A NOTE ON LARGE TIME BEHAVIOUR OF SOLUTIONS FOR VISCOUS CAHN-HILLIARD EQUATION

This article is devoted to the discussion of large time behaviour of solutions for viscous Cahn-Hilliard equation with spatial dimension n 〈 5. Some results on global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial value are established.

Global Existence and Blow-up of Solutions to Multi-dimensional (n ≤ 5) Viscous Cahn-Hilliard Equation

In this paper we consider the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5, and established global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial data.

Viscous Cahn-Hilliard Equation with Concentration Dependent Mobility

This paper is devoted to viscous Cahn-Hiliiard equation with concentration dependent mobility. Some results on the existence, uniqueness and large time behavior are established.

Spectral Method for a Class of Cahn-Hilliard Equation with Nonconstant Mobility

In this paper, we propose and analyze a full-discretization spectral approximation for a class of Cahn-Hilliard equation with nonconstant mobility. Convergenee analysis and error estimates are presented and numerical experiments are carried out.

Two Dimensional Cahn-Hilliard Equation with Concentration Dependent Mobility and Gradient Dependent Potential

In this paper we consider the initial boundary value problem of Cahn-Hilliard equation with concentration dependent mobility and gradient dependent potential. By the L^P type estimates and the theory of Morrey spaces,we prove the Holder continuity of the solutions.Then we obtain the existence of global classical solutions.The present work can be viewed as an extension to the previous work on the Cahn-Hilliard equation with concentration dependent mobility and potential.

Convective-diffusive Cahn-Hilliard Equation with Concentration Dependent Mobility

In this paper, we study the global existence of classical solutions for the convective-diffusive Cahn-Hilliard equation with concentration dependent mobility. Based on the Schauder type estimates, we establish the global existence of classicalsolutions.

Existence and Blow-up of Smooth Solutions for Cahn-Hilliard Equation with Inertial Term

The motivation of this paper is to systematically study how the inertial term affects the behavior of the solution of the Cahn-Hilliard equation.When there is an inertial term,the equation becomes a parabolic-hyperbolic one.To overcome the difficulties from large perturbation and the hyperbolicity,a few elaborate energy estimates should be given,which are based on choosing suitable space of the smooth solution,even some inequalites in negative Sobolev space.More precisely,for 1-dimensional(1D)non-viscous case and 1D,2D,and 3D viscous case,the global existence of the smooth solutions are given.Moreover,several blow-up results are established by using the convex method.The results exhibit the interplay between the viscosity and the inertial term for the behavior of the smooth solution.

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization,and by applying the Fourier spectral collocation to the spatial discretization.In addition,two stabilization terms in different forms are added for the sake of the numerical stability.We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme,combined with the rough error estimate and the refined estimate.By regarding the numerical solution as a small perturbation of the exact solution,we are able to justify the discrete?^(∞)bound of the numerical solution,as a result of the rough error estimate.Subsequently,the refined error estimate is derived to obtain the optimal rate of convergence,following the established?∞bound of the numerical solution.Moreover,the energy stability is also rigorously proved with respect to a modified energy.The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work,and the energy stability estimate has greatly improved the corresponding result therein.

Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applicable to a variety of free-energy potentials,including Ginzburg-Landau and Flory-Huggins,to general wetting boundary conditions,and to degenerate mobilities.Its central thrust is the upwind methodology,which we combine with a semi-implicit formulation for the freeenergy terms based on the classical convex-splitting approach.The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature,which allows to efficiently solve higher-dimensional problems with a simple parallelisation.The numerical schemes are validated and tested through a variety of examples,in different dimensions,and with various contact angles between droplets and substrates.

A Structure-Preserving JKO Scheme for the SizeModified Poisson-Nernst-Planck-Cahn-Hilliard Equations

In this paper,we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard(SPNPCH)equations derived from the free energy including electrostatic energies,entropies,steric energies,and Cahn-Hilliard mixtures.Based on the Jordan-Kinderlehrer-Otto(JKO)framework and the Benamou-Brenier formula of quadratic Wasserstein distance,the SPNPCH equations are transformed into a constrained optimization problem.By exploiting the convexity of the objective function,we can prove the existence and uniqueness of the numerical solution to the optimization problem.Mass conservation and unconditional energy-dissipation are preserved automatically by this scheme.Furthermore,by making use of the singularity of the entropy term which keeps the concentration from approaching zero,we can ensure the positivity of concentration.To solve the optimization problem,we apply the quasi-Newton method,which can ensure the positivity of concentration in the iterative process.Numerical tests are performed to confirm the anticipated accuracy and the desired physical properties of the developed scheme.Finally,the proposed scheme can also be applied to study the influence of ionic sizes and gradient energy coefficients on ion distribution.

A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.

Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit,Nesterov Accelerated Schemes

We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent(PAGD)method.We discretize these problems with a Fourier collocation method in space,and employ various second-order schemes in time.We observe a significant speedup with this solver when compared to the preconditioned gradient descent(PGD)method.With the PAGD solver,fully implicit,second-order-in-time schemes are not only feasible to solve the PFC and FCH equations,but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account.Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that,for the FCH experiments,the fully implicit schemes(midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver)perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision.For the PFC,the results are not as conclusive as in the FCH experiments,which,we believe,is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation.We also discuss some practical matters in applying the PAGD.We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters.The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

The Maxwell-Heaviside Equations Explained by the Theory of Informatons

In the articles “Newtons Law of Universal Gravitation Explained by the Theory of Informatons” and “The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons” the gravitational interaction has been explained by the hypothesis that information carried by informatons is the substance of gravitational fields, i.e. the medium that the interaction in question makes possible. From the idea that “information carried by informatons” is its substance, it has been deduced that—on the macroscopic level—a gravitational field manifests itself as a dual entity, always having a field- and an induction component (Egand Bg) simultaneously created by their common sources. In this article we will mathematically deduce the Maxwell-Heaviside equations from the kinematics of the informatons. These relations describe on the macroscopic level how a gravitational field (Eg, Bg) is generated by whether or not moving masses and how spatial and temporal changes of Egand Bgare related. We show that there is no causal link between Egand Bg.