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Discrete & Continuous Dynamical Systems - S

January 2021 , Volume 14 , Issue 1

Analysis of evolutionary and complex systems: Issue on the occasion of Alexander Mielke’s 60th birthday

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Preface
M. Liero, S. Reichelt, G. Schneider, F. Theil and M. Thomas
2021, 14(1): i-iv doi: 10.3934/dcdss.2020455 +[Abstract](191) +[HTML](46) +[PDF](109.85KB)
Abstract:
Optimal laminates in single-slip elastoplasticity
Sergio Conti and Georg Dolzmann
2021, 14(1): 1-16 doi: 10.3934/dcdss.2020302 +[Abstract](809) +[HTML](390) +[PDF](525.5KB)
Abstract:

Recent progress in the mathematical analysis of variational models for the plastic deformation of crystals in a geometrically nonlinear setting is discussed. The focus lies on the first time-step and on situations where only one slip system is active, in two spatial dimensions. The interplay of invariance under finite rotations and plastic deformation leads to the emergence of microstructures, which can be analyzed in the framework of relaxation theory using the theory of quasiconvexity. A class of elastoplastic energies with one active slip system that converge asymptotically to a model with rigid elasticity is presented and the interplay between relaxation and asymptotics is investigated.

Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity
Martin Kalousek, Joshua Kortum and Anja Schlömerkemper
2021, 14(1): 17-39 doi: 10.3934/dcdss.2020331 +[Abstract](636) +[HTML](293) +[PDF](479.66KB)
Abstract:

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization.

First, we show that our model possesses global in time weak solutions, thus extending work by Benešová et al. 2018. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local-in-time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property.

Cahn-Hilliard equation with capillarity in actual deforming configurations
Tomáš Roubíček
2021, 14(1): 41-55 doi: 10.3934/dcdss.2020303 +[Abstract](759) +[HTML](335) +[PDF](451.87KB)
Abstract:

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity (i.e. concentration gradient) term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Faedo-Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at an actual configuration allow for similar models and analysis.

Viscoelasticity with limiting strain
Yasemin Şengül
2021, 14(1): 57-70 doi: 10.3934/dcdss.2020330 +[Abstract](788) +[HTML](335) +[PDF](1481.05KB)
Abstract:

A self-contained review is given for the development and current state of implicit constitutive modelling of viscoelastic response of materials in the context of strain-limiting theory.

Adaptive time stepping in elastoplasticity
Sören Bartels and Jakob Keck
2021, 14(1): 71-88 doi: 10.3934/dcdss.2020323 +[Abstract](633) +[HTML](293) +[PDF](1043.65KB)
Abstract:

Using rate-independent evolutions as a framework for elastoplasticity, an a posteriori bound for the error introduced by time stepping is established. A time adaptive algorithm is devised and tested in comparison to a method with constant time steps. Experiments show that a significant error reduction can be obtained using variable time steps.

Rate-independent evolution of sets
Riccarda Rossi, Ulisse Stefanelli and Marita Thomas
2021, 14(1): 89-119 doi: 10.3934/dcdss.2020304 +[Abstract](809) +[HTML](362) +[PDF](732.57KB)
Abstract:

The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.

In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads
Dorothee Knees and Chiara Zanini
2021, 14(1): 121-149 doi: 10.3934/dcdss.2020332 +[Abstract](580) +[HTML](306) +[PDF](555.72KB)
Abstract:

We study a rate-independent system with non-convex energy in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized \begin{document}$ BV $\end{document}-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact.

