Schedule for: 20w5188 - Multiscale Models for Complex Fluids: Modeling and Analysis (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorov’s two-equation model of turbulence. The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stick–slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications.

We shall review recent results on the mathematical derivation of Einstein's formula and higher order corrections for the effective viscosity of a dilute suspension of neutrally buoyant particles.

We discuss the validity of the ergodic hypothesis (convergence of ergodic means) for solutions of the compressible Navier-Stokes systems with either non-homogeneous boundary conditions or driven by a stochastic forcing. In both cases we show convergence of the classical Krylov-Bolyubov method that gives rise to a statistical (stochastic) stationary solution of the problem.

We are interested in finding solutions of nonlinear differential equations describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We consider the corresponding Cauchy problem for the stress variable. Under the monotonicity assumption of the nonlinear constitutive function, we convert the problem to a new form for the strain variable and prove local-in-time existence of solutions.

We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.

The result is based on the joint work with Jose A. Carrillo and Ewelina Zatorska.

In this talk, we will discuss the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We will explain why the prediction of such a blow-up, via direct numerical experiments, is so difficult. We will describe how, in such a scenario, the solution becomes unstable as time approaches the blow-up time. The method use the relation between the vorticity of the solution, and the bi-characteristic amplitude solutions, which describe the evolution of the linearized Euler equation at high frequency. In the axisymmetric case, we can also study the instability of blow-up profiles.

This work was partially supported by the NSFDMS-1907981.

This a joint work with Misha Vishik and Laurent Lafleche.

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi and we construct infinitely many solutions.

This is a joint work with Elisabetta Chiodaroli, V\'aclav M\'acha and Sebastian Schwarzacher.

We consider the equations of a linear Maxwell fluid with spatially varying coefficients. Pure stress modes are solutions with zero velocity but nonzero stresses. We derive equations to characterize such solutions. In two dimensions, we find that under generic hypotheses only certain "trivial" solutions exist. In three dimensions, on the other hand, there exist nontrivial solutions. To get them, we derive a system of partial differential equations whose type (elliptic or hyperbolic) depends on the sign of the Gauss curvature of level surfaces of the relaxation time.

(joint work with Debanjana Mitra and Mythily Ramaswamy)

We discuss long time behavior of solutions to a non-linear second order integrodifferential convolution equation, in particular we focus on the speed of convergence to equilibrium. The key assumptions are that the convolution kernel is small and the non-linear operator satisfies the Lojasiewicz inequality.

The rheology of complex fluids such as polymeric liquids is highly non-Newtonian in nature and manifests itself as an extra stress component in the Cauchy stress tensor. At the purely macroscopic level, the extra stress tensor is linked to the velocity field through, say, a partial differential equation. An alternative approach consists in finding an expression for the macroscopic extra stress tensor in terms of the microscopic dynamics of the polymer chains. We present a thermodynamically based approach to the design of a class of such micro-macro models for dilute polymeric liquids and show that the thermodynamic background of the model naturally yields stability of the steady state when the fluid occupies an isolated vessel.

Instead of the abstract, please see the video on https://youtu.be/l85eQapJ_bA.

We show that under natural regularity assumptions, an abstract nonlinear parabolic evolution problem has a finite-dimensional attractor. Moreover, the long-time dynamics can be recast as a system of ODEs with exponentially decaying delay.

As an application, we consider a class of non-Newtonian fluids with dynamic boundary conditions.

We analyse the system of the form \begin{align*} {\partial}_t{\rho} +{\rm div \,}_x(\rho u) = 0\\ {\partial}_t(\rho u) +{\rm div \,}_x(\rho u\otimes u) + \nabla_x p = {\rm div \,}_x {\mathbb{S}}\label{secondequation}\\ {\rm div \,}_x(u) = 0 \end{align*} where $\rho$ is the mass density, $u$ denotes velocity field, ${\mathbb{S}}$ the stress tensor and $p$ is the pressure. We are interested in the measure-valued solutions to those equations and prove the mv-strong uniqueness property. This work bases its assumptions on the recent paper by Abbatiello and Feireisl [1], but differs from it in density dependency. Surprisingly the solutions are not defined by the Young measures, but by the similar tool to the so-called defect measure.

BIBLIOGRAPHY

[1] A. Abbatiello and E. Feireisl. On a class of generalized solutions to equations describing incompressible viscous fluids. Ann. Mat. Pura Appl. (4), 199(3):1183–1195, 2020.

