arXiv subjects

Athanasios E. Tzavaras

Publications and source records attributed to Athanasios E. Tzavaras.

At least 37 records · Page 2Linked to original sources

arXiv2022

Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion

A Type-I model of a multicomponent system of fluids with non-constant temperature is derived as the high-friction limit of a Type-II model via a Chapman-Enskog expansion. The asymptotic model is shown to fit into the general theory of hyperbolic-parabolic systems, by exploiting the entropy structure inherited through the asymptotic procedure. Finally, by deriving the relative entropy identity for the Type-I model, two convergence results for smooth solutions are presented, from the system with mass-diffusion and heat conduction to the corresponding system without mass-diffusion but including heat conduction and to its hyperbolic counterpart.

math.AP↗

arXiv2022

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 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 $H^2(Ω)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

math.AP↗

arXiv2021

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 to 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.

math.AP↗

arXiv2021

The relaxation limit of bipolar fluid models

This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid system and is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.

math.AP↗

arXiv2021

Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics

For an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws: we prove stability for the NSK system via a modified relative energy approach. We prove continuous dependence of solutions on initial data and convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, but compensating via higher-order gradients.

math.AP↗

arXiv2021

From gas dynamics with large friction to gradient flows describing diffusion theories

We study the emergence of gradient flows in Wasserstein distance as high friction limits of an abstract Euler flow generated by an energy functional. We develop a relative energy calculation that connects the Euler flow to the gradient flow in the diffusive limit regime. We apply this approach to prove convergence from the Euler-Poisson system with friction to the Keller-Segel system in the regime that the latter has smooth solutions. The same methodology is used to establish convergence from the Euler-Korteweg theory with monotone pressure laws to the Cahn-Hilliard equation.

math.AP↗

arXiv2021

Relative entropy in diffusive relaxation

We establish convergence in the diffusive limit from entropy weak solutions of the equations of compressible gas dynamics with friction to the porous media equation away from vacuum. The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy. The relative entropy method is also employed to establish convergence from entropic weak solutions of viscoelasticity with memory to the system of viscoelasticity of the rate-type.

math.AP↗

arXiv2020

Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy

We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored energies satisfying an Andrews-Ball condition, allowing for non convexity in a compact set. Existence of weak solutions with deformation gradients in $H^1$ is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.

math.AP↗

arXiv2020

An energy stable and positivity-preserving scheme for the Maxwell-Stefan diffusion system

We develop a new finite difference scheme for the Maxwell-Stefan diffusion system. The scheme is conservative, energy stable and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

math.NA↗

arXiv2020

A discrete variational scheme for isentropic processes in polyconvex thermoelasticity

We propose a variational scheme for the construction of isentropic processes of the equations of adiabatic thermoelasticity with polyconvex internal energy. The scheme hinges on the embedding of the equations of adiabatic polyconvex thermoelasticity into a symmetrizable hyperbolic system. We establish existence of minimizers for an associated minimization theorem and construct measure-valued solutions that dissipate the total energy. We prove that the scheme converges when the limiting solution is smooth.

math.AP↗

arXiv2018

High-friction limits of Euler flows for multicomponent systems

The high-friction limit in Euler-Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: The first-order correction system is shown to be of Maxwell-Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman-Enskog approximate system is proved in the weak-strong solution context for general Euler-Korteweg systems. Finally, the limit towards the zeroth-order system is shown for smooth solutions in the isentropic case and for weak-strong solutions in the Euler-Korteweg case. These results include the case of constant capillarities and multicomponent quantum hydrodynamic models.

math.AP↗

arXiv2018

Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure- valued weak versus strong uniqueness using the averaged relative entropy inequality.

math.AP↗

arXiv2018

A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

We embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system and derive a relative entropy identity in the extended variables. Following the relative entropy formulation, we prove the convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero. Also, convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. Finally, we establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.

math.AP↗

arXiv2017

On the construction and properties of weak solutions describing dynamic cavitation

We consider the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions $d =2, 3$ we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity. Our analysis treats both stress-free cavities and cavities with contents.

math.AP↗

arXiv2017

The problem of dynamic cavitation in nonlinear elasticity

The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.

math.AP↗

arXiv2016

Stability properties of the Euler-Korteweg system with nonmonotone pressures

We establish a relative energy framework for the Euler-Korteweg system with non-convex energy. This allows us to prove weak-strong uniqueness and to show convergence to a Cahn-Hilliard system in the large friction limit. We also use relative energy to show that solutions of Euler-Korteweg with convex energy converge to solutions of the Euler system in the vanishing capillarity limit, as long as the latter admits sufficiently regular strong solutions.

math.AP↗

arXiv2016

Emergence of coherent localized structures in shear deformations of temperature dependent fluids

Shear localization occurs in various instances of material instability in solid mechanics and is typically associated with Hadamard-instability for an underlying model. While Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. The latter is a nonlinear effect and its analysis is the main objective of this article. We consider a model that captures the main mechanisms observed in high strain-rate deformation of metals, and describes shear motions of temperature dependent non-Newtonian fluids. For a special dependence of the viscosity on the temperature, we carry out a linearized stability analysis around a base state of uniform shearing solutions, and quantitatively assess the effects of the various mechanisms affecting the problem: thermal softening, momentum diffusion and thermal diffusion. Then, we turn to the nonlinear model, and construct localized states - in the form of similarity solutions - that emerge as coherent structures in the localization process. This justifies a scenario for localization that is proposed on the basis of asymptotic analysis in \cite{KT}.

math.AP↗

arXiv2015

A comparison of macroscopic models describing the collective response of sedimenting rod-like particles in shear flows

We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity. This model has recently been derived by Helzel and Tzavaras in \cite{HT2015}. Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a so-called quasi-dynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.

physics.flu-dyn↗