The state of the art of the mathematical and numerical analysis of multi-scale models of complex fluids is reviewed. Issues addressed include well-posedness of the models, convergence analysis of the numerical methods, and the structure of stationary solutions of the Doi-Onsager equation.

This paper deals with a system of two equations which describes heatless adsorption of a gaseous mixture with two species. Using the hyperbolicity property of the system with respect to the $(x, t)$ variables, that is with $x$ as the evolution variable, we find all the entropy-flux pairs. Making use of a Godunov-type scheme we obtain an existence result of a weak entropy solution satisfying some BV regularity.

In this work an optimal control problem for a stationary quantum drift diffusion (QDD) model is analyzed. This QDD model contains four space-dependent observables: The non-negative particle density of electrons, the electrostatic potential, the quantum quasi-Fermi potential and the current density. The goal is to optimize the shape of quantum barriers in a quantum diode. Existence of optimal solutions is proved. Moreover, first-order necessary optimality conditions are derived.

We study the $2-D$ Smoluchowski equation governing the evolution of orientational distribution of rodlike molecules under an imposed weak shear. We first recover the well-known isotropic-to-nematic phase transition result: in the absence of flow the isotropic-nematic phase transition occurs at $U =2$ where $U$ is the normalized polymer concentration, representing the intensity of the Maier-Saupe interaction potential. Then we show that in the presence of an imposed weak shear there is a threshold $(U\sb 0\approx 2:41144646)$ for $U$: When $U < U\sb 0$, steady state solution exists; otherwise there is no steady state. Furthermore, we carry out multi-scale asymptotic anlaysis to study the slow time evolution driven by the weak shear. It is revealed that, to the leading order, the order parameter of the orientational distribution is invariant with respect to time whereas the angular velocity of the director is position-dependent. When $U < U\sb 0$, the director of the orientational distribution converges to a stable steady state position; when $U > U\sb 0$, the angular velocity of the director is always positive and the orientational distribution will not reach a steady state. Finally, the effect of weak shear on the phase diagram is investigated. It is found that the phase relation under weak shear can be obtained from the pure nematic phase relation through a simple algebraic transformation.

A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-upwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.

Discrete reflection-transmission acoustic models are introduced and analysed regarding their underlying physical properties. Namely, phenomena related to multiple scattering as encountered in the underlying continuous model. Moreover, the discrete models are designed so that computational experiments can be performed effciently. Each discrete model is expressed through its corresponding reflection-transmission matrix establishing a connection with lattice models encountered in the Physics literature. Their connections with the continuous acoustic model are discussed in detail. In particular we show how a Goupillaud medium can arise from a stable discretization of a more general random medium. Related physical phenomena are studied computationally. In particular reflection-transmission properties of waves in a rapidly varying random medim, which are valid over long propagation distances. By using a long computational domain, in a regime where we have Anderson localization, the energy of an incoherently scattered signal is entirely reflected back and, by time-reversal, completely recompressed into the smooth initial data. Experiments are also performed in a regime where separation of scales fails to hold. Another new result is the time-reversed refocusing, in reflection, of a wave train in the form of a bit stream. The bit stream is scrambled by reflection and unscrambled through time reversal. The robustness of the time-reversed refocusing phenomenon is outstanding.

We consider the Suliciu model which is a relaxation approximation of the $p$-system. In the case of the Dirichlet boundary condition we prove that the local smooth solution of the $p$-system is the zero limit of the Suliciu model solutions.

We construct degenerate quadratic models for the Euler equations which distinguish the stabilizing effect of an antisymmetry in the Lie structure of the Euler equations when this antisymmetry is accounted for versus when it is not. We derive a matrix, depending only on the mesh size, $N$, and the 2-wave strengths, whose powers propagate the 1- and 3-waves up to time $t$. We give sharp estimates for the magnitude of the largest eigenvalue of this matrix and conclude that the solution will decay for initial data of arbitrarily large total variation, of order the 4-th root of $N$ in the limit $N$ approaches infinity, when the antisymmetry is accounted for, and only for suffciently small total variation when it is not

We introduce a set of new interfacial energies for approximating the Euler number of level surfaces in the phase field (diffuse-interface) representation. These new formulae have simpler forms than those studied earlier in Q. Du, C. Liu and X. Wang, Retrieving topological information for phase field models, and do not contain higher order derivatives of the phase field function. Theoretical justifications are provided via formal asymptotic analysis, and practical validations are performed through numerical experiments. Relaxation and renormalization schemes are also developed to improve the robustness of the new energy functionals.