In multiscale models for polymeric fluids, the evolution of the polymer chain is usually modeled using an entropic force, computed from the free energy associated with the end-to-end vector. We will present results which aim at justifying under which circumstances such a dynamics is indeed close to the original dynamics based on the full-atom chain.

References:

F. Legoll and T. Lelièvre, Effective dynamics using conditional expectations, Nonlinearity, 23, 2131-2163, (2010).
F. Legoll, T. Lelièvre and S. Olla, Pathwise estimates for an effective dynamics, Stochastic Processes and their Applications, 127, 2841-2863, (2017).
F. Legoll, T. Lelièvre and U. Sharma, Effective dynamics for non-reversible stochastic differential equations: a quantitative study, Nonlinearity, 32(12), 4779, (2019).

We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely $L^1$-coercivity. We use the concept of renormalized solutions and higher differentiability techniques to prove existence and uniqueness of weak solution with $L^1$-integrable flux for all values of a positive model parameter $a$. If this parameter is smaller than $2/(d+1)$, where $d$ denotes the spatial dimension, we obtain higher integrability of the flux. We also relate the studied problem to problems in fluid mechanics.

Classical models describing the motion of Newtonian fluids, such as water, rely on the assumption that the Cauchy stress is a linear function of the symmetric part of the velocity gradient of the fluid. This assumption leads to the Navier-Stokes equations. It is known however that the framework of classical continuum mechanics, built upon an explicit constitutive equation for the Cauchy stress, is too narrow to describe inelastic behavior of solid-like materials or viscoelastic properties of materials. Our starting point in this work is therefore a generalization of the classical framework of continuum mechanics, called the implicit constitutive theory, which was proposed recently in a series of papers by K.R. Rajagopal. The underlying principle of implicit constitutive theory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and, in particular, linear) function of the symmetric part of the velocity gradient, one may allow a nonlinear, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian fluid flow models with implicit and possibly discontinuous power-law-like rheology.

We develop the analysis of finite element approximations of implicit power-law-like models for viscous in-compressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued, maximal monotone graph. Using a variety of weak compactness techniques, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the discretisation parameter, measuring the granularity of the finite element triangulation, tends to zero. A key new technical tool in our analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions.

The talk is based on a series of recent papers with Lars Diening and Tabea Tscherpel (Bielefeld), Christian Kreuzer (Dortmund), Alexei Gazca Orozco (Erlangen) and Patrick Farrell (Oxford).

Analysis of finite amplitude stability of fluid flows is a challenging task even if the fluid of interest is described using the classical mathematical models such as the Navier--Stokes--Fourier model. The issue gets more complicated when one has to deal with complex models for coupled thermomechanical behaviour of non-Newtonian fluids; in particular the viscoelastic rate-type fluids.
We show that the knowledge of thermodynamical underpinnings of these complex models can be gainfully exploited in the stability analysis. First we introduce general concepts that allow one to deal with thermodynamically isolated systems, and then we proceed to thermodynamically open systems. Next we document the applications of these concepts in the case of container flows (thermodynamically isolated systems), and in the case of flows in containers with non-uniformly heated walls (mechanically isolated but thermally open system). We end up with mechanically driven systems such as the Taylor--Couette flow.

We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time-periodic solution in the class of weak solutions satisfying the energy inequality.

This is a joint work with Eduard Feireisl.

In tis talk I will present a derivation of macroscopic model of interacting particles. The population of N particles evolve according to a diffusion process and interacts through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e. with O(N) particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics. The derivation combines the stochastic averaging (over time-scale parameter) and the many particles ($N\to \infty$) limits.

We study a two-species model of tissue growth describing dynamics under mechanical pressure and cell growth. The pressure is incorporated into the common fluid velocity through an elliptic equation, called Brinkman’s law, which accounts for viscosity effects in the individual species. Our aim is to establish the incompressible limit as the stiffness of the pressure law tends to infinity - thus demonstrating a rigorous bridge between the population dynamics of growing tissue at a density level and a geometric model thereof.

Joint work with B. Perthame (Sorbonne), M. Schmidtchen (TU Dresden) and N. Vauchelet (Paris 13).

