Energy Dissipation Inequality Research Articles

Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow

We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and 3, primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or Röger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak formulation of the contact angle at the intersection of the interface and the domain boundary. To incorporate these, we introduce a functional framework encoding a weak solution concept for Mullins–Sekerka flow essentially relying only on (i) a single sharp energy dissipation inequality in the spirit of De Giorgi, and (ii) a weak formulation for an arbitrary fixed contact angle through a distributional representation of the first variation of the underlying capillary energy. Both ingredients are intrinsic to the interface of the evolving phase indicator and an explicit distributional PDE formulation with potentials can be derived from them. The existence of weak solutions is established via subsequential limit points of the naturally associated minimizing movements scheme. Smooth solutions are consistent with the classical Mullins–Sekerka flow, and even further, we expect our solution concept to be amenable, at least in principle, to the recently developed relative entropy approach for curvature driven interface evolution.

Superelasticity degradation of NiTi shape memory alloy in wide ranges of temperature and loading level: Experimental observation and micromechanical constitutive model

In this work, cyclic tension-unloading tests with different peak strains (10%-18%) in the temperature range from 313 K to 393 K are performed to investigate the effects of temperature and loading level on the cyclic deformation behavior of NiTi shape memory alloy (SMA) wires. Experimental results demonstrate that the superelasticity degradation occurs during the cyclic deformation process, and such phenomenon aggravates with the rise of ambient temperature and loading level, which can be attributed to the complex interactions among the martensite transformation (MT), austenite plasticity (AP), martensite plasticity (MP) and the transformation-induced plasticity (TRIP). Then, a micromechanical cyclic constitutive model is proposed based on the framework of irreversible thermodynamics and Eshelby's inclusion theory. The martensite (M) phase is treated as mobile inclusions with alterable eigenstrain embedded in the austenite (A) phase matrix. The volume fractions of the M-phase, A-phase, and A-M interface-phase are introduced. The non-uniform stress fields in the three phases are estimated by employing the Mori-Tanaka's homogenization scheme and interfacial operator. The driving forces of the four types of the inelastic deformation mechanisms, MT, AP, MP, and TRIP are derived based on the proposed new Helmholtz free energy, instantaneous growth hypothesis of M domains, and the energy dissipation inequality. The inheritances of the plastic deformation induced by the movement of A-M interfaces during the repeated MT and its reverse are incorporated. Finally, to validate the predictive capability of the proposed model, the predicted results for the superelasticity degradation of NiTi SMA with various peak strains at different ambient temperatures are compared with the experimental ones. Moreover, the dominant plastic deformation mechanisms and the effects of AP, MP, TRIP on the cyclic deformation of NiTi SMA are discussed.

Cyclic degeneration of elastocaloric effect for NiTi shape memory alloy: Experimental observation and constitutive model

Based on the cyclic tension-unloading tests with various peak strains (4%-12%), the effect of loading level on the cyclic deformation and evolution of elastocaloric effect (eCE) for NiTi shape memory alloy (SMA) wires was investigated. Experimental results show that the transformation ratchetting and degeneration of eCE simultaneously occur during the cyclic deformation, and these two phenomena aggravate with the increase of loading level. Although the refrigeration capacity (maximum temperature drop in the reverse phase transformation process) exhibits a positive relativity with loading level in the 1st loading cycle, it changes non-monotonically with the variation of loading level in the steady-state cycle, which can be attributed to the interaction between the phase transformation and plasticity deformation. Moreover, it is found that the phase transformation between austenite (A) and martensite (M) phases occurs as a two-step process involving an intermediate phase, rhombohedral (R) phase. Then, a thermo-mechanically coupled cyclic constitutive model is constructed based on the fundamental laws of irreversible thermodynamics. Besides the thermo-elastic deformation, four different inelastic deformation mechanisms, i.e., R phase transformation (RT), martensite transformation (MT), plasticity in A and R phases (ARP), transformation-induced plasticity (TRIP) and their interactions are considered in the proposed model. The volume fractions of R phase, M phase, martensite affecting zone (MAZ) and the free zone (FZ) of mixed A + R phases are introduced. The thermodynamic driving forces for the four considered inelastic deformation mechanisms, i.e., RT, MT, ARP and TRIP are derived by the energy dissipation inequality and a new proposed Helmholtz free energy. Internal heat generation caused by the mechanical dissipation and the latent heat of phase transformation, and the heat balance equation are obtained by applying the conservation law of energy. The forward and reverse inheritances of dislocation (i.e., plastic deformation) caused by the moving martensite interfaces during MT are addressed. Adopting a first-order approximation of the constitutive equations and lumped analysis of heat transfer, the proposed constitutive model for a material point is extended to describe the overall responses of the whole wire. Finally, the predicted results for the cyclic deformation of NiTi SMA and correspondent evolution of eCE are compared against the experimental ones to validate the capability of the proposed model.

