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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 .

International audience; The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

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.

Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.

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

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.

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 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 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.

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 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.

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.

The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formulation, where the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes that have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. In addition, unlike the recently populated IEQ or SAV approaches using auxiliary variables, our resulting energy laws are formulated in the original variables. This is a significant improvement, as the modified energy laws with auxiliary variables sometimes deviate from the original energy law. Our proposed framework lays a foundation for designing decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, various numerical algorithms can be obtained given different splitting steps, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order temporal and spatial accuracy are verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.

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

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on $\mathcal{M}$ can be approximated by the classical nonlocal-interaction equation on $\mathbb{R}^d$ by adding an external potential which strongly attracts to $\mathcal{M}$. The proof relies on the Sandier--Serfaty approach to the $\Gamma$-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on $\mathcal{M}$. Uniqueness, on the other hand, is established using a stability argument. We also provide an another approximation to the interaction equation on $\mathcal{M}$, based on iterating approximately solving an interaction equation on $\mathbb{R}^d$ and projecting to $\mathcal{M}$. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics., Comment: 24 pages, 8 figures

In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish \begin{document}$ \Gamma $\end{document} -convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.

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.

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.

We study distributional solutions to the radially symmetric aggregation equation for power-law potentials. We show that distributions containing spherical shells form part of a basin of attraction in the space of solutions in the sense of “shifting stability." For spherical shell initial data, we prove the exponential convergence of solutions to equilibrium and construct some explicit solutions for specific ranges of attractive power. We further explore results concerning the evolution and equilibria for initial data formed from convex combinations of spherical shells.

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.

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 consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary $\Gamma$-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels., Comment: 48 pages

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.

For gradient flows of energies, both spectral renormalization (SRN) and energy landscape (EL) techniques have been used to establish slow motion of orbits near low-energy manifold. We show that both methods are applicable to flows induced by families of gradients and compare the scope and specificity of the results. The SRN techniques capture the flow in a thinner neighborhood of the manifold, affording a leading order representation of the slow flow via as projection of the flow onto the tangent plane of the manifold. The SRN approach requires a spectral gap in the linearization of the full gradient flow about the points on the low-energy manifold. We provide conditions on the choice of gradient under which the spectral gap is preserved, and show that up to reparameterization the slow flow is invariant under these choices of gradients. The EL methods estimate the magnitude of the slow flow, but cannot capture its leading order form. However the EL only requires normal coercivity for the second variation of the energy, and does not require spectral conditions on the linearization of the full flow. It thus applies to a much larger class of gradients of a given energy. We develop conditions under which the assumptions of the SRN method imply the applicability of the EL method, and identify a large family of gradients for which the EL methods apply. In particular we apply both approaches to derive the interaction of multi-pulse solutions within the 1+1D Functionalized Cahn-Hilliard (FCH) gradient flow, deriving gradient invariance for a class of gradients arising from powers of a homogeneous differential operator., Comment: 28 pages

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space \begin{document}$ L^2(0,1)^2 $\end{document} according to the classical theory by Brezis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the \begin{document}$ L^m $\end{document} -norms for all \begin{document}$ m\in [1,+\infty] $\end{document} and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

In this paper we study a gradient flow generated by the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the space of $Q$-tensors. This free energy density functional is composed of three quadratic terms as the elastic energy density part, and a singular potential in the bulk part that is considered as a natural enforcement of a physical constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approaches the physical boundary. We give a rigorous proof that for rather general initial data with possibly infinite free energy, the system has a unique strong solution after any positive time $t_0$. Furthermore, this unique strong solution detaches from the physical boundary after a sufficiently large time $T_0$. We also give estimate of the Hausdorff measure of the set where the solution touches the physical boundary and thus prove a partial regularity result of the solution in the intermediate stage $(0,T_0)$.

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in Jacobs and Léger [Numer. Math.146(2020) 513–544.]. to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedo\={g}lu-Otto and Laux-Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedo\={g}lu-Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations., Comment: improved version; numerical experiments included; to appear in Arch. Ration. Mech. Anal

We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the L p optimal transport with L p distance. For p = 1 , we derive the corresponding L 1 generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple L 1 minimization which is solved efficiently by a primal-dual algorithm. For p = 2 , we derive the L 2 generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampere equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation arising from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework., published version

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn-Hilliard functional coupled with a small deformation energy We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen-Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long time behaviour raft-like structures can emerge and stablise, and their parameter dependence is further explored., 27 pages, 11 figures

We consider nonlinear, nonhomogeneous elliptic Dirichlet equations driven by the sum of a p‐Laplacian and a Laplacian (so‐called (p, 2)‐equation). We are concerned with both cases 1 2. In the first one, the reaction f(z,x) is linear grow near ±∞ and resonant with respect to a nonprincipal nonnegative eigenvalue. In the second case, the reaction f(z,·) is (p−1)‐superlinear near ±∞ and has z‐dependent zeros of constant sign. Using variational methods together with flow invariance arguments, we establish the existence of nodal solutions.