Analytic solution for heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermophysical effects

An analytic approximate solution is presented for the natural convective dissipative heat transfer of an incompressible, third grade, non-Newtonian fluid flowing past an infinite porous plate embedded in a Darcy–Forchheimer porous medium. The mathematical model is developed in an $(x\text{,}y)$ coordinate system. Using a set of transformations, the momentum equation is rendered one-dimensional and a partly linearized heat conservation equation is derived. The viscoelastic formulation presented by Akyildiz [Akyildiz FT. A note on the flow of a third grade between heated parallel plates. Int J Non-Linear Mech 2001;36:349–52] is adopted, which generates lateral mass and viscoelastic terms in the heat conservation equation, as well as in the momentum equation. A number of special cases of the general transformed model are discussed. A homotopy analysis method (HAM) is implemented to solve, with appropriate boundary conditions, the coupled third-order, second degree ordinary differential equation for momentum and the second-order, fourth degree heat conservation equation.     

Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations

We study the existence and uniqueness of a weighted pseudo-almost periodic (mild) solution to the semilinear fractional equation ${?}_t^{\alpha }u=Au+{?}_t^{\alpha ?1}f(?,u)$, $1<\alpha <2$, where $A$ is a linear operator of sectorial negative type. This article also deals with the existence of these types of solutions to abstract partial evolution equations.     

Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2?\theta$ for any $\theta >0$. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.     

Convergence of damped inertial dynamics governed by regularized maximally monotone operators

In a Hilbert space setting, we study the asymptotic behavior, as time t goes to infinity, of the trajectories of a second-order differential equation governed by the Yosida regularization of a maximally monotone operator with time-varying positive index $\lambda (t)$. The dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient $\gamma (t)$. A suitable tuning of the parameters $\gamma (t)$ and $\lambda (t)$ makes it possible to prove the weak convergence of the trajectories towards zeros of the operator. When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch–Cabot [3], and Attouch–Peypouquet [8]. In this last paper, the authors considered the case $\gamma (t)=\frac{\alpha }{t}$, which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature.     

On implicit and explicit discretization schemes for parabolic SPDEs in any dimension

We study the speed of convergence of the explicit and implicit space–time discretization schemes of the solution $u(t,x)$ to a parabolic partial differential equation in any dimension perturbed by a space-correlated Gaussian noise. The coefficients only depend on $u(t,x)$ and the influence of the correlation on the speed is observed.     

The convergence of the Rényi entropy of the normalized sums of IID random variables

We prove that the Rényi entropy of order $\alpha$ ($\alpha >1$) of the normalized sums of IID random variables with continuous differentiable density is convergent to the Rényi entropy of order $\alpha$ of the standard Gaussian distribution, and obtain the corresponding rates of convergence.     

Fixed points of n-valued maps,the fixed point property and the case of surfaces—A braid approach

We study the fixed point theory of $n$-valued maps of a space $X$ using the fixed point theory of maps between $X$ and its configuration spaces. We give some general results to decide whether an $n$-valued map can be deformed to a fixed point free $n$-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of $X$ to study this problem. If $X$ is either the $k$-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for $n$-valued maps for all $n\ge 1$, and we prove the same result for even-dimensional spheres for all $n\ge 2$. If $X$ is the $2$-torus, we classify the homotopy classes of $2$-valued maps in terms of the braid groups of $X$. We do not currently have a complete characterisation of the homotopy classes of split $2$-valued maps of the $2$-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.