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Athanasios E. Tzavaras

Publications and source records attributed to Athanasios E. Tzavaras.

At least 19 recordsLinked to original sources
arXiv2026

Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes

These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow.

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arXiv2026

Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations

We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument.

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arXiv2026

Strong convergence of sequences with vanishing relative entropy

We show that under natural growth conditions on the entropy function, convergence in relative entropy is equivalent to $L_p$-convergence. The main tool is the theory of Young measures, in a form that accounts for the formation of concentrations in weak limits.

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arXiv2025

Variational structure of Fokker-Planck equations with variable mobility

We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation.

math.OC↗
arXiv2024

Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy

We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity $ν\rightarrow 0$ or when the dispersion coefficient $δ\rightarrow 0$. For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin \cite{jean1996remark} on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier-Stokes equations with bounded vorticity.

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arXiv2024

A bilinear fractional integral operator for Euler-Riesz systems

We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis of a tensor-valued bilinear fractional integral operator associated with Euler-Riesz systems modeling mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of the Euler-Riesz system that yields a gain in integrability for finite energy solutions through compensated integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.

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arXiv2024

Renormalized solutions for the Maxwell--Stefan system with an application to uniqueness of weak solutions

We give conditions that guarantee uniqueness of renormalized solutions for the Maxwell-Stefan system. The proof is based on an identity for the evolution of the symmetrized relative entropy. Using the method of doubling the variables we derive the identity for two renormalized solutions and use information on the spectrum of the Maxwell-Stefan matrix to estimate the symmetrized relative entropy and show uniqueness. We then show that weak solutions for the Maxwell-Stefan system have sufficient regularity to produce renormalized solutions. Combining the two results yields a uniqueness result for weak solutions of the Maxwell-Stefan system with bounded fluxes.

math.AP↗
arXiv2024

Alignment via friction for nonisothermal multicomponent fluid systems

The derivation of an approximate Class-I model for nonisothermal multicomponent systems of fluids, as the high-friction limit of a Class-II model is justified, by validating the Chapman-Enskog expansion performed from the Class-II model towards the Class-I model. The analysis proceeds by comparing two thermomechanical theories via relative entropy.

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arXiv2024

Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems

In the first part of this article we present some exact solutions for special hyperbolic-parabolic systems with sustained oscillations induced by the initial data, most notably the compressible Navier-Stokes system with non-monotone pressure. This part complements \cite{Tzavaras23} where such examples are extensively studied. The second part deals with the problem of homogenization for one-dimensional models describing phase transitions for viscoelastic materials . Ideas from the kinetic formulation of conservation laws are employed to derive effective equations that describe the propagation of oscillations.

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arXiv2024

Sustained Oscillations in Hyperbolic-Parabolic Systems

We construct examples of oscillating solutions with persistent oscillations for various hyperbolic-parabolic systems with singular diffusion matrices that appear in mechanics. These include, an example for the equations of nonlinear viscoelasticity of Kelvin-Voigt type with stored energy that violates rank-one convexity, which amounts to a time-dependent variant of twinning solutions. An example pertaining to the system of gas dynamics with thermal effects for a viscous, adiabatic gas. Finally, an example for the compressible Navier-Stokes system in one-space dimension with nonmonotone pressure function. We also study the existence of oscillating solutions for linear hyperbolic-parabolic systems with singular diffusion matrices.

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arXiv2024

Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system

The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores.

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arXiv2024

Zero-electron-mass and quasi-neutral limits for bipolar Euler-Poisson systems

We consider a set of bipolar Euler-Poisson equations and study two asymptotic limiting processes. The first is the zero-electron-mass limit, which formally results in a non-linear adiabatic electron system. In a second step, we analyse the combined zero-electron-mass and quasi-neutral limits, which together lead to the compressible Euler equations. Using the relative energy method, we rigorously justify these limiting processes for weak solutions of the two-species Euler-Poisson equations that dissipate energy, as well as for strong solutions of the limit systems that are bounded away from vacuum. This justification is valid in the regime of initial data for which strong solutions exist. To deal with the electric potential, in the first case we use elliptic theory, whereas in the second case we employ the theory of Riesz potentials and properties of the Neumann function.

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arXiv2023

Axisymmetric flows with swirl for Euler and Navier-Stokes equations

We consider the incompressible axisymmetric Navier-Stokes equations with swirl as an idealized model for tornado-like flows. Assuming an infinite vortex line which interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and axisymmetric Navier-Stokes equations. We are particularly interested in the connection of the two problems in the zero-viscosity limit. First, we construct a class of explicit stationary self-similar solutions for the axisymmetric Euler equations. Second, we consider the possibility of discontinuous solutions and prove that there do not exist self-similar stationary Euler solutions with slip discontinuity. This nonexistence result is extended to a class of flows where there is mass input or mass loss through the vortex core. Third, we consider solutions of the Euler equations as zero-viscosity limits of solutions to Navier-Stokes. Using techniques from the theory of Riemann problems for conservation laws, we prove that, under certain assumptions, stationary self-similar solutions of the axisymmetric Navier-Stokes equations converge to stationary self-similar solutions of the axisymmetric Euler equations as $ν\to0$. This allows to characterize the type of Euler solutions that arise via viscosity limits.

math.AP↗
arXiv2023

Self-similar axisymmetric flows with swirl

We consider an infinite vortex line in a fluid which interacts with a boundary surface as a simplified model for tornadoes. We study self-similar solutions for stationary axisymmetric Navier-Stokes equations and investigate the types of motion which are compatible with this structure when viscosity is non-negative. For viscosity equal to zero, we construct a class of explicit stationary solutions. We then consider solutions with slip discontinuity and show that they do not exist in this framework.

math.AP↗
arXiv2023

Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accurately described in a wide range of phase speeds. Undular bores are often described using the original Peregrine system which, even if not possessing traveling waves tends to provide accurate approximations for appropriate time scales. To explain this phenomenon, we show that the error between the solutions of the dissipative versus the non-dissipative Peregrine systems are proportional to the dissipation times the observational time.

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arXiv2022

Dispersive shocks in diffusive-dispersive approximations of elasticity and quantum-hydrodynamics

The aim is to assess the combined effect of diffusion and dispersion on shocks in the moderate dispersion regime. For a diffusive dispersive approximation of the equations of one-dimensional elasticity (or p-system), we study convergence of traveling waves to shocks. The problem is recast as a Hamiltonian system with small friction, and an analysis of the length of oscillations yields convergence in the moderate dispersion regime $\eps, δ\to 0$ with $δ= o(\eps)$, under hypotheses that the limiting shock is admissible according to the Liu E-condition and is not a contact discontinuity at either end state. A similar convergence result is proved for traveling waves of the quantum hydrodynamic system with artificial viscosity as well as for a viscous Peregrine-Boussinesq system where traveling waves model undular bores, in all cases in the moderate dispersion regime.

math.AP↗
arXiv2022

Optimization Methods for One Dimensional Elastodynamics

We propose a new approach for solving systems of conservation laws that admit a variational formulation of the time-discretized form, and encompasses the p-system or the system of elastodynamics. The approach consists of using constrained gradient descent for solving an implicit scheme with variational formulation, while discontinuous Galerkin finite element methods is used for the spatial discretization. The resulting optimization scheme performs well, it has an advantage on how it handles oscillations near shocks, and a disadvantage in computational cost, which can be partly alleviated by using techniques on step selection from optimization methods.

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