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Bilayer plates are compound materials that exhibit large bending deformations when exposed to environmental changes that lead to different mechanical responses in the involved materials. In this article a new numerical method that is suitable for simulating the isometric deformation induced by a given material mismatch in a bilayer plate is discussed. A dimensionally reduced formulation of the bending energy is discretized generically in an abstract setting and specified for discrete Kirchhoff triangles; convergence towards the continuous formulation is proved. A practical semi-implicit discrete gradient flow employing a linearization of the isometry constraint is proposed as an iterative method for the minimization of the bending energy; stability and a bound on the violation of the isometry constraint are proved. The incorporation of obstacles is discussed and the practical performance of the method is illustrated with numerical experiments involving the simulation of large bending deformations and investigation of contact phenomena.

In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the$L^2$-bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgsk-functional with Higgs self-interaction, we show that, provided$\dim (M), for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local$L^2$-derivative estimates, energy estimates and blow-up analysis.

We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to study several one- and two-dimensional systems, with a general free-energy functional accounting for external fields and inter-particle potentials, and placed in non-homogeneous thermal baths characterized by anisotropic, space-dependent and time-dependent properties.

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

We consider a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance | x i − x j | p + 2 for any p ≠ 0 . In this paper, we employ both attractive and repulsive couplings to study the asymptotic behavior of the system accompanied by both p > 0 (positive range) and p 0 (negative range), and this enables to yield richer dynamical phenomena. Firstly in an attractive regime, we focus on the emergence of the complete aggregation; however, the relaxation dynamics towards the aggregated state for the positive range differs from the one for the negative range. More precisely for p > 0 , the complete aggregation occurs with an algebraic rate O ( t − 1 / p ) . On the other hand for p 0 , the issue of global existence arises due to the singular interaction and is crucially related to the aggregation estimate. To this end, we show that the complete aggregation emerges in finite time and thus a solution exists until such a time. Lastly in a repulsive regime, we mainly consider the splay state for both positive and negative ranges, and several case studies are presented to obtain the qualitative insight. Finally, we compare our results with the case of p = 0 .

The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evol...

In the Stiefel manifold X n , k , we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n − 2 k . The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞ {t\to\infty} .

We consider the one-constant Landau-de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $$\mathbf{Q}$$ , which is constrained to be uniaxial: $$\mathbf{Q}= s (\mathbf{n}\otimes \mathbf{n}- d^{-1}\mathbf{I})$$ where $$\mathbf{n}$$ is a director field, $$s\in \mathbb {R}$$ is the degree of orientation, and $$d\ge 2$$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $$\Gamma $$ -convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method’s capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.

We study the systematic numerical approximation of a class of Allen–Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. and support the observation that their hybrid Allen–Cahn model avoids the problem of mesh-locking to a large extent.

We study the gradient flow of the Mobius energy introduced by O’Hara (Topology 30:2 (1991), 241–247). We will show a fundamental 𝜀-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He (

We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy., 27 pages

We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit l...

We propose numerical schemes for a class of Keller--Segel equations. The discretization is based on the gradient flow structure. The resulting first-order scheme is mass conservative, bound preserv...

In this paper, we develop a new square phase-field crystal model using the L 2 -gradient flow approach, where the total mass of atoms is conserved through a nonlocal Lagrange multiplier. We construct a fast, provably unconditionally energy stable, second-order scheme by using the recently developed SAV approach with the stabilization technique, where an extra stabilization term is added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with the classical Cahn–Hilliard type square phase-field crystal model and the non-stabilized SAV scheme for simulating some benchmark numerical examples, we demonstrate the robustness of the new model, as well as the stability and the accuracy of the developed scheme, numerically.

