The one-dimensional motion of compressible viscous and heat-conductive fluid is investigated in the half space. By examining the sign of fluid velocity prescribed on the boundary, initial boundary value problems with Dirichlet type boundary conditions are classified into three cases: impermeable wall problem, inflow problem and outflow problem. In this paper, the asymptotic stability of the rarefaction wave, boundary layer solution, and their combination is established for both the impermeable wall problem and the inflow problem under some smallness conditions. The proof is given by an elementary energy method.

A reflected diffusion process is proposed for modeling of viscoelastic fluids. In order to define the viscoelastic stress tensor, the reflected diffusion process is approximated by Itô diffusions with a penalization factor in the drift term. Convergence of the stress tensor approximation is proved and an expression for the limiting stress tensor in terms of the reflected process itself is provided.

Description of complex materials, in particular complex fluids, involves numerous computational challenges. Today, atomistic descriptions are too expensive from the computational point of view, motivating that this kind of analysis is restricted to extremely small systems. The next scale introduces some simplificative hypotheses, leading to coarse grained descriptions. At this level, Brownian dynamics simulations are usually employed. However, this level of description requires intensive computation resources with significant unfavorable impact on the simulation performances (CPU time). For these reasons sometimes kinetic theory descriptions are preferred. In those de- scriptions, the molecular conformation is described from a probability distribution function (PDF) whose evolution is governed by the Fokker-Planck equation. This approach, despite its mathematical simplicity introduces some computational challenges. First, the distribution function is defined in a multidimensional space, and then the associated partial differential equations must be solved in a multidimensional domain (some times involving thousands dimensions). Secondly, the analysis of transient models needs intensive computation, in particular when the system response under small amplitude oscillations (of high and very high frequency) is concerned. In some of our former works we proposed a technique based on the separated representation of the unknown field able to circumvent the curse of dimensionality. In this paper, we are addressing the second chal- lenging point, the one related to the transient behavior. For this purpose we propose a separated representation of transient models leading to a non-incremental strategy, allowing impressive CPU time savings.

We develop a kinetic theory of the biaxial phases for the bent-core or V-shaped liquid crystal polymer solution (VLCPS). The Brownian motion of the biaxial molecules, biaxial molecular excluded interaction, and the rotary convection are modeled explicitly. Using the hydrodynamic theory, we study the flow-driven orientational dynamics and the corresponding rheological response in selected regimes of plane shear flows, strength of intermolecular interaction, and the bent-angle of the molecule. We identify three steady state biaxial phases and three periodic motions at various regimes of the shear rate and strength of the intermolecular potential. Phase transition sequences among the orientational phases and motions are documented with respect to the strength of the intermolecular potential at fixed bent angles. The effect of the molecular configuration (or the bent- angle) of the V-shaped molecule on mesoscopic phases and their transition sequences is investigated as well.

In this work, we develop a reduced-basis approach for the deficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Itô stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offine, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.

We consider a polymeric fluid model, consisting of the incompressible Navier-Stokes equations coupled to a non-symmetric Fokker-Planck equation. First, the existence of steady states and the exponential convergence to them in relative entropy are proved for the linear Fokker-Planck equation in the Hookean case. The FENE model is also addressed, and the proof of the existence of stationary states and the convergence towards them in suitable weighted norms is given. Then, using the “entropy method” exponential convergence to the steady state is established for the coupled model in the Hookean case under some smallness assumption. The results continue and expand the analysis of B. Jourdain, C. Le Bris, T. Lelièvre and F. Otto, Arch. Rational Mech. Anal., 181, 97-148, 2006 in both the Hookean and the FENE models.

The dynamics of a membrane is a coupled system of a moving elastic surface and an incompressible membrane fluid. The difficulties in analyzing such a system include the nonlinearity of the curved space (geometric nonlinearity), the nonlinearity of the fluid dynamics (fluid nonlinearity), and the coupling to the surface incompressibility. In the two-dimensional case, the fluid vanishes and the system reduces to a coupling of a wave equation and an elliptic equation. Here we prove the local existence and uniqueness of the solution to the system by constructing a suitable discrete scheme and proving the compactness of the discrete solutions. The risk of blowing up due to the geometric nonlinearity is overcome by the bending elasticity.