Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces
We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence.
Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach
We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂w/∂t = -(-Δ) α/2 w + w 1+β in the critical dimension d = α/β. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as t → ∞. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of α-Laplacians with possibly different parameters a.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...