Convergence of optimal control problems governed by second kind parabolic variational inequalities

We consider a family of optimal control problems where the control variable is given by a boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem. This work solves the open problem left by the authors in IFIP TC7 CSMO2011

Unsteady 3D-Navier-Stokes System with Tresca's Friction Law

Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the " incompressibility limit " of compressible fluids, allowing to get better insights into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem

Existence result for a strongly coupled problem with heat convection term and Tresca's law.

International audienceWe study a problem describing the motion of an incompressible, nonisothermal and non-Newtonian uid, taking into account the heat convection term. The novelty here is that uid viscosity depends on the temperature, the velocity of the uid, and also of the deformation tensor, but not explicitly. The boundary conditions take into account the slip phenomenon on a part of the boundary of the domain. By using the notion of pseudo-monotone operators and xed point Theorem we prove an existence result of its weak solution

A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain D=\mathbb{R}^{n-1}\times\br^{+} for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.Comment: Accepted by Revista UMA, April 30 2019, in press. arXiv admin note: substantial text overlap with arXiv:1610.0168

Sur quelques problèmes elliptiques de type Kirchhoff et dynamique des fluides

Cette thèse est composée de deux parties indépendantes. La première est consacrée à l'étude de quelques problèmes elliptiques de type de Kirchhoff de la forme suivante : -M( Nul dx) u = f(x, u) x ; u(x) = o x où cRN, N >= 2, f une fonction de Carathéodory et M une fonction strictement positive et continue sur R+. Dans le cas où la fonction f est asymptotiquement linéaire à l infini par rapport à l'inconnue u, on montre, en combinant une technique de troncature et la méthode variationnelle, que le problème admet au moins une solution positive quand la fonction M est non décroissante. Et si f(x, u) = |u|p-1 u + g(x), où p >0, un paramètre réel et g une fonction de classe C1 et changeant de signe sur , alors sous certaines hypothèses sur M, il existe deux réels positifs . et . tels que le problème admet des solutions positives si 0 .. Dans la deuxième partie, on étudie deux problèmes soulevés en dynamique des fluides. Le premier est une généralisation d'un modèle décrivant la propagation unidirectionnelle dispersive des ondes longues dans un milieu à deux fluides. En écrivant le problème sous la forme d'une équation de point fixe, on montre l'existence d'au moins une solution positive. On montre ensuite sa symétrie et son unicité. Le deuxième problème consiste à prouver l'existence de la vitesse, la pression et la température d'un fluide non newtonien, incompressible et non isotherme, occupant un domaine borné, en prenant en compte un terme de convection. L originalité dans ce travail est que la viscosité du fluide ne dépend pas seulement de la vitesse mais aussi de la température et du module du tenseur des taux de déformations. En se basant sur la notion des opérateurs pseudo-monotones, le théorème de De Rham et celui de point fixe de Schauder, l'existence du triplet, (vitesse, pression, température) est démontréThis thesis consists of two independent parts. The first is devoted to the study of some elliptic problems of Kirchhoff-type in the following form : -M( Nul dx) u = f(x, u) x ; u(x) = o x where cRN, N >= 2, f is a Caratheodory function and M is a strictly positive and continuous function on R+. In the case where the function f is asymptotically linear at infinity with respect to the unknown u, we show, by combining a truncation technique and the variational method, that the problem admits a positive solution when the function M is nondecreasing. And if f(x, u) = |u|p-1 u + g(x) where p> 0, a real parameter and g is a function of class C1 and changes the sign in , then under some assumptions on M, there exist two positive real . and . such that the problem admits positive solutions if 0 .. In the second part, we study two problems arising in fluid dynamics. The first is a generalization of a model describing the unidirectional propagation of long waves in dispersive medium with two fluids. By writing the problem as a fixed point equation, we prove the existence of at least one positive solution. We then show its symmetry and uniqueness. The second problem is to prove the existence of the velocity, pressure and temperature of a non-Newtonian, incompressible and isothermal fluid, occupying a bounded domain, taking into account a convection term. The originality in this work is that the fluid viscosity depends not only on the velocity but also on the temperature and the modulus of deformation rate tensor. Based on the notion of pseudo-monotone operators, the De Rham theorem and the Schauder fixed point theorem, the existence of the triplet, (velocity, pressure, temperature) is shownST ETIENNE-Bib. électronique (422189901) / SudocSudocFranceF

A brief survey on lubrication problems with nonlinear boundary conditions

International audienceWe consider some lubrication problems in a thin domain with thickness of order $\varepsilon$, with mixed boundary conditions and subject to slip phenomenon on a part of the boundary. We study the existence and uniqueness results for the weak solution of each problem, then we establish the asymptotic behavior of its solutions, when the depth of the thin domain tends to zero

Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary.

International audienceMotivated by lubrication problems, we consider a micropolar uid ow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < " << 1. We prove the existence and uniqueness of a solution of this problem for any value of " and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when " tends to zero. Moreover we show that the limit velocity and micro-rotation elds are uniquely determined via auxiliary well-posed problems and the limit pressure is given as the unique solution of a Reynolds equation