Conley index of nontrivial invariant sets in a Hilbert space

We study flows defined in a Hilbert space by potential completely continuous fields Id-K(·), where K(·) is an operator close to a homogeneous one. The Conley index of the set of fixed points and separatrices joining them (a nontrivial invariant set) is defined for such flows. By using this index, we prove that the equation K(x) = x has infinitely many solutions of arbitrarily large norm provided that the potential φ: ?φ(·) = K(·) is coercive and has an even leading part. As a corollary, we justify the stability of an arbitrary finite number of solutions under small perturbations of the field. We show that the Conley index differs from the classical rotation theory of vector fields when proving existence theorems.     

Loss of Energy Concentration in Nonlinear Evolution Beam Equations

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation

$$\begin{aligned} u_{tt} + u_{xxxx} + f(u)= g(x, t) \end{aligned}$$

in bounded space–time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities f and for some forcing terms g, highlighting some of their structural properties and performing some numerical simulations.

     

Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders

Summary For the problem of hydrodynamical stability in an infinite cylindrical domain, we investigate all time-periodic solutions, not only spatially periodic ones, when a Hopf bifurcation occurs. When reflection symmetry is present, we show the existence of spatially quasiperiodic flows. We also show the existence of heteroclinic solutions connecting two symmetrically traveling waves that stay at each end of the cylinders (defect solutions). The technique we use rests on (i) a center manifold argument in a space of time-periodic vector fields, (ii) symmetry and normal form arguments for the reduced ordinary differential equation in two dimensions (without reflection symmetry) or in four dimensions (with reflection symmetry), and (iii) the integrability of the associated normal form. It then remains to prove a persistence result when we add the higher-order terms of the vector field.     

Stabilization and stability for the spherically symmetric Navier–Stokes–Poisson system

We consider global behaviour of viscous compressible flows with spherical symmetry driven by gravitation and an outer pressure, outside a hard core. For a general state function $p=p\left(\rho \right)$, we present global-in-time bounds for solutions with arbitrarily large data. For non-decreasing $p$, the $\omega$-limit set for the density $\rho$ is studied. For increasing $p$, uniqueness and static stability of the stationary solutions (including variational aspects) are investigated. Moreover, stabilization rate bounds toward the statically stable solutions are given and their nonlinear dynamical stability is shown.     

Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit

We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude number or “weakly unstable” limit \(F\rightarrow 2^+\) and for general values of the Froude number F, including the limit \(F\rightarrow +\infty \). In the former, \(F\rightarrow 2^+\), limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as \(F\rightarrow 2^+\) is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around \(F=2.3\) from weakly unstable to different, large-F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically \(2.5\le F\le 6.0\).     

Games with a permission structure - A survey on generalizations and applications

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.     

On the general problem of stability for impulsive differential equations

Criteria for stability, asymptotical stability and instability of the nontrivial solutions of the impulsive system

     

Algebraic and abelian solutions to the projective translation equation

Let \({{\mathrm x}=(x,y)}\). A projective two-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) \({(1-z)\phi({\mathrm x})=\phi(\phi({\mathrm x}z)(1-z)/z)}\), \({\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}}\). Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field \({\varpi(x,y)\bullet \varrho(x,y)}\) which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, such as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows us to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are described in terms of abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR), and the orbits of these flows can be transformed into algebraic curves \({x^{A}(x-y)^{B}y^{C}\equiv{\mathrm{const.}}}\) (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines \({y\equiv{\mathrm{const.}}}\) (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.     

On center singularity for compressible spherically symmetric nematic liquid crystal flows

The paper is concerned with a simplified system, proposed by Ericksen [6] and Leslie [20], modeling the flow of nematic liquid crystals. In the first part, we give a new Serrin's continuation principle for strong solutions of general compressible liquid crystal flows. Based on new observations, we establish a localized Serrin's regularity criterion for the 3D compressible spherically symmetric flows. It is proved that the classical solution loses its regularity in finite time if and only if, either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center.     

Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field

In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,

(i)

Couette flow

(ii)

Poiseuille flow and

(iii)

generalized Couette flow.