Mathematical modelling and numerical simulations of phase separation becomes much more involved if one component is a macromolecular compound. In this case, the large molecular relaxation time gives rise to a dynamic coupling between intra-molecular processes and the unmixing on experimentally relevant time scales, with interesting new phenomena, for which the name “viscoelastic phase separation” has been coined.
Our model of viscoelastic phase separation describes time evolution of the volume fraction of a polymer and the bulk stress leading to a strongly coupled (possibly degenerate) cross-diffusion system. The evolution of volume fraction is governed by the Cahn-Hilliard type equation, while the bulk stress is a parabolic relaxation equation. The system is further combined with the Navier-Stokes-Peterlin system, describing time evolution of the velocity and (elastic) conformation tensor.
Under some physically relevant assumptions on boundedness of model parameters we have proved that global in time weak solutions exist. Further, we have derived a suitable notion of the relative energy taking into account the non-convex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle and consequently the uniqueness of a weak solution in special cases.
Our extensive numerical simulations confirm robustness of the analysed model and the convergence of a suitable numerical scheme with respect to the relative energy.

States of matter (such as solid, liquid, etc) are characterized by different types of order associated with local invariances under different transformation groups. Recently, a new notion of topological order, popularized by the 2016 physics nobel prize awarded to Haldane, Kosterlitz and Thouless, has emerged. It refers to the global rigidity of the system arising in some circumstances from topological constraints. Topologically ordered states are extremely robust i.e. « topologically protected » against localized perturbations. Collective dynamics occurs when a system of self-propelled particles organizes itself into a coherent motion, such as a flock, a vortex, etc. Recently, the question of realizing topologically protected collective states has been raised. In this work, we consider a system of self-propelled solid bodies interacting through local full body alignment up to some noise. In the large-scale limit, this system can be described by hydrodynamic equations with topologically non-trivial explicit solutions. At the particle level, these solutions persist for a certain time but eventually decay towards a uniform flocking state, due to the stochastic nature of the particle system. We show numerically that the persistence time of these topologically non-trivial solutions is far longer than for topologically trivial ones, showing a new kind of « topological protection » of a collective state. To our knowledge, it is the first time that a hydrodynamic model guides the design of topologically non-trivial states of a particle system and allows for their quantitative analysis and understanding. In passing, we will raise fascinating mathematical questions underpinning the analysis of collective dynamics systems.
Joint work with Antoine Diez and Mingye Na (Imperial College London)

In this talk we consider the interaction of a Stokes/Navier-Stokes flow with a viscoelastic body. The elastic body is allowed to undergo large deformations (but no self-collisions). In order to handle this situation correctly, we devise a variational approximation scheme in the spirit of DeGiorgi to the combined problem. Moreover, by using a two-scale scheme, we also extend this approach to the hyperbolic regime including inertia of the solid body. These variational approaches allow us to prove proper energetic estimates while also controling the geometric restictions posed on the solid body and, eventually, to establish existence of weak solutions. This is joint work with Malte Kampschulte and Sebastian Schwarzacher (both Prague).

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic N−function, which may be possibly also dependent on the spatial variable, i.e., the homogenization process will change the characteristic function spaces at each step.

References:

[1] Bulíček, Miroslav; Gwiazda, Piotr; Kalousek, Martin; Świerczewska-Gwiazda, Agnieszka: Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces. Nonlinear Anal. 194 (2020), 111487, 34 pp.
[2] Bulíček, Miroslav; Gwiazda, Piotr; Kalousek, Martin; Świerczewska-Gwiazda, Agnieszka: Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces. Nonlinearity 32 (2019), no. 3, 1073–1110.

Several notions of weak or 'very weak' solutions have been suggested for the incompressible and compressible Euler systems, motivated by the lack of a satisfactory well-posedness theory for these equations in turbulent regimes. Surprisingly, the speaker and L. Székelyhidi showed in 2012 that dis- tributional and measure-valued solutions are in a sense the same, although the latter had been expected to be a much weaker notion. In this talk, we turn to the isentropic compressible Euler system, where the situation is fundamentally different.

Joint work with E. Chiodaroli, E. Feireisl, O. Kreml, and D. Gallenmüller.

We concern the system consisting of a moving body filled with a compressible fluid. We present several existence proofs, however, our main aim is to deal with the long-time behavior of the whole system.

Results presented during this work were done in collaboration with G. P. Galdi, S. Nečasová and B. She.

We study the homogenization of stationary compressible Navier–Stokes–Fourier system in a bounded three dimensional domain perforated with a large number of very tiny holes. Under suitable assumptions imposed on the smallness and distribution of the holes, we show that the homogenized limit system remains the same in the domain without holes.

We prove the existence of global weak solutions à la Leray for compressible Navier-Stokes equations with a pressure law which depends on the density and on time and space variables t and x. The assumptions on the pressure contain only locally Lipschitz assumption with respect to the density variable and some hypothesis with respect to the extra time and space variables. It may be seen as a first step to consider heat-conducting Navier-Stokes equations with physical laws such as the truncated virial assumption. The paper focuses on the construction of approximate solutions through a new regularized and fixed point procedure and on the weak stability process taking advantage of the new method introduced by the two first authors with a careful study of an appropriate regularized quantity linked to the pressure.
This is a joint work with D. Bresch and F. Wang.

