2,398 research outputs found
On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay
An initial-boundary value problem for a viscoelastic wave equation subject to
a strong time-localized delay in a Kelvin & Voigt-type material law is
considered. Transforming the equation to an abstract Cauchy problem on the
extended phase space, a global well-posedness theory is established using the
operator semigroup theory both in Sobolev-valued - and BV-spaces. Under
appropriate assumptions on the coefficients, a global exponential decay rate is
obtained and the stability region in the parameter space is further explored
using the Lyapunov's indirect method. The singular limit is
further studied with the aid of the energy method. Finally, a numerical example
from a real-world application in biomechanics is presented.Comment: 34 pages, 4 figures, 1 set of Matlab code
Global Well-Posedness and Exponential Stability for Heterogeneous Anisotropic Maxwell's Equations under a Nonlinear Boundary Feedback with Delay
We consider an initial-boundary value problem for the Maxwell's system in a
bounded domain with a linear inhomogeneous anisotropic instantaneous material
law subject to a nonlinear Silver-Muller-type boundary feedback mechanism
incorporating both an instantaneous damping and a time-localized delay effect.
By proving the maximal monotonicity property of the underlying nonlinear
generator, we establish the global well-posedness in an appropriate Hilbert
space. Further, under suitable assumptions and geometric conditions, we show
the system is exponentially stable.Comment: updated and improved versio
On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay
In the present paper, we consider a Cauchy problem for a linear second order
in time abstract differential equation with pure delay. In the absence of
delay, this problem, known as the harmonic oscillator, has a two-dimensional
eigenspace so that the solution of the homogeneous problem can be written as a
linear combination of these two eigenfunctions. As opposed to that, in the
presence even of a small delay, the spectrum is infinite and a finite sum
representation is not possible. Using a special function referred to as the
delay exponential function, we give an explicit solution representation for the
Cauchy problem associated with the linear oscillator with pure delay. In
contrast to earlier works, no positivity conditions are imposed.Comment: 20 pages, 2 figure
Solving the Linear 1D Thermoelasticity Equations with Pure Delay
We propose a system of partial differential equations with a single constant
delay describing the behavior of a one-dimensional thermoelastic
solid occupying a bounded interval of . For an initial-boundary
value problem associated with this system, we prove a global well-posedness
result in a certain topology under appropriate regularity conditions on the
data. Further, we show the solution of our delayed model to converge to the
solution of the classical equations of thermoelasticity as .
Finally, we deduce an explicit solution representation for the delay problem.Comment: 15 pages, 1 figur
On a Kelvin-Voigt viscoelasticwave equation with strong delay
An initial-boundary value problem for a viscoelastic wave equation subject to a strong timelocalized
delay in a Kelvin & Voigt-type material law is considered. Transforming the equation
to an abstract Cauchy problem on the extended phase space, a global well-posedness theory
is established using the operator semigroup theory both in Sobolev-valued C0- and BV-spaces.
Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained
and the stability region in the parameter space is further explored using the Lyapunov’s indirect
method. The singular limit Ď„ -> 0 is further studied with the aid of the energy method. Finally,
a numerical example from a real-world application in biomechanics is presented
Asymmetric flows of viscoelastic fluids in symmetric planar expansion geometries
The flow of viscoelastic liquids with constant shear viscosity through symmetric sudden expansions is studied
by numerical means. The geometry considered is planar and the constitutive model follows the modified FENE-CR
equation, valid for relative dilute solutions of polymeric fluids. For Newtonian liquids in a 1:3 expansion we predict the result that the flow becomes asymmetric for a Reynolds number (based on upstream mean velocity and channel height) of about 54, in agreement with previously published results. For the non-Newtonian case the transition depends on both the concentration and the extensibility parameters of the model, and the trend is for the pitch-fork bifurcation to occur at higher Reynolds numbers. Detailed simulations are carried out for increasing Reynolds number, at fixed concentration and Weissenberg number, and for increasing concentration at a fixed Reynolds number of 60. The results given comprise size and strength of the recirculation zones, bifurcation diagrams, and streamline plots
Globalwell-posedness and exponential stability for heterogeneous anisotropic Maxwell’s equations under a nonlinear boundary feedback with delay
We consider an initial-boundary value problem for the Maxwell’s system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver–Müller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable
Evolutionary Equations
This open access book provides a solution theory for time-dependent partial differential equations, which classically have not been accessible by a unified method. Instead of using sophisticated techniques and methods, the approach is elementary in the sense that only Hilbert space methods and some basic theory of complex analysis are required. Nevertheless, key properties of solutions can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated by a large variety of examples, showing the applicability of the approach of evolutionary equations in various fields. Additionally, a quantitative theory for evolutionary equations is developed. The text is self-contained, providing an excellent source for a first study on evolutionary equations and a decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical research
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