Journal of computational physics 436, 110165 (2021). doi:10.1016/j.jcp.2021.110165, Published by Elsevier, Amsterdam

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^+$ happens., Comment: 15 pages

In this paper, we study the ground and first excited states of the fractional Bose–Einstein condensates (BEC) which is modeled by fractional Gross–Pitaevskii (GP) equation. We first introduce the normalized gradient flow method and prove its energy diminishing property. Then the weighted shifted Grunwald–Letnikov difference (WSGD) method is used to discretize the Gross–Pitaevskii equation. The corresponding normalization and energy diminishing property for the semi-discrete scheme are proved. For the time discretization, we use the implicit integration factor (IIF) method which decouples the diffusion and nonlinear terms separately. Finally the numerical methods are applied to compute the ground and first excited states of fractional BEC with harmonic oscillator, harmonic-plus-optical lattice and box potential. Our numerical results show that the ground and excited states in fractional GP equation differ from those of the standard (non-fractional) GP equation.

The space of $$G_2$$ -structures is naturally stratified by those structures compatible with a fixed Riemannian metric. We study the restriction of the total torsion energy functional to these strata. Precisely we show the short-time existence of its negative gradient flow, we characterise the space of its critical points in terms of spinor fields and, finally, we describe the long-time behaviour of the homogeneous negative gradient flow.

In this letter, we present a novel class of arbitrarily high-order and unconditionally energy-stable algorithms for gradient flow models by combining the energy quadratization (EQ) technique and a specific class of Runge–Kutta (RK) methods, which is named the EQRK schemes. First of all, we introduce auxiliary variables to transform the original model into an equivalent system, with the transformed free energy a quadratic functional with respect to the new variables and the modified energy dissipative law is conserved. Then a special class of RK methods is employed for the reformulated system to arrive at structure-preserving time-discrete schemes. Along with rigorous proofs, numerical experiments are presented to demonstrate the accuracy and unconditionally energy-stability of the EQRK schemes.

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.

We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.

This work investigates the formation of singularities under the steepest descent $$L^2$$ -gradient flow of $${{\,\mathrm{{ W}}\,}}_{\lambda _1, \lambda _2}$$ with zero spontaneous curvature, i.e., the sum of the Willmore energy, $$\lambda _1$$ times the area, and $$\lambda _2$$ times the signed volume of an immersed closed surface without boundary in $$\mathbb {R}^3$$ . We show that in the case that $$\lambda _1>1$$ and $$\lambda _2=0$$ , any immersion develops singularities in finite time under this flow. If $$\lambda _1 >0$$ and $$\lambda _2 > 0$$ , embedded closed surfaces with energy less than $$\begin{aligned} 8\pi +\min \left\{ \left( 16 \pi \lambda _1^3\right) \bigg /\left( 3\lambda _2^2\right) , 8\pi \right\} \end{aligned}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than $$8 \pi $$ , the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that $$\lambda _2$$ is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For $$\lambda _1 >0$$ and $$\lambda _2 \ge 0$$ , they showed that embedded closed spheres with positive volume and energy close to $$4\pi $$ , i.e., close to the Willmore energy of a round sphere, converge to round points in finite time.

In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L 2 {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L 2 {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called L 2 {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the L 2 {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

In this paper we derive optimal algebraic-in-time relaxation rates to the kink for the Cahn-Hilliard equation on the line. We assume that the initial data have a finite distance---in terms of either a first moment or the excess mass---to a kink profile and capture the decay rate of the energy and the perturbation. Our tools include Nash-type inequalities, duality arguments, and Schauder estimates.

A novel algorithmic approach is presented for the problem of partitioning a connected undirected weighted graph under constraints such as cardinality or membership requirements or must-link and can...

We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([12, 11]). The proof is based on a Lojasiewicz-Simon inequality.

The Merriman–Bence–Osher (MBO) scheme, also known as diffusion generated motion or thresholding, is an efficient numerical algorithm for computing mean curvature flow (MCF). It is fairly well understood in the case of hypersurfaces. This paper establishes the first convergence proof of the scheme in codimension two. We concentrate on the case of the curvature motion of a filament (curve) in $${{\mathbb{R}}^3}$$ . Our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces (Esedoglu–Otto ’15) by means of an energy that approximates the Dirichlet energy of the state function. As long as a smooth MCF exists, we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this MCF. The current result that holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy. This is an improvement of an earlier approximate version.