We present global-in-time existence results for two cross-diffusion systems modeling heat-conducting fluid mixtures. Both models consist of the balance equations for the mass densities and temperature. The key difficulty is the nonstandard degeneracy in the diffusion (Onsager) matrices, i.e., ellipticity is lost when the fluid density or temperature vanishes. This problem is overcome in the first model by exploiting the volume-filling property of the mixture, leading to gradient estimates for the square root of the partial densities, and in the second model by compensated compactness and renormalization techniques from mathematical fluid dynamics.
The first model is joint work with C. Helmer, the second one with G. Favre, C. Schmeiser, and N. Zamponi.

In the talk, we discuss a response of the fluid on the boundary, which acts as a delayed slip due to material properties. In the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier-Stokes-like problems with dynamic slip boundary condition, which requires a proper generalisation of the Gelfand triplet and the corresponding function spaces setting.

It is a joint work with Anna Abbatiello and Miroslav Bulíček.

Amyloid fibrils are important biological structures associated with devastating human diseases such as Alzheimer disease, as well as have vital biological functions such as adhesion and biofilm formation. The division of amyloid protein fibrils is required for the propagation of the amyloid state and is an important contributor to their stability, pathogenicity, and normal function. We apply asymptotic results on the fragmentation equation to develop an inverse problem approach, allowing us to compare the division stability of amyloid fibrils and estimate their division features (fragmentation rate and kernel).

This is a joint work with Magali Tournus, Miguel Escobedo and Wei-Feng Xue.

We deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a critical power index $p=\frac{11}{5}$ we show that a Gehring type argument is applicable which allows to improve regularity of any weak solution. Improving further the regularity of weak solutions along a regularity ladder allows to show that actually solution belongs to a uniqueness class provided data of the problem are sufficiently smooth.

We also briefly discuss if the similar technique is applicable to critical Convective Brinkman-Forchheimer equation.

Viscoelastic fluids often exhibit high sensitivity of material properties on temperature changes. Nevertheless, the available mathematical theory for these fluids concerns only models that are isothermal or that are simplified in other ways. For example, one can find existence theories in 2D, for small data, with only the corotational derivative, with only the spherical part of the elasticity tensor etc. In the talk, we introduce an existence theory without any of these assumptions and treat a rather general class of Johnson-Segalman-like models including full thermal evolution. To avoid the well-known ill-posedness of the corresponding PDE system, we modify the ``elastic part'' of the dissipation of the fluid far from the equilibrium, while preserving thermodynamic compatibility of the model. This way, we are able to prove the existence of a global-in-time weak solution for any initial datum with finite total energy and entropy.

The Keller-Segel model for bacterial chemotaxis supports travelling wave solutions which have been described in the literature as both linearly stable and unstable and in the case of linear consumption (conditionally) nonlinearly stable. We reconcile this apparent contradiction by locating the essential spectrum, absolute spectrum and point spectrum of the linear operators associated with the travelling wave solutions. We derive conditions for the spectral (in)stability of the travelling wave solutions and the critical parameters that indicate a transition from a transient to absolute instability. Furthermore, we show that the absolute spectrum deforms as the consumption is changed, illustrating a connection between the constant, sublinear and linear cases.

The nonlocal Mullins-Sekerka flow can be seen as the $H^{-\frac12}$-gradient flow of the so called sharp-interface Ohta-Kawaski energy. In this talk we will show that three-dimensional periodic configurations that are strictly stable with respect to this energy are exponentially stable also for the nonlocal Mullins-Sekerka flow. This result is contained in a joint paper with E. Acerbi, M. Morini and V. Julin

We will consider a system of evolutionary PDEs that describe a model of viscoelastic bodies exhibiting a certain strain-limiting property. Namely, working in the small strain setting, we ask that a sum of the linearised strain and the strain rate is given by some function $F$ acting on the Cauchy stress tensor, where this function $F$ is nonlinear and bounded. These models come from the much larger class of implicit constitutive relations. We will show the existence and uniqueness of global-in-time large-data weak solutions to this strain-limiting problem by first proving the existence of solutions to a broader class of models. This broader class replaces the bounded function $F$ on the stress by one that experiences some level of polynomial growth. Using a suitable approximation of the strain-limiting problem by these problems with growth, we are able to deduce suitable a priori bounds that allow us to prove the existence of a solution to our original problem. The main issue is that the stress tensor, and thus approximations of the stress, are initially seen to be bounded a priori only in $L^1$. However, we are able to circumvent such an issue without introducing any notion of measure-valued solutions, and in particular, we obtain a satisfactory existence theory for these problems under some suitable assumptions on the data.

