Hölder Continuity of Solutions to the Kinematic Dynamo Equations

We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.     

A chaotic system with Hölder continuity

This paper presents a new chaotic system with infinitely many equilibria. The new system contains two system parameters and a nonlinear term which does not satisfy Lipschitz continuity but does satisfy $\frac{1}{2}$ -Hölder continuity condition. The complicated dynamics are studied through theoretical analysis and numerical simulation. Synchronization for two identical systems by a piecewise linear feedback controller is investigated based on Lyapunov stability criteria.     

Hölder stability in Type III thermoelastodynamics

This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We here assume that the mass density is positive and the thermal conductivity tensor is positive definite. However, we do not assume the positivity of any other tensor. In this situation, we obtain Hölder continuous dependence results on the supply terms. We also sketch how to prove the continuous dependence on the initial data.     

On Nonperiodic Euler Flows with Hölder Regularity

In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding \({1/3}\). This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space \({L_t^\infty B_{3,\infty}^{1/3}}\) due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than \({1/5}\). The main result of the present paper shows that any given smooth Euler flow can be perturbed in \({C^{1/5-\epsilon}_{t,x}}\) on any pre-compact subset of \({\mathbb{R}\times \mathbb{R}^3}\) to violate energy conservation. Furthermore, the perturbed solution is no smoother than \({C^{1/5-\epsilon}_{t,x}}\). As a corollary of this theorem, we show the existence of nonzero \({C^{1/5-\epsilon}_{t,x}}\) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.     

The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity

This study is concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. For the solutions of this problem we prove a result of Hölder’s-type stability on the supply terms. We impose middle restrictions on the thermoelastic coefficients, which are common in continuum mechanics. For the same conditions we propose a continuous dependence result with regard to the initial data.     

Continuity and Invariance of the Sacker–Sell Spectrum

The Sacker–Sell (also called dichotomy or dynamical) spectrum \(\varSigma \) is a fundamental concept in the geometric, as well as for a developing bifurcation theory of nonautonomous dynamical systems. In general, it behaves merely upper-semicontinuously and a perturbation theory is therefore delicate. This paper explores an operator-theoretical approach to obtain invariance and continuity conditions for both \(\varSigma \) and its dynamically relevant subsets. Our criteria allow to avoid nonautonomous bifurcations due to collapsing spectral intervals and justify numerical approximation schemes for \(\varSigma \).     

Higher Regularity of H?lder Continuous Solutions of Parabolic Equations with Singular Drift Velocities

Motivated by an equation arising in magnetohydrodynamics, we prove that H?lder continuous weak solutions of a nonlinear parabolic equation with singular drift velocity are classical solutions. The result is proved using the space?Ctime Besov spaces introduced by Chemin and Lerner (J Differ Equ 121(2):314?C328, 1995), combined with energy estimates, without any minimality assumption on the H?lder exponent of the weak solutions.     

Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws

General results about maps of convex sets in ? ⁿ are proved. We outline their extensions to an infinite-dimensional context. Such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps. Here, we explore applications only in the finite-dimensional context. More specifically, we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite-difference schemes applied to systems of conservation laws in several space variables. In particular, we establish a final characterization of the invariant regions under the Lax-Friedrichs scheme and also give sufficient conditions for the local invariance. Further, we give sufficient conditions for the global and local invariance of regions under flux-splitting finite-difference schemes. An example of the multi-dimensional Euler equations for non-isentropic gas dynamics is discussed.     

The Devil’s Staircase Dimensions and Measure-Theoretical Entropy of Maps with Horizontal Gap

This work elucidates the measure-theoretical entropy and dimensions of a unimodal map with a horizontal gap. The measure-theoretical entropy and dimensions of the F_t (which is defined later) are shown to form a devils staircase structure with respect to the gap size t. Pesins formula for gap maps is also considered.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Optimal control of the transmission dynamics of tuberculosis

This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A tuberculosis model which incorporates the essential biological and epidemiological features of the disease such as exogenous reinfection and chemoprophylaxis of latently infected individuals, and treatment of the infectious is developed and rigorously analyzed. Based on this continuous model, the tuberculosis control is formulated and solved as an optimal control theory problem, indicating how a control term on the chemoprophylaxis should be introduced in the population to reduce the number of individuals with active TB. The feedback control law has been proved to be capable of reducing the number of individuals with active TB. An advantage is that the proposed scheme accounts for the energy wasted by the controller and the closed-loop performance on tracking. Numerical results show the performance of the optimization strategy.     

Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation

Nonlinear Dynamics - The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and Kupershmidt’s bi-Hamiltonian (Kupershmidt in Phys...     

Constrained Optimal Design of Columns Against Buckling∗

The buckling loads of straight, simply supported columns, loaded by axial thrust are maximized, subject to the restriction that the available volume of structural material is specified. In addition, a constraint is placed upon the maximum allowable prebuckling stress. Analytic solutions are presented for a variety of different cross-sectional geometries.     

Optimal Synthesis of Frameworks under Elastic and Plastic Performance Constraints Using Discrete Sections *, †

ABSTRACT

A computer-based method is presented for minimum-weight design of planar frameworks under service and ultimate performance conditions using discrete member sections. Service-load conditions ensure acceptable elastic stresses and displacements, and ultimate-load conditions ensure adequate safety against plastic collapse. The sizes of the discrete sections, which can be of any type; e.g., WF, HSS, etc., may be constrained to satisfy fabrication requirements related to member continuity and structure symmetry. The design method is iterative in nature and remarkably efficient. The number of iterations is generally small and almost totally independent of the complexity of the structure. Several comparative designs for simple truss and frame structures are presented to illustrate features of the method.     

Synthesis of Optimal Feedback Control of the Crane–Load System

International Applied Mechanics - The problems of the optimal control of a mechanical crane–load system are stated in three cases: acceleration, deceleration, and full cycle of motion. The...     

Elastizitätsuntersuchungen an Hölzern

Ohne Zusammenfassung     

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

Nonlinear Dynamics - All possible nonlocal versions of the derivative nonlinear Schrödinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the...     

Constrained Optimal Design of Circular Plates Against Buckling∗

Several problems are investigated in which the buckling loads of structures are maximized, subject to the restriction that the volume of structural material is specified. In addition, a constraint is placed upon either the maximum allowable prebuckling stress, or the minimum allowable thickness. The structures analyzed are flat, axisymmetric circular plates, loaded by uniform radial thrust, and assumed to buckle axisymmetrically. The plates are investigated for both sandwich and solid cross-sectional geometries.