We show that weak solutions of degenerate Navier-Stokes equations converge\nto the strong solutions of the pressureless Euler sys tem with linear drag term\, Newtonian\nrepulsion and quadratic confinement . The proof is based on the relative entropy method\nusing the artificial velocity formulation for the one-dimensional Navier-Stokes system.\n
\n\nThe result is based on the joint work with Jose A. Carrillo and Eweli na Zatorska.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexis Vasseur (University of Texas at Austin) DTSTART:20201123T170000Z DTEND:20201123T172000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/6 DESCRIPTION:Title: Instability of finite time blow-ups for incompressible Euler\nby Alexis Vasseur (University of Texas at Austin) as part of BIRS work shop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbs tract\nIn this talk\, we will discuss the interaction between the stabi lity\, and the propagation of regularity\, for solutions to the incompress ible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We will explain wh y the prediction of such a blow-up\, via direct numerical experiments\, is so difficult. We will describe how\, in such a scenario\, the solution be comes unstable as time approaches the blow-up time. The method use the rel ation between the vorticity of the solution\, and the bi-characteristic am plitude solutions\, which describe the evolution of the linearized Euler e quation at high frequency. In the axisymmetric case\, we can also study th e instability of blow-up profiles. \n
\nThis work was partially supp orted by the NSFDMS-1907981. \n
\nThis a joint work with Misha Vishi k and Laurent Lafleche.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Ondřej Kreml (Czech Academy of Sciences) DTSTART:20201123T172500Z DTEND:20201123T174500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/7 DESCRIPTION:Title: Non-uniqueness of admissible weak solutions to the compressibl e Euler equations with smooth initial data\nby Ondřej Kreml (Czech Ac ademy of Sciences) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe consider the isentropi c Euler equations of gas dynamics in the whole two-dimensional space and w e prove the existence of a $C^\\infty$ initial datum which admits infinite ly many bounded admissible weak solutions. Taking advantage of the relatio n between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed R iemann state at some time instant $T > 0$. In order to continue the soluti on after the formation of the discontinuity\, we adjust and apply the theo ry developed by De Lellis and Székelyhidi and we construct infinitely man y solutions.\n
\n\nThis is a joint work with Elisabetta Chiodaroli\, V\\'aclav M\\'acha and Sebastian Schwarzacher. \n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Renardy (Virginia Tech) DTSTART:20201123T175000Z DTEND:20201123T182000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/8 DESCRIPTION:Title: Pure stress modes for linear viscoelastic flows with variable coefficients\nby Michael Renardy (Virginia Tech) as part of BIRS works hop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbst ract\nWe consider the equations of a linear Maxwell fluid with spatiall y 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 "t rivial" solutions exist. In three dimensions\, on the other hand\, there e xist nontrivial solutions. To get them\, we derive a system of partial dif ferential equations whose type (elliptic or hyperbolic) depends on the sig n of the Gauss curvature of level surfaces of the relaxation time. \n
\ n\n\n(joint work with Debanjana Mitra and Mythily Ramaswamy)\n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Tomas Barta (Charles University) DTSTART:20201124T130000Z DTEND:20201124T132000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/9 DESCRIPTION:Title: Decay of solutions to integrodifferential equations\nby To mas Barta (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe discuss long time behavior of solutions to a non-linear second order integrodifferentia l convolution equation\, in particular we focus on the speed of convergenc e to equilibrium. The key assumptions are that the convolution kernel is s mall and the non-linear operator satisfies the Lojasiewicz inequality.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark Dostalik (Charles University) DTSTART:20201124T132500Z DTEND:20201124T134500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/10 DESCRIPTION:Title: Thermodynamically consistent derivation of a micro-macro mode l for dilute polymeric fluids\nby Mark Dostalik (Charles University) a s part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling an d Analysis\n\n\nAbstract\nThe rheology of complex fluids such as polymeric liquids is highly non-Newtonian in nature and manifests itself as an extr a 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 thermodynami cally based approach to the design of a class of such micro-macro models f or dilute polymeric liquids and show that the thermodynamic background of the model naturally yields stability of the steady state when the fluid oc cupies an isolated vessel.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Piotr Mucha (University of Warsaw) DTSTART:20201124T135000Z DTEND:20201124T141000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/11 DESCRIPTION:Title: Flows initiated by ripped density\nby Piotr Mucha (Univer sity of Warsaw) as part of BIRS workshop: Multiscale Models for Complex Fl uids: Modeling and Analysis\n\n\nAbstract\nInstead of the abstract\, pleas e see the video on https://youtu.be /l85eQapJ_bA.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Dalibor Pražák (Charles University) DTSTART:20201124T150000Z DTEND:20201124T152000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/12 DESCRIPTION:Title: A finite-dimensional reduction of dissipative dynamical syste ms\nby Dalibor Pražák (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract \nWe show that under natural regularity assumptions\, an abstract nonli near parabolic evolution problem has a finite-dimensional attractor. Moreo ver\, the long-time dynamics can be recast as a system of ODEs with expone ntially decaying delay.\n
