Abstract The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L 2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1+ t)-3 4 in L 2-norm, while the individual momentum of the charged particles converges at the optimal rate (1+ t)-1 4 which is slower than the rate (1+ t)-3 4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that almost counteracts the influence of the electric field so that the total density and

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L 1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.