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Time-Reversible Dynamics in a System of two Coupled Active Rotators
No full textWe study two coupled active rotators with Kuramoto-type coupling and focus our attention to specific transitional regimes where the coupling is neither attractive nor repulsive. We show that certain such situations at the edge of synchronization can be characterized by the existence of a time-reversal symmetry of the system. We identify two different cases with such a time-reversal symmetry. The first case is characterized by a non-reciprocal attractive/repulsive coupling. The second case is a reciprocal coupling exactly at the edge between attraction and repulsion. We give a detailed description of possible different types of dynamics and bifurcations for both cases. In particular, we show how the time-reversible coupling can induce both oscillation death and oscillation birth to the active rotators. Moreover, we analyse the coexistence of conservative and dissipative regions in phase space, which is a typical feature of systems with a time-reversal symmetry. We show also, how perturbations breaking the time-reversal symmetry and destroying the conservative regions can lead to complicated types of dissipative dynamics such as the emergence of long-period cycles showing a bursting-like behavior
Predicting Disordered Regions Driving Phase Separation of Proteins under Variable Salt Concentration
No full textWe determine the intrinsically disordered regions (IDRs) of phase separating proteins and investigate their impact on liquid-liquid phase separation (LLPS) with a random-phase approx- imation (RPA) that accounts for variable salt concentration. We focus on two proteins, PGL-3 and FUS, known to undergo LLPS. For PGL-3 we predict that an IDR near the C-terminus pro- motes LLPS, which we validate through direct comparison with in vitro experimental results. For the structurally more complex protein FUS the role of the low complexity (LC) domain in LLPS is not as well understood. Apart from the LC domain we here identify two IDRs, one near the N-terminus and another near the C-terminus. Our RPA analysis of these domains predict that, surprisingly, the IDR at the N-terminus (aa 1-285) and not the LC domain promotes LLPS of FUS by comparison to in vitro experiments under physiological temperature and salt conditions
Breakdown of Electroneutrality in Polyelectrolyte Gels
No full textMathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length
Probabilistic maximization of time-dependent capacities in a gas network
Get PDFThe determination of free technical capacities belongs to the core tasks of a gas network owner. Since gas loads are uncertain by nature, it makes sense to understand this as a probabilistic problem as far as stochastic modeling of available historical data is possible. Future clients, however, don?t have a history or they do not behave in a random way, as is the case, for instance, in gas reservoir management. Therefore, capacity maximization turns into an optimization problem with uncertainty-related constrained which are partially of probabilistic and partially of robust (worst case) type. While previous attempts to solve this problem had be devoted to models with static (time-independent) gas flow, we aim at considering here transient gas flow subordinate to a PDE (Euler equations). The basic challenge here is two-fold: first, a proper way of joining probabilistic constraints to the differential equations has to be found. This will be realized on the basis of the so-called spherical-radial decomposition of Gaussian random vectors. Second, a suitable characterization of the worst-case load behaviour of future customers has to be figured out. It will be shown, that this is possible for quasi-static flow and can be transferred to the transient case. The complexity of the problem forces us to constrain ourselves in this first analysis to simple pipes or to a V-like structure of the network. Numerical solutions are presented and show that the differences between quasi-static and transient solutions are small, at least in these elementary examples
Pontryagin's principle for some probabilistic control problems
Get PDFIn this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin's optimality principle
Periodic Lp estimates by R-boundedness: Applications to the Navier--Stokes equations
No full textGeneral evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic Lp estimates of maximal regularity type are established from R-bounds of the family of solution operators (R-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier--Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data
Analysis of a drift-diffusion model for perovskite solar cells
Get PDFThis paper deals with the analysis of an instationary drift-diffusion model for perovskite solar cells including Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite layer. The free energy functional is related to this choice of the statistical relations. Exemplary simulations varying the mobility of the ionic vacancy demonstrate the necessity to include the migration of ionic vacancies in the model frame. To prove the existence of weak solutions, first a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval is considered. Its solvability follows from a time discretization argument and passage to the time-continuous limit. Applying Moser iteration techniques, a priori estimates for densities, chemical potentials and the electrostatic potential of its solutions are derived that are independent of the regularization level, which in turn ensure the existence of solutions to the original problem
Optimal Temperature Distribution for a Nonisothermal Cahn--Hilliard System with Source Term
No full textIn this note, we study the optimal control of a nonisothermal phase field system of Cahn--Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn--Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the so-called ``strict separation property'', which states that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this separation property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions
Solving a spectral problem for large-area photonic crystal surface-emitting lasers
No full textWe present algorithms for constructing and resolving spectral problems for novel photonic crystal surface-emitting lasers with large emission areas, given by first-order PDEs with two spatial dimensions. These algorithms include methods to overcome computer-arithmetic-related challenges when dealing with huge and small numbers. We show that the finite difference schemes, constructed using relatively coarse numerical meshes enable accurate estimation of several major, optical modes, which are essential in practical applications
Calculation of Optical Modes in Large Emission Area Photonic Crystal Surface-Emitting Lasers
No full textWe discuss numerical challenges in constructing and resolving spectral problems for photonic crystal surface-emitting lasers with large (up to several mm2) emission area. We show that finite difference schemes with moderate and large domain discretization steps provide sufficient accuracy of several major (lowest-threshold) modes of particular device designs