Mathematical Models and Methods in Applied Sciences

In this paper we consider a simplified model for fluid–solid interaction in one space dimension. The fluid is assumed to be governed by the viscous Burgers equation. It is coupled with a finite number of solid masses in the form of point particles, which share the velocity of the fluid and are accelerated by the jump in velocity gradient of the fluid on both sides, which replaces here the standard pressure jump of Navier–Stokes models. We prove global existence and uniqueness of solutions. This requires proving that the solid particles never collide in finite time, a key fact that follows from suitable a priori estimates together with uniqueness results for ordinary differential equations.

We also describe the asymptotic behavior of solutions as t → ∞, extending previous results established for a single solid mass. The evolution of the relative position of the particles is examined in terms of the strength of the convection term.

The possible 2D analogues of these results in the context of Navier–Stokes equations are open problems.

This paper is devoted to study the global existence of a solution of bounded variation to the initial value problem for a system of conservation laws with artificial viscosity. The method of finite difference schemes of implicit type is used. We prove that the difference approximations converge to a weak solution of the problem. Moreover, this weak solution is really a classical solution according to the Oleĭnik's argument.

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

Diffusion approximations are obtained for space inhomogeneous linear transport models with reflection boundary conditions. The collision kernel is not required to satisfy any balance condition and the scattering kernel on the boundary is general enough to include all examples of boundary conditions known to the authors (with conservation of the number of particles) and, in addition, to model the Debye sheath. The mathematical approach does not rely on Hilbert expansions, but rather on martingale and stochastic averaging techniques.

This work is devoted to the modeling of space charge dominated particle beams in the paraxial approximation with several types of external focusing fields (uniform, periodic and alternating gradient). A solid mathematical background for numerical beam simulation is established. The Kapchinsky–Vladimirsky (KV) distribution which can be matched exactly or numerically to the focusing channel is first studied. Then the matched KV beam is used to approximately match arbitrary beams. Moreover, Waterbag and Maxwell–Boltzmann beams are studied to give analytical solution for code validation. Finally, numerical simulations are presented in different configurations.