Stefan System Research Articles

Global existence of weak solutions and weak–strong uniqueness for nonisothermal Maxwell–Stefan systems *

The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell–Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modelling. Furthermore, a conditional weak–strong uniqueness property is shown by using the relative entropy method.

Safe PDE Backstepping QP Control With High Relative Degree CBFs: Stefan Model With Actuator Dynamics

High-relative-degree control barrier functions (hi-rel-deg CBFs) play a prominent role in automotive safety and in robotics. In this paper we launch a generalization of this concept for PDE control, treating a specific, physically-relevant model of thermal dynamics where the boundary of the PDE moves due to a liquid-solid phase change—the so-called Stefan model. The familiar QP design is employed to ensure safety but with CBFs that are infinite-dimensional (including one control barrier “functional”) and with safe sets that are infinite-dimensional as well. Since, in the presence of actuator dynamics, at the boundary of the Stefan system, this system's main CBF is of relative degree two, an additional CBF is constructed, by backstepping design, which ensures the positivity of all the CBFs without any additional restrictions on the initial conditions. It is shown that the “safety filter” designed in the paper guarantees safety in the presence of an arbitrary operator input. This is similar to an automotive system in which a safety feedback law overrides—but only when necessary—the possibly unsafe steering, acceleration, or braking by a vigorous but inexperienced driver. Simulations have been performed for a process in metal additive manufacturing, which show that the operator's heat-and-cool commands to the Stefan model are being obeyed but without the liquid ever freezing.

Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems

A Maxwell–Stefan system for fluid mixtures with driving forces depending on Cahn–Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The nonconvex part of the energy is assumed to have a bounded Hessian. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive-definiteness of the matrix on a subspace and using the Bott–Duffin matrix inverse. The global existence of weak solutions and a weak–strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding H^2(\Omega) bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

Weak-Strong Uniqueness for Maxwell--Stefan Systems

The weak-strong uniqueness for Maxwell--Stefan systems and some generalized systems is proved. The corresponding parabolic cross-diffusion equations are considered in a bounded domain with no-flux boundary conditions. The key points of the proofs are various inequalities for the relative entropy associated with the systems and the analysis of the spectrum of a quadratic form capturing the frictional dissipation. The latter task is complicated by the singular nature of the diffusion matrix. This difficulty is addressed by proving its positive definiteness on a subspace and using the Bott--Duffin matrix inverse. The generalized Maxwell--Stefan systems are shown to cover several known cross-diffusion systems for the description of tumor growth and physical vapor deposition processes.

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.

Control of the Stefan System and Applications: A Tutorial

The so-called Stefan system describes the dynamical model of the liquid–solid phase change in materials ranging from water and ice in the polar caps to metal casting and additive manufacturing (3D printing). The mathematical structure is given by a partial differential equation (PDE) with a moving boundary governed by a scalar ordinary differential equation. Because of the system's moving-boundary nature, control of the Stefan model is unconventional even within the class of otherwise challenging PDE control problems. The second decade of the twenty-first century has witnessed remarkable advances in control design for the Stefan system. Such advances carry significant potential in several areas of technology. In this article, we briefly review the principal literature on control of the Stefan model, along with the associated basics of the PDE analysis of the model and select applications. Principal ideas from our work on control design, stability analysis, and the maintenance of physical phase constraints are given sufficient attention and tutorial treatment so that the article can serve as a self-contained point of entry into the growing subject of boundary control of the Stefan system using the method of PDE backstepping.

