Exponential Behavior of Solutions to Stochastic Integrodifferential Equations with Distributed Delays

In this work, we study the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates a C₀-semigroup.     

Weak solutions to stochastic 3D Navier-Stokes-α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier-Stokes-α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier-Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier-Stokes-α model.     

Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials

Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated fields.     

A remark on the Kazhikhov–Smagulov type model: The vanishing initial density

This work deals with the global existence of weak solutions for a Kazhikhov–Smagulov type system with a density which may or not vanish. Our model is formally equivalent to the physical compressible model with Fick’s law, in contrast to those in previous works. This model may be used for addressing environmental problems such as propagation of pollutants and avalanche modelling. We also explain why this system may be seen as a physical regularization of the standard nonhomogeneous incompressible Navier–Stokes equations and we give an existence result with an initial density less regular but away from the vacuum.     

Calculation of solvation free energies of charged solutes using mixed cluster/continuum models

We derive a consistent approach for predicting the solvation free energies of charged solutes in the presence of implicit and explicit solvents. We find that some published methodologies make systematic errors in the computed free energies because of the incorrect accounting of the standard state corrections for water molecules or water clusters present in the thermodynamic cycle. This problem can be avoided by using the same standard state for each species involved in the reaction under consideration. We analyze two different thermodynamic cycles for calculating the solvation free energies of ionic solutes: (1) the cluster cycle with an n water cluster as a reagent and (2) the monomer cycle with n distinct water molecules as reagents. The use of the cluster cycle gives solvation free energies that are in excellent agreement with the experimental values obtained from studies of ion-water clusters. The mean absolute errors are 0.8 kcal/mol for H(+) and 2.0 kcal/mol for Cu(2+). Conversely, calculations using the monomer cycle lead to mean absolute errors that are >10 kcal/mol for H(+) and >30 kcal/mol for Cu(2+). The presence of hydrogen-bonded clusters of similar size on the left- and right-hand sides of the reaction cycle results in the cancellation of the systematic errors in the calculated free energies. Using the cluster cycle with 1 solvation shell leads to errors of 5 kcal/mol for H(+) (6 waters) and 27 kcal/mol for Cu(2+) (6 waters), whereas using 2 solvation shells leads to accuracies of 2 kcal/mol for Cu(2+) (18 waters) and 1 kcal/mol for H(+) (10 waters).     

A Characterization of Schur Multipliers Between Character-Automorphic Hardy Spaces

We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech.      

Strong solution for a stochastic model of two-dimensional second grade fluids: Existence,uniqueness and asymptotic behavior

We investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of R²${\mathbb{R}}^2$. Global existence and uniqueness of strong probabilistic solution is established. In contrast to previous results on this model we show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem. We also give two results on the long time behavior of the solution. Mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time-independent second grade fluids equations if the deterministic part of the external force does not depend on time. If the deterministic forcing term explicitly depends on time, then the strong probabilistic solution decays exponentially in mean square.     

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of $\Sigma$-convergence for stochastic processes.