Homotopy analysis method (HAM) is used to solve the non-linear differential equation analytically. It is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants (non-Newtonian parameters) which is different from the model of Oldroyd 3-constant fluid. Numerical solutions are also given and compared with the solutions by HAM.     

Rotating hydromagnetic flow in the presence of a horizontal magnetic field

Following Hollerbachs work (Geophys. Astrophys.Fluid Dynam., 1996) we investigate the hydromagnetic flow in a region x 0, – < y < , 0 < z < 1 bounded by three electrically insulating rigid walls. The rotation vector is in the z-direction while the applied uniform magnetic field B₀ is in the x-direction. Antisymmetric and symmetric cases are considered and analytical solutions are obtained for all the field variables for both the transition field regime (E^1/2 E^1/3) and strong magnetic field regime ( E^1/3) where (= B²/) is Elsasser number. Emphasis is put on the physical aspects of the problem and the meridional cir-culation pattern of electric currents. Unlike the case where a separate magnetic boundary layer exists to close the meridional electric current flux when the rotation vector and applied magnetic field are aligned, it is found that no such layer exists in the present problem; the electric currents generated in the interior and in the boundary layer regions have to be closed through interior region only. The transition field regime is characterized by the Stewartsons double layer structure with the noted exception that the outer Stewartson layer O(E/)^1/2 is weak. In addition, sub-boundary layers with an axial scale equal to the corresponding boundary layer scale develop at z=0,1 for each layer. In the large magnetic field regime, while the layer which replaces the inner Stewartson layer O(E^1/3) satisfies the boundary condition on u-field, the thin (E/)^1/2 layer is necessary to satisfy the boundary condition on v and w fields.     

Specialization results and ramification conditions

Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T) with group G such that E/k is regular, we produce some specializations of E/k(T) at points t0 ∈ P¹(k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior — ramified or unramified — at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T).     

New stress and velocity fields for highly frictional granular materials

The idealized theory for the quasi-static flow of granular materialswhich satisfy the Coulomb–Mohr hypothesis is considered.This theory arises in the limit as the angle of internal frictionapproaches /2, and accordingly these materials may be referredto as being ‘highly frictional’. In this limit,the stress field for both two-dimensional and axially symmetricflows may be formulated in terms of a single nonlinear second-orderpartial differential equation for the stress angle. To obtainan accompanying velocity field, a flow rule must be employed.Assuming the non-dilatant double-shearing flow rule, a furtherpartial differential equation may be derived in each case, thistime for the streamfunction. Using Lie symmetry methods, a completeset of group-invariant solutions is derived for both systems,and through this process new exact solutions are constructed.Only a limited number of exact solutions for gravity-drivengranular flows are known, so these results are potentially importantin many practical applications. The problem of mass flow througha two-dimensional wedge hopper is examined as an illustration.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Mathematical modeling of a thin viscous layer adhering to an infinite plate withdrawn at an angle

We consider the problem of a thin viscous layer adhering to an inclined plate. Systems of boundary-layer equations and capillary-statics equations are used. The solutions of the two systems are matched at a certain point, which ensures a smooth profile of the free surface of the liquid.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 57–61, 1992.     

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

On the heat flow on metric measure spaces: existence, uniqueness and stability

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some ${\lambda\in\mathbb {R}}$ . Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ?converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.     

Higher order Galerkin methods in time for free surface flows

In many engineering applications, free surface or two-phase flows are discretized in time with an explicit decoupling of geometry and fluid flow. Such a strategy leads to a capillary CFL condition of the form [3]. For the case of surface tension dominated flows (i.e. high Weber number We) this can dictate infeasibly small time steps. As an alternative we suggest a Galerkin method in time based on the discontinuous Galerkin method of first order (dG(1)). For this choice, an energy estimate can be proved [7], so unconditional stability of the method is given. While for ODEs or parabolic PDEs the method is of third order at the discrete points in time t_n [4], in the case of free surface flows second order convergence can still be achieved. Numerical examples using the Arbitrary Lagrangian Eulerian (ALE) method for both capillary one-phase and two-phase flow demonstrate this convergence order. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)