Enrique Fernández-Cara

Null and approximate controllability for weakly blowing up semilinear heat equations

Abstract We consider the semilinear heat equation in a bounded domain of R d , with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that |f(s)| grows slower than |s|log3/2(1+|s|) as |s|→∞ . For instance, this condition is fulfilled by any function f growing at infinity like |s|logp(1+|s|) with 1 2 , null controllability does not hold. The problem remains open when f behaves at infinity like |s|logp(1+|s|) , with 3/2≤p≤2 . Results of the same kind are proved in the context of approximate controllability.

On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient

We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of

Why viscous fluids adhere to rugose walls: A mathematical explanation

{\mathbb R}^N

The Differentiability of the Drag with Respect to the Variations of a Lipschitz Domain in a Navier--Stokes Flow

with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximately controllable at any time if the nonlinear term

Some Controllability Results for the N -Dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls

f( y, \nabla y)

The convergence of two numerical schemes for the Navier-Stokes equations

grows slower than

SOME CONTROL RESULTS FOR SIMPLIFIED ONE-DIMENSIONAL MODELS OF FLUID-SOLID INTERACTION

|y| \log^{3/2}(1+ |y| + |\nabla y|) + |\nabla y| \log^{1/2}(1+ |y| + |\nabla y|)

Null controllability of the Burgers system with distributed controls

at infinity (generally, in this case, in the absence of control, blow-up occurs). The proofs use global Carleman estimates, parabolic regularity, and the fixed point method.

Controllability results for linear viscoelastic fluids of the Maxwell and Jeffreys kinds

Abstract The main purpose of this paper is to justify rigorously the following assertion: A viscous fluid cannot slip on a wall covered by microscopic asperities because, due to the viscous dissipation, the surface irregularities bring to rest the fluid particles in contact with the wall. In mathematical terms, this corresponds to an asymptotic property established in this paper for any family of fields that slip on oscillating boundaries and remain uniformly bounded in the H1-norm.

On the approximate controllability of a stochastic parabolic equation with a multiplicative noise

This paper is concerned with the computation of the drag