Formal Gradient-flow Structure Research Articles

Gradient Flows for Coupling Order Parameters and Mechanics

We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution, and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally brief, we construct an extensible functional analytical framework and a discretization approach that preserves the underlying variational structure. We consider phase separation and swelling of gels and then study stationary states of multiphase systems with surface tension and contact lines and show the robustness of the general approach for large deformations. We highlight differences between compressible and incompressible models and discuss issues of the sharp-interface limit for different magnitudes of the Cahn–Hilliard mobility.

On multi-species diffusion with size exclusion

We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it provides a weak–strong stability estimate for a wide range of coefficients, which had been missing so far.In order to achieve the results mentioned above, we exploit the formal gradient-flow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the full-interaction case, where all cross-diffusion coefficients are non-zero. Those are crucial to obtain the minimal Sobolev regularity needed for a weak–strong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the full-interaction case.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

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.

Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

AbstractAn implicit Euler finite-volume scheme for a cross-diffusion system modeling biofilm growth is analyzed by exploiting its formal gradient-flow structure. The numerical scheme is based on a two-point flux approximation that preserves the entropy structure of the continuous model. Assuming equal diffusivities the existence of non-negative and bounded solutions to the scheme and its convergence are proved. Finally, we supplement the study by numerical experiments in one and two space dimensions.

Exponential Time Decay of Solutions to Reaction-Cross-Diffusion Systems of Maxwell\u2013Stefan Type

The large-time asymptotics of weak solutions to Maxwell–Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. The key elements of the proof are the existence of a unique detailed-balance equilibrium and the derivation of an inequality relating the entropy and the entropy production. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. The idea is to enlarge the space of n partial concentrations by adding the total concentration, viewed as an independent variable, thus working with n+1 variables. Further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex-balance systems.

An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems

As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in Zhang (2017) [42] in many scenarios. Hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

Energy-transport systems for optical lattices: Derivation, analysis, simulation

Energy-transport equations for the transport of fermions in optical lattices are formally derived from a Boltzmann transport equation with a periodic lattice potential in the diffusive limit. The limit model possesses a formal gradient-flow structure like in the case of the energy-transport equations for semiconductors. At the zeroth-order high-temperature limit, the energy-transport equations reduce to the whole-space logarithmic diffusion equation which has some unphysical properties. Therefore, the first-order expansion is derived and analyzed. The existence of weak solutions to the time-discretized system for the particle and energy densities with periodic boundary conditions is proved. The difficulties are the nonstandard degeneracy and the quadratic gradient term. The main tool of the proof is a result on the strong convergence of the gradients of the approximate solutions. Numerical simulations in one-space dimension show that the particle density converges to a constant steady state if the initial energy density is sufficiently large, otherwise the particle density converges to a nonconstant steady state.

Nonlinear degenerate cross-diffusion systems with nonlocal interaction

We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications. Assuming a uniform “coerciveness” assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan–Kinderlehrer–Otto scheme. Our approach allows to consider nonlocal interaction terms not necessarily yielding a formal gradient flow structure.

Well-posedness of evolution equations with time-dependent nonlinear mobility: A modified minimizing movement scheme

AbstractWe prove the existence of nonnegative weak solutions to a class of second- and fourth-order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space {{{\mathbb{R}}^{d}}}, for arbitrary {d\geq 1}. Exploiting a very formal gradient flow structure, the cornerstone of our proof is a modified version of the classical minimizing movement scheme for gradient flows. The mobility function is required to satisfy – at each time point separately – the conditions by which one can define a modified Wasserstein distance on the space of probability densities with finite second moment. The explicit dependency on the time variable is assumed to be at least of Lipschitz regularity. We also sketch possible extensions of our result to the case of bounded spatial domains and more general mobility functions.

Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L 2-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.

The gradient flow structure for incompressible immiscible two-phase flows in porous media

We show that the widely used model governing the motion of two incompressible immiscible fluids in a possibly heterogeneous porous medium has a formal gradient-flow structure. More precisely, the fluid composition is governed by the gradient flow of some non-smooth energy. Starting from this energy together with a dissipation potential, we recover the celebrated Darcy–Muskat law and the capillary pressure law governing the flow thanks to the steepest descent condition for the energy. Our interpretation does not require the introduction of any algebraic transformation like, e.g., the global pressure or the Kirchhoff transform, and can be transposed to the case of more phases.

The boundedness-by-entropy method for cross-diffusion systems

The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.

Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak–Keller–Segel model

This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic–elliptic semilinear Patlak–Keller–Segel system in a ball of RN, exhibiting a critical mass phenomenon for N≥2. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case. We stress that this covers in particular the classical Keller–Segel system with N=2, well known for its critical mass 8π, and that the result improves on the known results even for this most studied problem. A key tool is an associated one-dimensional degenerate parabolic problem. The proof exploits its formal gradient flow structure ut=−∇F[u(t)] on an “infinite dimensional Riemannian manifold”. In particular, we show a new Hardy type inequality, equivalent to the strict convexity of F at any steady state of subcritical mass, which heuristically explains the exponential speed of convergence.

Dynamic Stability of Equilibrium Capillary Drops

We investigate a model for contact angle motion of quasi-static capillary drops resting on a horizontal plane. We prove global in time existence and long time behavior (convergence to equilibrium) in a class of star-shaped initial data for which we show that topological changes of drops can be ruled out for all times. Our result applies to any drop which is initially star-shaped with respect to a small ball inside the drop, given that the volume of the drop is sufficiently large. For the analysis, we combine geometric arguments based on the moving-plane type method with energy dissipation methods based on the formal gradient flow structure of the problem.