Towards implementation of PDE control for Stefan system: Input-to-state stability and sampled-data design

The Stefan system is a representative model for a liquid–solid phase change which describes the dynamics of a material’s temperature profile and the liquid–solid interface position. Our previous work designed a boundary feedback control to stabilize the phase interface position modeled by the Stefan system. This paper resolves two issues our previous work did not study, that are, the robustness analysis under the unknown heat loss and the digital control action. First, we introduce the one-phase Stefan problem with a heat loss by modeling a 1-D diffusion Partial Differential Equation (PDE) dynamics of the liquid temperature and the interface position governed by an Ordinary Differential Equation (ODE) with a time-varying disturbance. We focus on the closed-loop system under the control law proposed in our previous work, and show an estimate of L2 norm in a sense of Input-to-State Stability (ISS) with respect to the unknown heat loss. Second, we consider the sampled-data control of the one-phase Stefan problem without the heat loss, by applying Zero-Order-Hold (ZOH) to the control law in our previous work. We prove that the closed-loop system under the sampled-data control law satisfies the global exponential stability in the spatial L2 norm. Analogous ISS result for the two-phase Stefan problem which incorporates the dynamics of the solid phase is also obtained. Numerical simulation verifies our theoretical results for showing the robust performance under the heat loss and the digital control implemented to vary at each sampling time.

Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method

The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong–Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brzeźniak and coworkers. Three-species Maxwell–Stefan systems and n-species biofilm models are examples that satisfy the general assumptions.

Stability of the Maxwell–Stefan System in the Diffusion Asymptotics of the Boltzmann Multi-species Equation

We investigate the diffusion asymptotics of the Boltzmann equation for gaseous mixtures, in the perturbative regime around a local Maxwellian vector whose fluid quantities solve a flux-incompressible Maxwell–Stefan system. Our framework is the torus and we consider hard-potential collision kernels with angular cutoff. As opposed to existing results about hydrodynamic limits in the mono-species case, the local Maxwellian we study here is not a local equilibrium of the mixture due to cross-interactions. By means of a hypocoercive formalism and introducing a suitable modified Sobolev norm, we build a Cauchy theory which is uniform with respect to the Knudsen number $$\varepsilon $$ . In this way, we shall prove that the Maxwell–Stefan system is stable for the Boltzmann multi-species equation, ensuring a rigorous derivation in the vanishing limit $$\varepsilon \rightarrow 0$$ .

Analysis of Maxwell–Stefan systems for heat conducting fluid mixtures

The global-in-time existence of bounded weak solutions to the Maxwell–Stefan–Fourier equations in Fick–Onsager form is proved. The model consists of the mass balance equations for the partial mass densities and the energy balance equation for the total energy. The diffusion and heat fluxes depend linearly on the gradients of the thermo-chemical potentials and the gradient of the temperature and include the Soret and Dufour effects. The cross-diffusion system exhibits an entropy structure, which originates from the thermodynamic modeling. The lack of positive definiteness of the diffusion matrix is compensated by the fact that the total mass density is constant in time. The entropy estimate yields the a.e. positivity of the partial mass densities and temperature. Also diffusion matrices are considered that degenerate for vanishing partial mass densities.

Energy Storage in Paraffin: A PDE Backstepping Experiment

This article proposes a novel control algorithm of a thermal phase-change process and shows its experimental verification using paraffin as a phase-change material (PCM). The core problem is to design a boundary feedback control for the “Stefan system” that describes the time evolution of the temperature profile in the liquid phase, which is associated with the time evolution of a position of liquid-solid phase interface, for the sake of stabilizing the interface position at a chosen set point. First, we design the continuous-time full-state feedback control law by means of the PDE backstepping method, which, in the absence of a demand for accelerated convergence, can also be arrived at by the energy-shaping method, and rigorously prove the stability of the closed-loop system under sufficiently small heat loss. Next, the control law is refined via observer-based output feedback under sampled-data measurements of the surface temperature and the phase interface position so that the control algorithm is practically implementable. Then, we conducted an experiment under a constant input to calibrate unknown parameters involved with the heat loss. Finally, the proposed model-based boundary feedback control algorithm is implemented in the experiment of melting paraffin. The experiment was successful: the convergence of the phase interface to the set point was achieved.