The spectrum of delay differential equations with multiple hierarchical large delays
Stefan Ruschel and Serhiy Yanchuk
2021, 14(1): 151-175 doi: 10.3934/dcdss.2020321 +[Abstract](747) +[HTML](309) +[PDF](2823.62KB)
Abstract:

We prove that the spectrum of the linear delay differential equation \begin{document}$ x'(t) = A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n}) $\end{document} with multiple hierarchical large delays \begin{document}$ 1\ll\tau_{1}\ll\tau_{2}\ll\ldots\ll\tau_{n} $\end{document} splits into two distinct parts: the strong spectrum and the pseudo-continuous spectrum. As the delays tend to infinity, the strong spectrum converges to specific eigenvalues of \begin{document}$ A_{0} $\end{document}, the so-called asymptotic strong spectrum. Eigenvalues in the pseudo-continuous spectrum however, converge to the imaginary axis. We show that after rescaling, the pseudo-continuous spectrum exhibits a hierarchical structure corresponding to the time-scales \begin{document}$ \tau_{1}, \tau_{2}, \ldots, \tau_{n}. $\end{document} Each level of this hierarchy is approximated by spectral manifolds that can be easily computed. The set of spectral manifolds comprises the so-called asymptotic continuous spectrum. It is shown that the position of the asymptotic strong spectrum and asymptotic continuous spectrum with respect to the imaginary axis completely determines stability. In particular, a generic destabilization is mediated by the crossing of an \begin{document}$ n $\end{document}-dimensional spectral manifold corresponding to the timescale \begin{document}$ \tau_{n} $\end{document}.

Global Hopf bifurcation in networks with fast feedback cycles
Bernold Fiedler
2021, 14(1): 177-203 doi: 10.3934/dcdss.2020344 +[Abstract](648) +[HTML](262) +[PDF](546.07KB)
Abstract:

Autonomous sustained oscillations are ubiquitous in living and nonliving systems. As open systems, far from thermodynamic equilibrium, they defy entropic laws which mandate convergence to stationarity. We present structural conditions on network cycles which support global Hopf bifurcation, i.e. global bifurcation of non-stationary time-periodic solutions from stationary solutions. Specifically, we show how monotone feedback cycles of the linearization at stationary solutions give rise to global Hopf bifurcation, for sufficiently dominant coefficients along the cycle.

We include four example networks which feature such strong feedback cycles of length three and larger: Oregonator chemical reaction networks, Lotka-Volterra ecological population dynamics, citric acid cycles, and a circadian gene regulatory network in mammals. Reaction kinetics in our approach are not limited to mass action or Michaelis-Menten type.

Orthogonality of fluxes in general nonlinear reaction networks
D. R. Michiel Renger and Johannes Zimmer
2021, 14(1): 205-217 doi: 10.3934/dcdss.2020346 +[Abstract](661) +[HTML](283) +[PDF](415.11KB)
Abstract:

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

Weak sequential stability for a nonlinear model of nematic electrolytes
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna and Arghir Zarnescu
2021, 14(1): 219-241 doi: 10.3934/dcdss.2020366 +[Abstract](589) +[HTML](221) +[PDF](592.8KB)
Abstract:

In this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal.

The evolution is described by a system coupling a Nernst-Planck system for the ions concentrations with a Maxwell's equation of electrostatics governing the evolution of the electrostatic potential, a Navier-Stokes equation for the velocity field, and a non-smooth Allen-Cahn type equation for the nematic director field.

We focus on the two-species case and prove apriori estimates that provide a weak sequential stability result, the main step towards proving the existence of weak solutions.

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials
Pierluigi Colli, Gianni Gilardi and Jürgen Sprekels
2021, 14(1): 243-271 doi: 10.3934/dcdss.2020213 +[Abstract](993) +[HTML](423) +[PDF](602.97KB)
Abstract:

Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn–Hilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible double-well potentials driving the phase separation process modeled by the Cahn–Hilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and first-order optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the so-called "deep quench" method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful first-order necessary optimality conditions.

Contraction and regularizing properties of heat flows in metric measure spaces
Giulia Luise and Giuseppe Savaré
2021, 14(1): 273-297 doi: 10.3934/dcdss.2020327 +[Abstract](640) +[HTML](322) +[PDF](658.93KB)
Abstract:

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorovich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow.

When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci curvature bounds in the setting of \begin{document}$ \mathrm{RCD}(K, \infty) $\end{document} metric measure spaces. As a byproduct, when \begin{document}$ K\ge0 $\end{document} we will also find new estimates for the asymptotic decay of the solution.

