Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients

This is the post-print version of the Article. The official published version can be found at the links below - Copyright @ 2011 Birkhäuser Boston.Some segregated systems of direct localized boundary-domain integral equations (LBDIEs) associated with several transmission problems for scalar PDEs with variable coefficients are formulated and analyzed for a bounded domain composed of two subdomains with a coefficient jump over the interface. The main results established in the paper are the LBDIE equivalence to the original transmission problems and the invertibility of the corresponding localized boundary-domain integral operators in corresponding Sobolev spaces function spaces.This research was supported by the EPSRC grant EP/H020497/1: ”Mathematical analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” and partly by the Georgian Technical University grant in the case of the third author

Analysis of segregated boundary-domain integral equations for mixed variable-coefficient BVPs in exterior domains

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2011 Birkhäuser Boston.Some direct segregated systems of boundary–domain integral equations (LBDIEs) associated with the mixed boundary value problems for scalar PDEs with variable coefficients in exterior domains are formulated and analyzed in the paper. The LBDIE equivalence to the original boundary value problems and the invertibility of the corresponding boundary–domain integral operators are proved in weighted Sobolev spaces suitable for exterior domains. This extends the results obtained by the authors for interior domains in non-weighted Sobolev spaces.The work was supported by the grant EP/H020497/1 ”Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients” of the EPSRC, UK

Generalized Smoluchowski equation with correlation between clusters

In this paper we compute new reaction rates of the Smoluchowski equation which takes into account correlations. The new rate K = KMF + KC is the sum of two terms. The first term is the known Smoluchowski rate with the mean-field approximation. The second takes into account a correlation between clusters. For this purpose we introduce the average path of a cluster. We relate the length of this path to the reaction rate of the Smoluchowski equation. We solve the implicit dependence between the average path and the density of clusters. We show that this correlation length is the same for all clusters. Our result depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di + Dj)(rj + ri)d-2, which vanishes for d = 1 and is valid up to logarithmic correction for d = 2, is the usual rate found with the Smoluchowski model without correlation (where ri is the radius and Di is the diffusion constant of the cluster). We compute a new rate: the correlation rate K_{i,j}^{C} (D_i+D_j)(r_j+r_i)^{d-1}M{\big(\frac{d-1}{d_f}}\big) is valid for d \leq 1(where M(\alpha) = \sum+\infty i=1i\alphaNi is the moment of the density of clusters and df is the fractal dimension of the cluster). The result is valid for a large class of diffusion processes and mass radius relations. This approach confirms some analytical solutions in d 1 found with other methods. We also show Monte Carlo simulations which illustrate some exact new solvable models

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement

Nonlinear Instability for Nonhomogeneous Incompressible Viscous Fluids

We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, this is different from the previous known results.Comment: 24 page