American Institute of Mathematical Sciences

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Evolution Equations & Control Theory

March 2016 , Volume 5 , Issue 1

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2016, 5(1): 1-36 doi: 10.3934/eect.2016.5.1 +[Abstract](1916) +[PDF](698.9KB)
Abstract:
We discuss the notion of the well productivity index (PI) for the generalized Forchheimer flow of fluid through porous media. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, [5,3,4]. Here we generalize our results from [4] in case of arbitrary order of the nonlinearity of the flow. In case of the compressible gas flow the mathematical model of the PI is studied for the first time. In contrast to slightly compressible fluid the PI stays almost'' constant for a long period of time, but then it blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The greater'' are the initial reserves, the larger is this critical value. We present numerical and theoretical results for the time asymptotic of the PI and its stability with respect to the initial data.
2016, 5(1): 37-59 doi: 10.3934/eect.2016.5.37 +[Abstract](1695) +[PDF](491.7KB)
Abstract:
In this paper we study the behavior of the energy and the $L^{2}$ norm of solutions of the wave equation with localized linear damping in exterior domain. Let $u$ be a solution of the wave system with initial data $\left( u_{0},u_{1}\right)$. We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:
1. If $(u_{0},u_{1})$ belong to $H_{0}^{1}( \Omega) \times L^{2}( \Omega ) ,$ then the total energy $E_{u}(t) \leq C_{0}(1+t) ^{-1}I_{0}$ and $\Vert u(t) \Vert _{L^{2}}^{2}\leq C_{0}I_{0},$ where \begin{equation*} I_{0}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}. \end{equation*}    2. If the initial data $\left( u_{0},u_{1}\right)$ belong to $H_{0}^{1}\left( \Omega \right) \times L^{2}\left( \Omega \right)$ and verifies \begin{equation*} \left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}<+\infty , \end{equation*} then the total energy $E_{u}\left( t\right) \leq C_{2}\left( 1+t\right) ^{-2}I_{1}$ and $\left\Vert u\left( t\right) \right\Vert _{L^{2}}^{2} \leq C_{2} \left( 1+t\right) ^{-1}I_{1},$ where \begin{equation*} I_{1}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}+\left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}^{2} \end{equation*} and \begin{equation*} d\left( x\right) =\left\{ \begin{array}{lc} \left\vert x\right\vert & d\geq 3, \\ \left\vert x\right\vert \ln \left( B\left\vert x\right\vert \right) & d=2, \end{array} \right. . \end{equation*} with $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.
2016, 5(1): 61-103 doi: 10.3934/eect.2016.5.61 +[Abstract](1743) +[PDF](734.0KB)
Abstract:
We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedness and regularity of bounded solutions, we also establish the existence of finite-dimensional global attractors and also derive basic conditions for blow-up.
2016, 5(1): 105-134 doi: 10.3934/eect.2016.5.105 +[Abstract](2264) +[PDF](599.9KB)
Abstract:
We consider the stochastic linear quadratic optimal control problem for state equations of the Itô-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.
2016, 5(1): 135-145 doi: 10.3934/eect.2016.5.135 +[Abstract](1618) +[PDF](336.5KB)
Abstract:
We study the Cauchy problem of the relativistic Nordström-Vlasov system. Under some additional conditions, total energy for weak solutions with BV scalar field are shown to be conserved.
2016, 5(1): 147-184 doi: 10.3934/eect.2016.5.147 +[Abstract](1804) +[PDF](747.3KB)
Abstract:
In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations.
2016, 5(1): 185-199 doi: 10.3934/eect.2016.5.185 +[Abstract](1430) +[PDF](458.7KB)
Abstract:
We study the free dynamic operator $\mathcal{A}$ which arises in the study of a heat-viscoelastic structure model with highly coupled boundary conditions at the interface between the heat domain and the contiguous structure domain. We use Baiocchi's characterization on the interpolation of subspaces defined by a constrained map [1], [16,p 96] to identify a relevant subspace $V_0$ of both $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^∗)^{\frac{1}{2}})$, which is sufficient to determine the optimal regularity of the interface (boundary) $\to$ interior map $\mathcal{A}^{-1} \mathcal{B}_N$ from the interface to the energy space. Here, $\mathcal{B}_N$ is the (boundary) control operator acting at the interface in the Neumann boundary conditions.

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