Single-boundary control of the two-phase Stefan system

This paper presents the control design of the two-phase Stefan problem. The two-phase Stefan problem is a representative model of liquid–solid phase transition by describing the time evolutions of the temperature profile, which is divided by subdomains of liquid and solid phases as the liquid–solid moving interface position. The mathematical formulation is given by two diffusion partial differential equations (PDEs) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) driven by the Neumann boundary values of both PDE states, resulting in a nonlinear coupled PDE–ODE–PDE system. We design a state feedback control law by means of energy-shaping to stabilize the interface position to a desired setpoint by using single boundary heat input. We prove that the closed-loop system under the control law ensures some conditions for model validity, and the global exponential stability estimate is shown in the spatial L2 norm. Furthermore, the robustness of the closed-loop stability with respect to the uncertainties of the physical parameters is shown. Numerical simulation is provided to illustrate the desired performance of the proposed control law in comparison to the control design for the one-phase Stefan problem.

Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System

The Navier–Stokes–Maxwell–Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.

On the relaxation of the Maxwell–Stefan system to linear diffusion

In this note, we rigorously prove the relaxation limit of the Maxwell–Stefan system to a system of heat equations when all binary diffusion coefficients tend to the same positive value.

Analysis of an Incompressible Navier–Stokes–Maxwell–Stefan System

The Maxwell–Stefan equations for the molar fluxes, supplemented by the incompressible Navier–Stokes equations governing the fluid velocity dynamics, are analyzed in bounded domains with no-flux boundary conditions. The system models the dynamics of a multicomponent gaseous mixture under isothermal conditions. The global-in-time existence of bounded weak solutions to the strongly coupled model and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell–Stefan diffusion matrix, the cross-diffusion terms, and the different molar masses of the fluid components. The key idea of the proof is the use of a new entropy functional and entropy variables, which allows for a proof of positive lower and upper bounds of the mass densities without the use of a maximum principle.

Existence Analysis of Maxwell--Stefan Systems for Multicomponent Mixtures

Maxwell--Stefan systems describing the dynamics of the molar concentrations of a gas mixture with an arbitrary number of components are analyzed in a bounded domain under isobaric, isothermal conditions. The systems consist of mass balance equations and equations for the chemical potentials, depending on the relative velocities, supplemented with initial and homogeneous Neumann boundary conditions. Global-in-time existence of bounded weak solutions to the quasi-linear parabolic system and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell--Stefan diffusion matrix, the cross-diffusion coupling, and the lack of standard maximum principles. Key ideas of the proofs are the Perron--Frobenius theory for quasi-positive matrices, entropy-dissipation methods, and a new entropy variable formulation allowing for the proof of nonnegative lower and upper bounds for the concentrations.

Étude d'un système non linéaire de Boussinesq–Stefan

We give a few existence results for solutions for a class of Boussinesq–Stefan systems, with suitable conditions on the forcing terms in the right-hand side of the momentum equation depending on the temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Finite element approximation of a Stefan problem with degenerate Joule heating

We consider a fully practical finite element approximation of the following degenerate system subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan system. In addition to showing stability bounds, we prove (subsequence) convergence of our finite element approximation in two and three space dimensions. The latter is non-trivial due to the degeneracy in σ(u) and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.

Analysis and application of a continuation method for a self-similar coupled Stefan system

This work deals with a continuation method for computing solutions to a self-similar two-component Stefan system in which the diffusion coefficients depend on the concentrations. The procedure computes a one-parameter homotopy connecting the known solution of a simplified problem (when the parameter is zero) to the solution of the problem at hand (when the parameter is one). Local convergence of the method and local existence and uniqueness of solutions for the original system are proven. Also, several examples coming from precipitant-driven protein crystal growth are discussed. The most interesting of these is a Stefan problem containing a porous media equation that corresponds to the liquid phase being in a meta-stable state near the spinodal region. The bifurcation code AUTO is used in the computations.

State and free boundary estimations for stochastic two-phase Stefan systems

In this paper, we consider the state and free boundary estimation problems for stochastic two-phase Stefan systems. First, the existence and uniqueness properties of the solution to the state equation with white noise coefficients are discussed by using the finite difference method with respect to time and spatial variables. Secondly, the state and free boundary estimator dynamics are derived by using martingale representation theory. Finally, representative sample runs obtained by digital simulation experiments are shown, including a simple approximation method for realizing non-linear estimator dynamics.