arXiv subjects

Athanasios Tzavaras

Publications and source records attributed to Athanasios Tzavaras.

8 recordsLinked to original sources

arXiv2026

Equilibration versus Localization in a Diffusion-Relaxation System

We consider a diffusion-relaxation system and investigate the conditions on parameters leading to equilibration versus localization. When the diffusion is dominant, solutions converge toward homogeneous equilibria. By contrast, when the effective diffusion is weak, localization emerges. Such behaviors have been studied for various models through formal asymptotic arguments and linearized stability analysis, but rigorous understanding of the associated nonlinear phenomena remains limited, particularly in higher dimensions. In the equilibration regime, we establish convergence toward constant equilibria by exploiting an energy dissipation structure and invariant-region estimates. In the localization regime, we study self-similar solutions and transform the problem of their existence into an autonomous dynamical system. The existence of self-similar profiles associated to localizing solutions is reduced to the construction of a heteroclinic orbit for an autonomous dynamical system. Their existence is obtained through an application of geometric singular perturbation theory. Our analysis provides a rigorous characterization of the transition from equilibration to localization.

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arXiv2023

Non-isothermal multicomponent flows with mass diffusion and heat conduction

A type-I model of non-isothermal multicomponent systems of gases describing mass diffusive and heat conductive phenomena is presented. The derivation of the model and a convergence result among thermomechanical theories in the smooth regime are discussed. Furthermore, the global-in-time existence of weak solutions and the weak-strong uniqueness property are established for the corresponding system with zero barycentric velocity.

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arXiv2019

Localization in adiabatic shear flow via geometric theory of singular perturbations

We study localization occurring during high speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition among Hadamard instability (caused by softening response) and the stabilizing effects of strain-rate hardening. We consider a hyperbolic-parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition. The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is four dimensional but has a fast-slow structure with respect to a small parameter capturing the strength of strain-rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space.

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arXiv2017

Conservation of energy for the Euler-Korteweg equations

In this article we study the principle of energy conservation for the Euler-Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.

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arXiv2017

Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

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arXiv2016

Existence of localizing solutions in plasticity via geometric singular perturbation theory

Shear bands are narrow zones of intense shear observed during plastic deformations of metals at high strain rates. Because they often precede rupture, their study attracted attention as a mechanism of material failure. Here, we aim to reveal the onset of localization into shear bands using a simple model developed from viscoplasticity. We exploit the properties of scale invariance of the model to construct a family of self-similar focusing solutions that capture the nonlinear mechanism of shear band formation. The key step is to de-singularize a reduced system of singular ordinary differential equations and reduce the problem into the construction of a heteroclinic orbit for an autonomous system of three first-order equations. The associated dynamical system has fast and slow time scales, forming a singularly perturbed problem. The geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincaré-Bendixson theorem to construct a heteroclinic orbit.

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arXiv2016

Localization in inelastic rate dependent shearing deformations

Metals deformed at high strain rates can exhibit failure through formation of shear bands, a phenomenon often attributed to Hadamard instability and localization of the strain into an emerging coherent structure. We verify formation of shear bands for a nonlinear model exhibiting strain softening and strain rate sensitivity. The effects of strain softening and strain rate sensitivity are first assessed by linearized analysis, indicating that the combined effect leads to Turing instability. For the nonlinear model a class of self-similar solutions is constructed, that depicts a coherent localizing structure and the formation of a shear band. This solution is associated to a heteroclinic orbit of a dynamical system. The orbit is constructed numerically and yields explicit shear localizing solutions.

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arXiv2011

Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

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