Numerical approximation of von Kármán viscoelastic plates
Manuel Friedrich, Martin Kružík and Jan Valdman
2021, 14(1): 299-319 doi: 10.3934/dcdss.2020322 +[Abstract](799) +[HTML](317) +[PDF](2586.7KB)
Abstract:

We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates [44]. Computational experiments exploiting C1 finite elements are provided, too.

Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system
Karoline Disser
2021, 14(1): 321-330 doi: 10.3934/dcdss.2020326 +[Abstract](588) +[HTML](301) +[PDF](403.28KB)
Abstract:

We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper \begin{document}$ L^{\infty} $\end{document}-bounds that exploits the entropic gradient structure of the system.

Existence of weak solutions for a sharp interface model for phase separation on biological membranes
Helmut Abels and Johannes Kampmann
2021, 14(1): 331-351 doi: 10.3934/dcdss.2020325 +[Abstract](614) +[HTML](304) +[PDF](502.2KB)
Abstract:

We prove existence of weak solutions of a Mullins-Sekerka equation on a surface that is coupled to diffusion equations in a bulk domain and on the boundary. This model arises as a sharp interface limit of a phase field model to describe the formation of liqid rafts on a cell membrane. The solutions are constructed with the aid of an implicit time discretization and tools from geometric measure theory to pass to the limit.

Threshold phenomenon for homogenized fronts in random elastic media
Patrick W. Dondl and Martin Jesenko
2021, 14(1): 353-372 doi: 10.3934/dcdss.2020329 +[Abstract](655) +[HTML](296) +[PDF](474.87KB)
Abstract:

We consider a model for the motion of a phase interface in an elastic medium, for example, a twin boundary in martensite. The model is given by a semilinear parabolic equation with a fractional Laplacian as regularizing operator, stemming from the interaction of the front with its elastic environment. We show that the presence of randomly distributed, localized obstacles leads to a threshold phenomenon, i.e., stationary solutions exist up to a positive, critical driving force leading to a stick-slip behaviour of the phase boundary. The main result is proved by an explicit construction of a stationary viscosity supersolution to the evolution equation and is based on a percolation result for the obstacle sites. Furthermore, we derive a homogenization result for such fronts in the case of the half-Laplacian in the pinning regime.

Perturbed minimizing movements of families of functionals
Andrea Braides and Antonio Tribuzio
2021, 14(1): 373-393 doi: 10.3934/dcdss.2020324 +[Abstract](672) +[HTML](278) +[PDF](472.64KB)
Abstract:

We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a (\begin{document}$ \Gamma $\end{document}-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters \begin{document}$ \varepsilon $\end{document} and \begin{document}$ \tau $\end{document}, governing energy and time scales, respectively. We characterize the extreme cases when \begin{document}$ \varepsilon/\tau $\end{document} and \begin{document}$ \tau/ \varepsilon $\end{document} converges to \begin{document}$ 0 $\end{document} sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of \begin{document}$ \varepsilon $\end{document} and \begin{document}$ \tau $\end{document}. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.

Effective diffusion in thin structures via generalized gradient systems and EDP-convergence
Thomas Frenzel and Matthias Liero
2021, 14(1): 395-425 doi: 10.3934/dcdss.2020345 +[Abstract](570) +[HTML](293) +[PDF](748.68KB)
Abstract:

The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker–Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin-de Donder kinetics.

Stochastic homogenization of $ \Lambda $-convex gradient flows
Martin Heida, Stefan Neukamm and Mario Varga
2021, 14(1): 427-453 doi: 10.3934/dcdss.2020328 +[Abstract](690) +[HTML](348) +[PDF](552.58KB)
Abstract:

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a \begin{document}$ \Lambda $\end{document}-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the \begin{document}$ p $\end{document}-Laplace operator with \begin{document}$ p\in (1, \infty) $\end{document}. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (\begin{document}$ \Lambda $\end{document}-)convex functionals.

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