Modeling ternary mixtures by mean-field theory of polyelectrolytes: Coupled Ginzburg–Landau and Swift–Hohenberg equations

Morales, Marco A.; Rojas, J. F.; Torres, Isaac; Rubio, Eduardo

doi:10.1016/j.physa.2011.08.054

Cited by 9 publications

(6 citation statements)

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“…The chapter has laid out the present state of the art of the application of continuation techniques to the class (7) of nonlinear kinetic equations for one-and twodimensional domains. While we have focused on such equations for a single scalar field, there are many more related equations when allowing for several coupled scalar fields [47,68,26,89,60,87,107,90,122] or indeed coupled scalar, vector and tensor fields [114,85,95]. Complicated problems involving several scalar fields are dealt with in many of the tutorials and demos coming with PDE2PATH, and research papers using PDE2PATH, in 1d, 2d and 3d, for instance…”

Section: Discussionmentioning

confidence: 99%

Continuation for Thin Film Hydrodynamics and Related Scalar Problems

Engelnkemper

Gurevich

Uecker

et al. 2018

Computational Methods in Applied Sciences

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This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn-and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dipcoating. Through the different examples we illustrate how to employ the numerical tools provided by the packages AUTO07P and PDE2PATH to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.Published as: Engelnkemper, S., Gurevich, S.V.,

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Section: Discussionmentioning

confidence: 99%

Continuation for Thin Film Hydrodynamics and Related Scalar Problems

Engelnkemper

Gurevich

Uecker

et al. 2018

Computational Methods in Applied Sciences

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“…λ > 0, α , θ 1 , g are constants. A new mechanochemical model can also be the model of ternary mixtures 1 . In this context, u represents the concentration difference of two polymeric solutions, and v is the difference of the surface electrostatic potential that can be formed from a ternary mixture.…”

Section: Introductionmentioning

confidence: 99%

“…The system ()–() without the control terms B 1 w , B 2 μ was firstly introduced by Morales, Rojas, Torres and Rubio 1 to model ternary mixtures by using the theory of pattern formation and of polyelectrolytes, with mean‐field approximations. Recently, Morales, Rojas, Oliveros and Hernández 2 found that this model can also be interpreted as the skin coating of some vertebrate marine animals in dermis, epidermis and basal layers.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Morales, Rojas, Oliveros and Hernández 2 found that this model can also be interpreted as the skin coating of some vertebrate marine animals in dermis, epidermis and basal layers. While there exist some numerical simulations of the new mechanochemical model (see previous studies 1,2 ), there are still only one results of the mathematical analysis of this model. Liu and Liu 3 proved the existence of attractor for the mechanochemical model.…”

Section: Introductionmentioning

confidence: 99%

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Optimal control of a new mechanochemical model with state constraint

Liu

Zhang

2021

Math Methods in App Sciences

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In this paper, a state‐constrained optimal control problem is considered for the new mechanochemical model in biological patterns modelling the skin coating of some vertebrate marine animals. The model consists of the time‐dependent Ginzburg–Landau equation for the concentration difference of at least two pigments u coupled with the Swift–Hohenberg equation for the difference of dermal cellular densities of at least two types of cells. The objective is to force the state to be as close as possible to a desired state. Considering that the cost functional is discontinuous, which along with state constraint, we apply a new penalty functional to approximate the cost functional, in this case, so we first prove the existence of optimal control. Then we derive the necessary optimality conditions for the approximating optimal control problem. Finally, the necessary optimality conditions is obtained by considering the limits of the results for the approximating optimal control problem.

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“…for (x, t) ∈ Ω ×(0, ∞), where Ω ⊂ R n (n ≤ 3) is a bounded domain with smooth boundary, and all the parameters are arbitrarily given positive constants. It was Morales, Rojas, Torres and Rubio [10] who first derived the system (1.1)- (1.2), which is to model ternary mixtures by using the theory of pattern formation and of polyelectrolytes, with mean-field approximations. Recently, Morales, Rojas, Oliveros and Hernández derived a new mechanochemical model based on Coupled Ginzburg-Landau and Swift-Hohenberg equations in biological patterns of marine animals [9].…”

Section: Introductionmentioning

confidence: 99%