Asymptotic Analysis - Volume 1, issue 1

Authors: Frank, L.S.

Article Type: Other

DOI: 10.3233/ASY-1988-1101

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 1-1, 1988

Authors: Lions, J.-L.

Article Type: Research Article

Abstract: Homogenization theory allows to “replace” a “complicated” operator with rapidly varying coefficients by a “simple one”. This procedure applies to evolution operators of hyperbolic type or of Petrowsky's type. One can study for these operators the exact controllability problem—in particular, using HUM (Hilbert Uniqueness Method) introduced in [5]. A general program is to study the following question: What happens to exact controllability during homogenization procedure? The present paper is a first (and small) result in this program

DOI: 10.3233/ASY-1988-1102

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 3-11, 1988

Authors: Perthame, B.

Article Type: Research Article

Abstract: We consider the impulse control of a reflected diffusion. It has been proved that the long run average cost for this problem solves the ergodic Quasi-Variational inequality (Q.V.I.) \begin{equation}\left\{\begin{array}{l}\int_{\Omega}\triangledown u_{k}\triangledown (v-u_{k})dx\geq {\int_{\Omega}}(f-\lambda _{k})\cdot(v-u_{k})dx,\\\forall_{v}\in H^{1}(\Omega),\,v\leq k+Mu_{k}\,\mathrm{a.e.;}\quad u_{k}\inH^{1}(\Omega),\,u_{k}\leq k+Mu_{k}\,\mathrm{a.e.,}\quad \int_{\Omega}u_{k}dx=0,\end{array}\right.\end{equation} Mu(x)=infess{c0 (ξ)+u(x+ξ);ξ≥0,x+ξ∈Ω}. We prove uniform bounds in H1 ∩L∞ on uk and we show that, extracting a subsequence if necessary, (uk ,λk ) converge as k→0 to a solution of (1)0 . We also study the uniqueness of (u0 ,λ0 ), and we prove that it is false in general although the complete sequence λk …converges to the maximal λ0 such that (1)0 admits a solution. Show more

DOI: 10.3233/ASY-1988-1103

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 13-21, 1988

Authors: Ghidaglia, J.M. | Temam, R.

Article Type: Research Article

Abstract: It is known that for dissipative evolution equations, the long time behavior of the solutions is generally described by a compact attractor to which all solutions converge, while such a result is not true for conservative equations of Hamiltonian type. In this paper we consider a partly dissipative system corresponding to the equations of slightly compressible fluids and investigate the long time behavior of their solutions. Despite the lack of compactness and smoothing effect for the pressure variable, the existence of a global attractor is shown and its fractal dimension is estimated.

DOI: 10.3233/ASY-1988-1104

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 23-49, 1988

Authors: Frank, L.S. | Heijstek, J.J.

Article Type: Research Article

Abstract: It is shown that the continuation by zero for families of Sobolev type spaces H(s),ε of vectorial order s=(s1 ,s2 ,s3 ) is a continuous linear mapping uniformly with respect to the parameter ε∈(0,1], provided that |s2 |<½,|s2 +s3 |<½ and the boundary ∂U of the open set U where the functions are defined is a C∞ -manifold. This result is needed in the theory of pseudodifferential coercive (elliptic) singular perturbations.

DOI: 10.3233/ASY-1988-1105

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 51-60, 1988

Authors: Wendt, W.D.

Article Type: Research Article

Abstract: The reduced problem is formulated for singularly perturbed pseudodifferential equations with unknown potentials and with general boundary conditions. Several classes of problems of this type have previously been investigated in [2,5,6,7]. The stability theory [5,7] and the asymptotic analysis [6,7] for coercive singularly perturbed problems without potentials is extended to the above class of problems. Systems of singular integral equations on the half line were investigated in [8], using Wiener–Hopf factorization. Wiener–Hopf matrix operators without small parameter have been introduced and investigated in [10,12].

DOI: 10.3233/ASY-1988-1106

Citation: Asymptotic Analysis, vol. 1, no. 1, pp. 61-93, 1988