Attractors for Nonautonomous Nonhomogeneous Navier-Stokes Equations

In this paper our aim is to derive an upper bound on the dimension of the attractor of the family of processes associated to the Navier-Stokes equations with nonhomogeneous boundary conditions depending on time. We consider two-dimensional flows with prescribed quasiperiodic (in time) tangential velocity at the boundary, and obtain an upper bound which is polynomial with respect to the viscosity

Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well

Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions

This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalpphase-field model with singular potentials and dynamic boundary conditions.Thanks to a suitable definition of solutions, coinciding with the strong ones underproper assumptions on the bulk and surfacepotentials, we are able to get dissipative estimates, leading tothe existence of the global attractor with finite fractal dimension,as well as of an exponential attractor

On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions

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Experimental Method Calibration (MECr): A new relative method for heat flux sensor calibration

Heat flux sensor calibration is often expensive, but it is also a fundamental step to obtain valid results. According to the needs, several calibration procedures could be considered. If experimental interpretations focus on heat flux values comparison rather than true values, a relative calibration with high accuracy can be performed. This paper focuses on a new relative calibration-type method of heat flux sensors (HFSs). In general, the HFSs were originally (even recently) factory and in-house-calibrated, but this method was used to calibrate them to an exact common reading to help in performing accurate experiments and eliminating errors that could result from differences between the HFSs. The developed methodology was adapted from a secondary calibration-type (i.e., absolute) generalization. This method aimed to present a simple, low cost, and accurate way to calibrate heat flux sensors. The proposed calibration method was relative, which means that it did not use any calibration reference. However, an absolute calibration adaptation of this method can easily be performed. In the method presented in this paper, calibration was made via sensor sensitivity correction. The experiment consisted of imposing an identical and homogeneous heat flux across four heat flux sensors during a sufficiently long time to be able to statistically subtract any unwanted influences, such as convective and radiative variation. More precisely, temperatures and heat flux levels of 30-35 °C and 60-70 W.m-2 , respectively, were used. Then, new sensor sensitivity values were found by basic statistical data manipulations. Although the focus of this methodology was for contenting thermopile-based HFSs that used differential temperature measurements, the set-up can be adapted for other types of HFSs. In this paper, the set-up, its conception, and guidelines for using this method are presented

Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle \K is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite

Uniqueness and regularity for the Navier--Stokes--Cahn--Hilliard system

The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier--Stokes equations, which are coupled with the Cahn--Hilliard equation associated to the Ginzburg--Landau free energy with physically relevant logarithmic potential. This model is studied in bounded smooth domains in $\mathbb{R}^d$, $d=2$, and $d=3$ and is supplemented with a no-slip condition for the velocity, homogeneous Neumann boundary conditions for the order parameter and the chemical potential, and suitable initial conditions. We study uniqueness and regularity of weak and strong solutions. In a two-dimensional domain, we show the uniqueness of weak solutions and the existence and uniqueness of global strong solutions originating from an initial velocity ${\it u}_0 \in {\mathbf{V}}_\sigma$, namely, $\textbf{\textit{u}}_0\in \mathbf{H}_0^1(\Omega)$ such that $\mathrm{div}\, {\it u}_0=0$. In addition, we prove further regularity properties and the validity of the instantaneous separation property. In a three-dimensional domain we show the existence and uniqueness of local strong solutions with initial velocity ${\it u}_0 \in {\mathbf{V}}_\sigma$

Interzone short wave radiative couplings through windows and large openings : proposal of a simplified model

International audienceA simplified model of indoor short wave radiation couplings adapted to multi-zone simulations is proposed, thanks to a simplifying hypothesis and to the introduction of an indoor short wave exchange matrix. The specific properties of this matrix appear useful to quantify the thermal radiation exchanges between the zones separated by windows or large openings. Integrated in CODYRUN software, this module is detailed and compared to experimental measurements carried out on a real scale tropical buildin

Global attractors for Cahn-Hilliard equations with non constant mobility

We address, in a three-dimensional spatial setting, both the viscous and the standard Cahn-Hilliard equation with a nonconstant mobility coefficient. As it was shown in J.W. Barrett and J.W. Blowey, Math. Comp., 68 (1999), 487-517, one cannot expect uniqueness of the solution to the related initial and boundary value problems. Nevertheless, referring to J. Ball's theory of generalized semiflows, we are able to prove existence of compact quasi-invariant global attractors for the associated dynamical processes settled in the natural "finite energy" space. A key point in the proof is a careful use of the energy equality, combined with the derivation of a "local compactness" estimate for systems with supercritical nonlinearities, which may have an independent interest. Under growth restrictions on the configuration potential, we also show existence of a compact global attractor for the semiflow generated by the (weaker) solutions to the nonviscous equation characterized by a "finite entropy" condition