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

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

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