An energy stable linear numerical method for thermodynamically consistent modeling of two-phase incompressible flow in porous media

In this paper, we consider numerical approximation of a thermodynamically consistent model of two-phase flow in porous media, which obeys an intrinsic energy dissipation law. The model under consideration is newly-developed, so there is no energy stable numerical scheme proposed for it at present. This model consists of two nonlinear degenerate parabolic equations and a saturation constraint, but lacking an independent equation for the pressure, so for the purpose of designing efficient numerical scheme, we reformulate the model forms as well as the free energy function, and further prove the corresponding energy dissipation inequality. Based on the alternative reformulations, using the invariant energy quadratization approach and subtle semi-implicit treatments for the pressure and saturation, we for the first time propose a linear and energy stable numerical method for this model. The fully discrete scheme is devised combining the upwind approach for the phase mobilities and the cell-centered finite difference method. The unique solvability of numerical solutions and unconditional energy stability are rigorously proved for both the semi-discrete time scheme and the fully discrete scheme. Moreover, the scheme can guarantee the local mass conservation for both phases. We also show that the upwind mobility approach plays an essential role in preserving energy stability of the fully discrete scheme. Numerical results are presented to demonstrate the performance of the proposed scheme.

On nonnegative solutions for the Functionalized Cahn–Hilliard equation with degenerate mobility

The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M(u) that is zero for u≤0. Assuming the initial data u0(x) is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.

Brakke\u2019s inequality for the thresholding scheme

We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke’s notion of mean curvature flow. Our proof is based on a localized version of the minimizing movements interpretation of Esedoğlu and the second author. We apply De Giorgi’s variational interpolation to the thresholding scheme and pass to the limit in the resulting energy-dissipation inequality. The result is conditional in the sense that we assume the time-integrated energies of the approximations to converge to those of the limit.

On Patlak-Keller-Segel system for several populations: A gradient flow approach

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.

Convergence to equilibria of global solutions to a degenerate quasilinear Keller–Segel system

This paper is devoted to the longtime behavior of global weak solutions to initial-boundary value problems of the following degenerate quasilinear Keller–Segel system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u^m-\nabla \cdot (u\nabla v), &{}\quad x\in \Omega ,\;t>0\\ v_t=\Delta v-v+\gamma u, &{}\quad x\in \Omega ,\;t>0. \end{array}\right. }\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (0.1) \end{aligned}$$ Here $$\Omega \subset \mathbb {R}^n$$ is a bounded domain, $$\gamma >0$$ and $$m>\max \{1,2-\frac{2}{n}\}$$ . First, we establish the existence of global weak solution which additionally satisfies an energy dissipation inequality. Thanks to the energy inequality, we prove that the global weak solution will converge to an equilibrium as time goes to infinity if $$m\ge 2$$ and $$\gamma <\frac{m}{m-1} M_0^{m-2}$$ , where $$M_0$$ is the average mass of cells. The proof is based on an application of a slightly modified Lojasiewicz–Simon inequality of non-smooth type, where the requirement on compactness of the trajectory is weakened compared with those in Feireisl et al. (J Differ Equ 236:551–569, 2007), Jiang and Zhang (Asymptot Anal 65:79–102, 2009). Moreover, in the special case $$m=2$$ and $$\gamma <2$$ , convergence toward trivial solution $$(M_0,\gamma M_0)$$ is verified by an alternative straightforward way.

