A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.     

Continuous maximal regularity on uniformly regular Riemannian manifolds

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on M in conjunction with maximal regularity and the implicit function theorem.     

Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space

This paper treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by p-potential function subject to slip boundary conditions in R_+~3. Making use of the Oseentype approximation to this model and the L~∞-truncation method, one can establish the existence theorem of weak solutions for p-potential flow with p ∈(8/5, 2] provided that large initial are regular enough.     

Scalarization of \epsilon -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of \(\epsilon \) -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of \(\epsilon \) -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.     

On a class of hemivariational inequalities at resonance

We consider a class of noncoercive hemivariational inequalities involving the p-Laplacian at resonance. We use the unilateral growth condition so the energy functional is nonsmooth, nonconvex and its effective domain does not coincide with the whole space . To avoid this difficulty we study the problem in finite-dimensional spaces using the mountain-pass theorem for locally Lipschitz functionals and then we pass to the limit to obtain the existence of solutions.     

Steady flow for incompressible fluids in domains with unbounded curved channels

We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential Equations 252 (2012), no. 6, 3873-3898), propose a definition for domains with unbounded curved channels which encompasses domains with an unbounded boundary, domains with nozzles, and domains with a boundary being a punctured surface, and argue on the existence of steady flowfor incompressible fluids with arbitrary fluxes in such domains.     

Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow–Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow–Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.     

On Morse--Smale Diffeomorphisms with Four Periodic Points on Closed Orientable Manifolds

We study Morse--Smale diffeomorphisms of n-manifolds with four periodic points which are the only periodic points. We prove that for n= 3 these diffeomorphisms are gradient-like and define a class of diffeomorphisms inevitably possessing a nonclosed heteroclinic curve. For n 4, we construct a complete conjugacy invariant in the class of diffeomorphisms with a single saddle of codimension one.     

An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems

We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is a linearly constrained semismoothly differentiable (SC¹) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange method, is proportional to 1/r, and the rate of multiplier iterates is proportional to  $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. As the objective function of the dual problem is a SC¹ function involving the projection operator onto the cone of symmetrically semi-definite matrices, the analysis requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and properties of the projection operator in the symmetric-matrix space. Furthermore, the semismooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, which is proven to have global convergence and local quadratic rate. Finally numerical results, implemented by the augmented Lagrangian method, are reported.     

Uniqueness of almost periodic-in-time solutions to Navier�CStokes equations in unbounded domains

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier?CStokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier?CStokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w)}$ within the class of solutions that have sufficiently small ${L^{\infty}(L^{3}_w)}$ -norm. In this paper, we show that a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ is unique within the class of all almost periodic-in-time solutions in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ . The proof of the present uniqueness theorem is based on the method of dual equations.     

Branching Rules for Finite-Dimensional $\mathcal {U}_{q}(\mathfrak {su}(3))$-Representations with Respect to a Right Coideal Subalgebra

We consider the quantum symmetric pair \((\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})\) where \(\mathcal {B}\) is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of \(\mathcal {B}\) are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of \(\mathcal {U}_{q}(\mathfrak {su}(3))\) to \(\mathcal {B}\) decomposes multiplicity free into irreducible representations of \(\mathcal {B}\). Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.     

Incompressible non‐Newtonian fluids: time asymptotic behaviour of weak solutions

We study the asymptotic behaviour in time of incompressible non‐Newtonian fluids in the whole space assuming that initial data also belong to L¹. Firstly, we consider the weak solution to the power‐law model with non‐zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with p?. Secondly, we are interested in the asymptotic behaviour of weak solutions to the second grade model, and finally, we deal with the asymptotic behaviour in time of weak solutions to a simplified model of viscoelastic fluids of the Oldroyd type. Copyright © 2006 John Wiley & Sons, Ltd.     

A variational principle for the Navier-Stokes equation

Motivated from Arnold's variational characterization of the Euler equation in terms of geodesic families of diffeomorphisms, a variational principle for the motion of incompressible viscous fluids is presented. A volume preserving diffusion process with drift velocity field subject to the Navier-Stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations.     

Large eddy simulation turbulence model with Young measures

We show an alternative proof for the existence of weak solutions to equations describing turbulent flows of fluids. The proof proposed by one of the authors in a previous paper (cf. [A. Świerczewska, Large Eddy Simulation. Existence of Stationary Solutions to a Dynamical Model (submitted for publication). Preprint TU-Darmstadt no. 2314, http://wwwbib.mathematik.tu-darmstadt.de/Math-Net/Preprints/Listen/shadow/pp2314.html]) based on more classical methods. We will use Young measures, which allow us to shorten significantly the limiting procedure in the nonlinear terms and generalize the statement.     

New Aspects For Bifurcation Problems in Continuous Mechanics

Consider a semilinear eigenvalue problem where λ ∈ R , the linear operator is defined in a real Hilbert space H and : H → H is generaly a nonlinear perturbation. We can define a coincidence degree of the pair ( ) under some conditions weaker than the ones when the classical coincidence degree was defined. Our final purpose is to extend the results to the case of the operators from the Banach space X into its dual X*, using the representation theorem due to Browder and Ton. We use these results to study resonance problems in mechanics of continua, such as the buckling in finite elastostatics and the steady state flow of incompressible fluids. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudo-stable and Pseudo-unstable Fiber Bundles for Dynamic Equations on Measure Chains

Invariant fiber bundles are the generalization of invariant manifolds from classical discrete or continuous dynamical systems to non-autonomous dynamic equations on measure chains. In this paper, we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called "Hadamard-Perron-Theorem" for hyperbolic finite-dimensional diffeomorphisms to pseudo-hyperbolic time-dependent non-regressive dynamic equations in Banach spaces. The proof of their smoothness uses a fixed point theorem of Vanderbauwhede-Van Gils.     

On the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system with shear thickening

We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.     

Strong solutions to an Oldroyd‐B model with slip boundary conditions via incompressible limit

In this paper, we establish the local existence of strong solutions to an Oldroyd‐B model for the incompressible viscoelastic fluids in a bounded domain , via the incompressible limit. The main idea is to derive the uniform estimates with respect to the Mach number for the linearized system of compressible Oldroyd equations. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence and Dimension Properties of a Global <Emphasis Type="Italic">B</Emphasis>-Pullback Attractor for a Cocycle Generated by a Discrete Control System

We consider cocycles on finite-dimensional manifolds generated by discrete-time control systems. Frequency conditions for the existence of a global B-pullback attractor for such cocycles considered over a general base system on a metric space are given. Upper bounds for the Hausdorff dimension of the global B-pullback attractor of a discrete cocycle are obtained using the transfer function of the linear part of the cocycle and the discrete Kalman–Yakubovich–Popov frequency theorem.