We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.

We study the small Deborah number limit of the Doi–Onsager equation in the case when hydrodynamic effects are neglected. This is a Smoluchowski-type equation that describes the dynamics of nematic liquid crystals at a molecular level, by characterizing the evolution of a number density function, depending upon both particle position x ∈ R d ( d = 2 , 3 ) and orientation vector m ∈ S 2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S 2 . This flow is a special case of the gradient flow of the well known Oseen–Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions. The proof relies on the strong compactness of the corresponding Q-tensor (namely the second moment), a detailed analysis of the linearized operator near the limit local equilibrium distribution, and an energy dissipation estimate.

A proper mathematical model given by nonlinear Poisson–Nernst–Planck (PNP) equations which describe electrokinetics of charged species is considered. The model is generalized with entropy variables associating the pressure and quasi-Fermi electro-chemical potentials in order to adhere to the law of conservation of mass. Based on a variational principle for suitable free energy, the generalized PNP system is endowed with the structure of a gradient flow. The well-posedness theorems for the mixed formulation (using the entropy variables) of the gradient-flow problem are provided within the Gibbs simplex and supported by a-priori estimates of the solution.

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto’s Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are, however, restricted to the 1D case of evolution of loops.

A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.

We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii), and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimisation of free energy functionals.

In this paper, we study a doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow. In the special case p=2 our theory includes the classical Yamabe flow on a bounded domain in Euclidean space. Our main aim is to prove the global existence of the p-Sobolev flow together with its qualitative properties., arXiv admin note: text overlap with arXiv:2103.15259

In this paper we study the $$L^2$$ -gradient flow of the penalized elastic energy on networks of q-curves in $$\mathbb {R}^{n}$$ for $$q \ge 3$$ . Each curve is fixed at one end-point and at the other is joint to the other curves at a movable q-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov’s theory on linear parabolic systems and Banach fixed point theorem in proper Holder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time representation of stochastic gradient descent. The stochastic gradient process is a dynamical system that is coupled with a continuous-time Markov process living on a finite state space. The dynamical system—a gradient flow—represents the gradient descent part, the process on the finite state space represents the random subsampling. Processes of this type are, for instance, used to model clonal populations in fluctuating environments. After introducing it, we study theoretical properties of the stochastic gradient process: We show that it converges weakly to the gradient flow with respect to the full target function, as the learning rate approaches zero. We give conditions under which the stochastic gradient process with constant learning rate is exponentially ergodic in the Wasserstein sense. Then we study the case, where the learning rate goes to zero sufficiently slowly and the single target functions are strongly convex. In this case, the process converges weakly to the point mass concentrated in the global minimum of the full target function; indicating consistency of the method. We conclude after a discussion of discretisation strategies for the stochastic gradient process and numerical experiments.

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment., 33 pages, comments are welcome

The Cahn--Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. In recent times, various dynamic boundary conditions have been introduced to model interactions of the materials with the boundary more precisely. To take long-range interactions of the materials into account, we propose a new model consisting of a nonlocal Cahn--Hilliard equation subject to a nonlocal dynamic boundary condition that is also of Cahn--Hilliard type and contains an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal total free energy with respect to a suitable inner product of order $H^{-1}$ which contains both bulk and surface contributions. The total free energy is considered as nonlocal since it comprises convolutions in the bulk and on the surface of the phase-field variables with certain interaction kernels. The main difficulties arise from defining a suitable kernel on the surface and from handling the resulting boundary convolution. In the main model, the chemical potentials in the bulk and on the surface are coupled by a Robin type boundary condition depending on a specific relaxation parameter related to the rate of chemical reactions. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when the relaxation parameter tends to zero or infinity. By this approach, we also obtain weak and strong well-posedness of the corresponding limit systems.

