On the motion of an elastic solid inside of an incompressible viscous fluid

The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now. In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.Comment: 72 pages, to appear in Archive for Rational Mechanics and Analysi

On the stability of non-symmetric equilibrium figures of a rotating viscous incompressible liquid

We consider a classical problem of stability of equilibrium figures of a liquid rotating uniformly as a rigid body about a fixed axis. We connect the problem of stability with the behavior for large t of solutions of an evolution problem governing the motion of an isolated liquid mass whose initial data are slight perturbations of the regime of a rigid rotation. The main attention is given to the case when the figure is not rotationally symmetric; in this case the regime of a rigid rotation defines a periodic solution of the above-mentioned nonstationary problem. It is proved that a sufficient condition of stability is the positivity of the second variation of the energy functional in an appropriate function space

A simple proof of the linear instability of rotating liquid drops

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Nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

In cylindrical domain, we consider the nonstationary flow with prescribed inflow and outflow, modelled with Navier-Stokes equations under the slip boundary conditions. Using smallness of some derivatives of inflow function, external force and initial velocity of the flow, but with no smallness restrictions on the inflow, initial velocity neither force, we prove existence of solutions in $W^{2,1}_2.

Analysis of the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations

The velocity tracking problem for the evolutionary Navier–Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch2, the L2(ΩT ) error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795–820], we extend our results to the case of L1(ΩT ) type functionals that allow sparse controls.This author was partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-

On the interaction between quasilinear elastodynamics and the Navier-Stokes equations

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.Comment: 43 pages, to appear in Archive for Rational Mechanics and Analysi

A Model of Porous Catalyst Accounting for Incipiently Non-isothermal Effects*

An approximate model accounting for incipiently non-isothermal effects is derived from a well-known model of porous catalyst for appropriate, realistic limiting values of the parameters. In this limit, the original model is a singularly perturbed, m-D reaction–diffusion system, and the approximate model is given by the m-D heat equation with nonlinear boundary condition, coupled with infinitely many (ifm2) 1-D semilinear parabolic equations, one for each point of the boundary of the spatial domain. Some limiting cases are still considered in the approximate model that lead to further simplifications

Counteraction to information influence in social networking services by means of fuzzy logic system

The article describes a decision support system based on fuzzy inference aimed to automate the procedure of choosing a model of formalizing the interaction between actors in virtual communities of social networking services and synergistic management of such processes. The developed system aims to increase the effectiveness of counteracting threats to information security of the state in social networking services. The mathematical apparatus of the fuzzy set theory and the Mamdani algorithm are the basis for the functioning of the decision support system. The usage of the developed fuzzy inference system will remove the ambiguity of information security expertise in the course of choosing approaches to formalization and the model of synergistic management of actors’ interaction in the conditions of incomplete information and ambiguous assessment of the state information security threat in social networking services

Existence of global strong solutions in critical spaces for barotropic viscous fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of the global existence of strong solutions for initial data close from a constant state having critical Besov regularity. In a first time, this article show the recent results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to \textit{kill} the coupling between the density and the velocity as in \cite{H2}. We study so a new variable that we call effective velocity. In a second time we improve the results of \cite{CD} and \cite{CMZ} by adding some regularity on the initial data in particular $\rho_{0}$ is in $H^{1}$. In this case we obtain global strong solutions for a class of large initial data on the density and the velocity which in particular improve the results of D. Hoff in \cite{5H4}. We conclude by generalizing these results for general viscosity coefficients