In this talk, I will discuss with the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. In particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. Of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.

It is a joint work with Mingwen Fei and Xiaoming Wang.

In continuum models for non-perfect fluids, viscoelastic stresses have often been introduced as extra-stresses of purely-dissipative (entropic) nature, similarly to viscous stresses that induce motions of infinite propagation speed. A priori, it requires only one to couple an evolution equation for the (extra-)stress with the momentum balance. In many cases, the apparently-closed resulting system is often not clearly well-posed, even locally in time. The procedure also raises questions about how to encompass transition toward alastic solids. A noticeable exception is K-BZK theory where one starts with a purely elastic fluids. Viscoelasticity then results from dissipative (entropic) stresses due to the relaxation of the fluids'"memory". That K-BKZ approach is physically appealing, but mathematically quite difficult because integrals are introduced to avoid material ('natural') configurations. We propose to introduce viscoelastic stress starting with hyperelastic fluids (like K-BKZ) and evolving material configurations (unlike K-BKZ). At the price of an enlarged system with an additional material-metric variable, one can define well-posed (compressible) motions with finite propagation speed through a system of conservation laws endowed with a "contingent entropy" (like in standard polyconvex elastodynamics).

We present large-data existence result for weak solutions to a steady compressible Navier-Stokes-Fourier system for chemically reacting fluid mixtures. General free energies satisfying some structural assumptions are considered, with a pressure containing a $\gamma$-power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the density in $L^{\gamma}$ with $\gamma>3/2$, the effective viscous flux identity, and uniform bounds related to Feireisl's oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

Rate-type fluid models involving the stress and its observer-invariant time derivatives of higher order are used to describe a large class of viscoelastic mixtures - geomaterials like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as SBR. A standard model that belongs to the category of viscoelastic rate-type fluid models of the second order is the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the first order. This viewpoint allows one to develop the whole hierarchy of generalized models of a Burgers type. We study one such generalization. Carrying on the study by Masmoudi (2011), who briefly proved the weak sequential stability of weak solutions to the Giesekus model, we prove long time and large data existence of weak solutions to a mixture of two Giesekus models in two spatial dimensions.

We analyse fast reaction limit in the reaction-diffusion system \begin{align*} \partial_t u^{\varepsilon} &= \frac{v^{\varepsilon} - F(u^{\varepsilon})}{\varepsilon}, \\ \partial_t v^{\varepsilon} &= \Delta v^{\varepsilon} + \frac{F(u^{\varepsilon}) - v^{\varepsilon}}{\varepsilon}, \end{align*} with nonmonotone reaction function $F$. As speed of reaction tends to infinity, the concentration of non-diffusing component $u^{\varepsilon}$ exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component $v^{\varepsilon}$, a result that is not obvious from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov [2]. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions and provides a weak formulation for the evolution of the Young measure.

This is a joint work with Beno\^\i t Perthame (Paris) [1]

[1] B. Perthame, J. Skrzeczkowski. Fast reaction limit with nonmonotone reaction function. arXiv: 2008.11086, submitted.
[2] P. I. Plotnikov. Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity. Differ. Uravn., 30:4 (1994), 665--674; Differ. Equ., 30:4 (1994), 614--622.

The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing parameter, $\alpha> 0$, a given length scale parameter, so that $\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In this talk I will derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, I will show that, for fixed viscosity, $\nu>0$, as the regularizing parameter $\alpha$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact is also supported by numerical observations, which provides an additional evidence that, for small values of the regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.

We consider the Kelvin-Voigt model for viscoelasticity and prove propagation of $H^1$-regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored energy satisfies the Andrews-Ball condition, in particular allowing for a non-monotone stress. By contrast, a counterexample indicates that for non-monotone stress-strain relations (even in 1-d) initial oscillations of the strain lead to solutions with sustained oscllations. In addition, in two space dimensions, we prove that the weak solutions with deformation gradient in $H^1$ are in fact unique, providing a striking analogy to the 2D Euler equations with bounded vorticity.

(joint work with K. Koumatos (U. of Sussex), C. Lattanzio and S. Spirito (U. of L’Aquila)).

We consider pressureless compressible Euler equations driven by nonlocal repulusion-attraction and alignment forces. Our attention is directed to measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a weak-strong uniqueness principle.