Global Existence Results for Viscoplasticity at Finite Strain

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

A Gradient Flow Approach to the Porous Medium Equation with Fractional Pressure

We consider a family of fractional porous media equations, recently studied by Caffarelli and V\'azquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.

A coupled theory for chemically active and deformable solids with mass diffusion and heat conduction

To analyse the frequently encountered thermo-chemo-mechanical problems in chemically active material applications, we develop a thermodynamically-consistent continuum theory of coupled deformation, mass diffusion, heat conduction and chemical reaction. Basic balance equations of force, mass and energy are presented at first, and then fully coupled constitutive laws interpreting multi-field interactions and evolving equations governing irreversible fluxes are constructed according to the energy dissipation inequality and the chemical kinetics. To consider the essential distinction between mass diffusion and chemical reactions in affecting free energy and dissipations of a highly coupled system, we regard both the concentrations of diffusive species and the extent of reaction as independent state variables. This new formulation then distinguishes between the energy contribution from the diffusive species entering the solid and that from the subsequent chemical reactions occurring among these species and the host solid, which not only interact with stresses or strains in different manners and on different time scales, but also induce different variations of solid microstructures and material properties. Taking advantage of this new description, we further establish a specialized isothermal model to predict precisely the transient chemo-mechanical response of a swelling solid with a proposed volumetric constraint that accounts for material incompressibility. Coupled kinetics is incorporated to capture the volumetric swelling of the solid caused by imbibition of external species and the simultaneous dilation arised from chemical reactions between the diffusing species and the solid. The model is then exemplified with two numerical examples of transient swelling accompanied by chemical reaction. Various ratios of characteristic times of diffusion and chemical reaction are taken into account to shed light on the dependency on kinetic time scales of evolution patterns for a diffusion-reaction controlled deformable solid.

The exponential formula for the wasserstein metric

A recurring obstacle in the study of Wasserstein gradient flow is the lack of convexity of the square Wasserstein metric. In this paper, we develop a class of transport metrics that have better convexity properties and use these metrics to prove an Euler−Lagrange equation characterizing Wasserstein discrete gradient flow. We then apply these results to give a new proof of the exponential formula for the Wasserstein metric, mirroring Crandall and Liggett’s proof of the corresponding Banach space result [M.G. Crandall and T.M. Liggett, Amer. J. Math. 93 (1971) 265–298]. We conclude by using our approach to give simple proofs of properties of the gradient flow, including the contracting semigroup property and energy dissipation inequality.

Mechanistic procedure for parameter determination of multiplicative decomposition based constitutive models

By exploiting the Clausius–Planck local energy dissipation inequality, a large strain, three-dimensional constitutive model has been developed for the monotonic and cyclic response prediction of various asphaltic materials. The model consists of a Zener non-linear, visco-elastic component acting in series with a stress dependent viscous component. A novel computational scheme has been developed for solution of the coupled system of equations expressing the interdependent response of these two in series components. An explicit, mechanistic, parameter determination procedure is presented for the laboratory determination of all necessary model parameters. Examples of model parameter determination and utilisation for prediction of the response of a recycled asphalt mix and a stone mastic asphalt mix are presented.

A free energy satisfying finite difference method for Poisson–Nernst–Planck equations

In this work we design and analyze a free energy satisfying finite difference method for solving Poisson–Nernst–Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.

A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows

We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant. To cite this article: F. Bouchut et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).