and prove local-in-time existence and uniqueness of classical solutions to the Cauchy problem. For special choices of initial data, global-in-time existence of these solutions is also shown. Finally, we provide an estimate which, independent of the choice of initial data, yields additional global-in-time regularity of the associated field.

2014, 3(4): 699-711
doi: 10.3934/eect.2014.3.699

*+*[Abstract](3995)*+*[PDF](396.7KB)**Abstract:**

In this paper the selfadjointness problem for Schrödinger operators $Au = -div(a\nabla u)+Vu$ in $\mathbb{R}^N$ $(N\in\mathbb{N})$ posed by Kato in [5] and its $L^p$-generalization ($1< p <\infty$) are dealt with. Under $|a(x)|\leq k(1+|x|)^{l+2}$ and $V(x)\geq c|x|^{l}$, the precise lower bounds of $c$ for (essential) selfadjointness in $L^2$ and $m$-sectoriality in $L^p$ of minimal and maximal realizations of $A$ are given. The proof is based on the method in Davies [1,Example 3.5]. This result is a (negative) answer to Kato's selfadjointness problem, and asserts that the lower bounds of $c$ stated in [7,Section 5] for $p=2$ and in [12,Section 3] for general $p$, are precise.

2014, 3(4): 713-738
doi: 10.3934/eect.2014.3.713

*+*[Abstract](1661)*+*[PDF](498.1KB)**Abstract:**

In this paper, we consider the Bresse system with frictional damping terms. We investigated the relationship between the frictional damping terms, the wave speeds of propagation and their influence on the decay rate of the solution. We proved that in many cases the solution enjoys the decay property of regularity-loss type. We introduced a new assumption on the wave speeds that controls the behavior of the solution of the Bresse system. In addition, when the coefficient $l $ goes to zero, we showed that the solution of the Bresse system decays faster than the one of the Timoshenko system. This result seems to be the first one to give the decay rate of the solution of the Bresse system in unbounded domain.

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