Search

We consider the equations governing the motion of incompressible second grade fluids in a bounded two-dimensional domain with Navier-slip boundary conditions. We first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus α in the L∞-H1 and the L2-H2 time–space norms. Next, we study their asymptotic behavior when α tends to zero, prove that they converge to regular solutions of the Navier–Stokes equations and give the rate of convergence in terms of α.

We study optimal control problems of systems describing the flow of incompressible shear-thinning fluids. Taking advantage of regularity properties of the flows, we derive necessary optimality conditions under a restriction on the optimal control.

We consider a non-Newtonian fluid governed by stationary, incompressible Navier–Stokes equations with shear-dependent viscosity. Using a fixed point argument in an appropriate functional setting, we establish the existence of a strong solution for small and suitably regular data. Uniqueness results are obtained under similar conditions.

This article is concerned with the equations governing the steady motion of a viscoelastic incompressible second-order fluid in a bounded domain. A new proof of existence and uniqueness of strong solutions is given. In addition, using appropriate finite element methods to approximate a coupled equivalent problem, sharp error estimates are obtained using a fixed point argument. The method is applied to the two-dimensional lid-driven cavity problem, at low Reynolds number and in a certain range of values of the viscoelastic parameters, to analyze the combined effects of inertia and viscoelasticity on the flow. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

We consider time optimal control problems governed by semilinear parabolic equations with Dirichlet boundary controls in the presence of a target state constraint. To establish optimality conditions for the terminal time $T$, we define a new Hamiltonian functional. Due to regularity results for the state and the adjoint state variables, this Hamiltonian belongs to $L_{l o c}^r(0,T)$ for some $r>1$. By proving that it satisfies a differential equation corresponding to an optimality condition for $T$, we deduce that it belongs to $W^{1,1}(0,T)$. This result answers to the question: how to define Hamiltonian functionals for infinite dimensional problems with variable endpoints (see [10], p. 282 and p. 595).

This paper is concerned with steady, fully developed motion of a Navier--Stokes fluid with shear-dependent viscosity in a curved pipe under a given axial pressure gradient. We establish existence and uniqueness results, derive appropriate estimates, and prove a characterization of the secondary flows. The approximation, with respect to the curvature ratio, of the full governing systems by some Dean-like equation is studied.

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.

In this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints. Since no convexity condition is assumed on data of the problem, we define a relaxed control problem, prove the existence of relaxed solutions, and give some relaxation results. By adapting the penalty method of Berkovitz, we prove a Pontryagin's minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition.

In this paper we study a minimax boundary control problem with pointwise state constraints for parabolic equations under unknown disturbances in the initial condition. We establish necessary optimality conditions.