774 research outputs found
Injection-suction control for Navier-Stokes equations with slippage
We consider a velocity tracking problem for the Navier-Stokes equations in a
2D-bounded domain. The control acts on the boundary through a injection-suction
device and the flow is allowed to slip against the surface wall. We study the
well-posedness of the state equations, linearized state equations and adjoint
equations. In addition, we show the existence of an optimal solution and
establish the first order optimality condition.Comment: 23 page
Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions
This work is devoted to the study of non-Newtonian fluids of grade three on
two-dimensional and three-dimensional bounded domains, driven by a nonlinear
multiplicative Wiener noise. More precisely, we establish the existence and
uniqueness of the local (in time) solution, which corresponds to an addapted
stochastic process with sample paths defined up to a certain positive stopping
time, with values in the Sobolev space H^3. Our approach combines a cut-off
approximation scheme, a stochastic compactness arguments and a general version
of Yamada-Watanabe theorem. This leads to the existence of a local strong
pathwise solution.Comment: 33 page
Weak solution for 3D-stochastic third grade fluid equations
UID/MAT/00297/2019This article studies the stochastic evolution of incompressible non-Newtonian fluids of differential type. More precisely, we consider the equations governing the dynamic of a third grade fluid filling a three-dimensional bounded domain O, perturbed by a multiplicative white noise. Taking the initial condition in the Sobolev space H2 (O), and supplementing the equations with a Navier slip boundary condition, we establish the existence of a global weak stochastic solution with sample paths in L∞ (0, T; H2 (O)).publishersversionpublishe
Optimal control of third grade fluids with multiplicative noise
This work aims to control the dynamics of certain non-Newtonian fluids in a
bounded domain of , perturbed by a multiplicative Wiener
noise, the control acts as a predictable distributed random force, and the goal
is to achieve a predefined velocity profile under a minimal cost. Due to the
strong nonlinearity of the stochastic state equations, strong solutions are
available just locally in time, and the cost functional includes an appropriate
stopping time. First, we show the existence of an optimal pair. Then,we show
that the solution of the stochastic forward linearized equation coincides with
the G\^ateaux derivative of the control-to-state mapping, after establishing
some stability results. Next, we analyse the backward stochastic adjoint
equation; where the uniqueness of solution holds only when . Finally, we
establish a duality relation and deduce the necessary optimality conditions.Comment: 36 pages,submitte
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