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. The MBP plays a crucial role in understanding the physical meaning and the wellposedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge–Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen–Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen–Cahn type of equations.

We consider the mass preserving $L^2$-gradient flow of the strong scaling of the functionalized Cahn Hilliard gradient flow and establish the nonlinear stability of a manifold comprised of quasi-equilibrium bilayer \muckmucks up to the manifold's boundary. In the limit of thin but non-zero interfacial width, $\varepsilon\ll1,$ the bilayer manifold is parameterized by meandering modes that describe the interfacial evolution and "pearling" modes that control the structure of the profile near the interface. The pearling modes are weakly damped and can lead to the dynamic rupture of the interface. Amphiphilic interfaces can lengthen to decrease energy. We introduce an implicitly defined parameterization of the interfacial shape that uncouples this growth from the parameters describing the shape and introduce a nonlinear projection onto the manifold from a surrounding neighborhood. The bilayer manifold has asymptotically large but finite dimension tuned to maximize normal coercivity while preserving the wave-number gap between the meandering and the pearling modes. Modulo a pearling stability assumption, we show that the manifold attracts nearby orbits into a tubular neighborhood about itself so long as the interfacial shape remains sufficiently smooth and far from self-intersection. In a companion paper, arXiv:1907.02196, we identify open sets of initial data whose orbits converge to circular equilibrium after a significant transient, and derive a singularly perturbed interfacial evolution comprised of motion against curvature regularized by an asymptotically weak Willmore term., arXiv admin note: text overlap with arXiv:1907.02196

We consider a one-dimensional discrete particle system of two species coupled through nonlocal interactions driven by the Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the $$L^m$$ -norms of a piecewise constant reconstruction of the density using the particle trajectories.

International audience; The problem of minimizing the sum, or composition, of two objective functions is a frequent sight in the field of optimization. In this article, we are interested in studying relations between the discrete-time gradient descent algorithms used for optimization of such functions and their corresponding gradient flow dynamics, when one of the functions is in particular time-dependent. It is seen that the subgradient of the underlying convex function results in differential inclusions with time-varying maximal monotone operator. We describe an algorithm for discretization of such systems which is suitable for numerical implementation. Using appropriate tools from convex and functional analysis, we study the convergence with respect to the size of the sampling interval. As an application, we study how the discretization algorithm relates to gradient descent algorithms used for constrained optimization.

We define the elastic energy of smooth immersed closed curves in $${\mathbb {R}}^n$$ as the sum of the length and the $$L^2$$ -norm of the curvature, with respect to the length measure. We prove that the $$L^2$$ -gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Łojasiewicz–Simon inequality, which is at the core of the proof, in a quite concise and versatile way.

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance., arXiv admin note: text overlap with arXiv:1908.10246

International audience; Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such problems which preserve the variational structure and have second order accuracy in space. We propose an interior point method to solve the discrete variational problem, providing an efficient and robust algorithm. We present two applications to test the scheme and show its order of convergence.

We propose and analyze a linearly stabilized semi-implicit diffusive Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme, the nonlinear bulk force is treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation. Rigorous error analysis is carried out for the fully discrete scheme. When the time step-size and the space step-size are small enough, second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of $1/\varepsilon$. {Here $\varepsilon$ is the thickness of the interface}. Numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme., arXiv admin note: text overlap with arXiv:1710.03604, arXiv:1708.09763

We study a fourth-order, singular, nonlinear PDE model for surface relaxation. A weak solution for the model is defined using an inequality formulation. A numerical scheme based on semi-implicit time stepping, mixed finite elements and regularization is proposed to approximate the PDE model. We investigate the convergence of the scheme with respect to the discretization and the regularization parameters. Finally, formal arguments show that the model can be viewed as a gradient flow with respect to an appropriate Riemannian metric.

We consider a class of Fokker--Planck equations with linear diffusion and superlinear drift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case, solutions will blow up in $L^\infty$ in finite time and---understood in a generalised, measure sense---they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if $d\ge3$., Accepted version