\n\nAs an application\, we consider a clas s of non-Newtonian fluids with dynamic boundary conditions.\n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Woznicki (University of Warsaw) DTSTART:20201124T152500Z DTEND:20201124T154500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/13 DESCRIPTION:Title: Mv-strong uniqueness for density dependent\, incompressible\, non-Newtonian fluids\nby Jakub Woznicki (University of Warsaw) as par t of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Ana lysis\n\n\nAbstract\n\nWe analyse the system of the form\n\\begin{align *}\n {\\partial}_t{\\rho} +{\\rm div \\\,}_x(\\rho u) = 0\\\\\n {\\pa rtial}_t(\\rho u) +{\\rm div \\\,}_x(\\rho u\\otimes u) + \\nabla_x p = {\ \rm div \\\,}_x {\\mathbb{S}}\\label{secondequation}\\\\\n {\\rm div \\ \,}_x(u) = 0\n\\end{align*}\nwhere $\\rho$ is the mass density\, $u$ denot es velocity field\, ${\\mathbb{S}}$ the stress tensor and $p$ is the press ure. We are interested in the measure-valued solutions to those equations and prove the mv-strong uniqueness property. This work bases its assumptio ns on the recent paper by Abbatiello and Feireisl [1]\, but differs from i t in density dependency. Surprisingly the solutions are not defined by the Young measures\, but by the similar tool to the so-called defect measure. \n
\n\n\n
\nIn mul tiscale models for polymeric fluids\, the evolution of the polymer chain i s usually modeled using an entropic force\, computed from the free energy associated with the end-to-end vector. We will present results which aim a t justifying under which circumstances such a dynamics is indeed close to the original dynamics based on the full-atom chain.\n
\n\nClassical 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 fl uid. This assumption leads to the Navier-Stokes equations. It is known how ever that the framework of classical continuum mechanics\, built upon an e xplicit constitutive equation for the Cauchy stress\, is too narrow to des cribe inelastic behavior of solid-like materials or viscoelastic propertie s of materials. Our starting point in this work is therefore a generalizat ion of the classical framework of continuum mechanics\, called the implici t constitutive theory\, which was proposed recently in a series of papers by K.R. Rajagopal. The underlying principle of implicit constitutive theor y in the context of viscous flows is the following: instead of demanding t hat the Cauchy stress is an explicit (and\, in particular\, linear) functi on of the symmetric part of the velocity gradient\, one may allow a nonlin ear\, implicit and not necessarily continuous relationship between these q uantities. The resulting general theory therefore admits non-Newtonian flu id flow models with implicit and possibly discontinuous power-law-like rhe ology.\n
\n\nWe develop the analysis of finite element approximation s of implicit power-law-like models for viscous in-compressible fluids. Th e Cauchy stress and the symmetric part of the velocity gradient in the cla ss 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 co nverges 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 counte rpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions.\n
\n\nThe talk is based on a series of recent papers with Lars Diening an d Tabea Tscherpel (Bielefeld)\, Christian Kreuzer (Dortmund)\, Alexei Gazc a Orozco (Erlangen) and Patrick Farrell (Oxford).\n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Vít Průša (Charles University) DTSTART:20201124T175000Z DTEND:20201124T181000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/17 DESCRIPTION:Title: Thermodynamics of viscoelastic rate-type fluids and its impli cations for stability analysis\nby Vít Průša (Charles University) a s part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling an d Analysis\n\n\nAbstract\nAnalysis of finite amplitude stability of fluid flows is a challenging task even if the fluid of interest is described usi ng the classical mathematical models such as the Navier--Stokes--Fourier m odel. The issue gets more complicated when one has to deal with complex mo dels for coupled thermomechanical behaviour of non-Newtonian fluids\; in p articular the viscoelastic rate-type fluids.\nWe consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain drive n by time periodic inflow/outflow boundary conditions. We show that the pr oblem admits a time-periodic solution in the class of weak solutions satis fying the energy inequality. \n
\n\nThis is a joint work with Eduard Feireisl.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Ewelina Zatorska (University College London) DTSTART:20201125T132500Z DTEND:20201125T134500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/19 DESCRIPTION:Title: On the dynamical network of interacting particles: from micro to macro\nby Ewelina Zatorska (University College London) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis \n\n\nAbstract\nIn this talk I will present a derivation of macroscopic mo del of interacting particles. The population of N particles evolve accordi ng to a diffusion process and interacts through a dynamical network. In tu rn\, 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 n etwork's dynamics is much faster than the particles' dynamics. The deriva tion combines the stochastic averaging (over time-scale parameter) and th e many particles ($N\\to \\infty$) limits.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Tomasz Dębiec (University of Warsaw) DTSTART:20201125T135000Z DTEND:20201125T141000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/20 DESCRIPTION:Title: Incompressible limit for a two-species model with coupling th rough Brinkman’s law.\nby Tomasz Dębiec (University of Warsaw) as p art of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and A nalysis\n\n\nAbstract\nWe study a two-species model of tissue growth de scribing dynamics under mechanical pressure and cell growth. The pressure is incorporated into the common fluid velocity through an elliptic equatio n\, called Brinkman’s law\, which accounts for viscosity effects in the individual species. \nOur 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 de nsity level and a geometric model thereof.\n
\n\nJoint work with B. Perthame (Sorbonne)\, M. Schmidtchen (TU Dresden) and N. Vauchelet (Paris 13).
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Mária Lukácová-Medvidová (Johannes Gutenberg-Universität Main z) DTSTART:20201125T150000Z DTEND:20201125T152000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/21 DESCRIPTION:Title: Viscoelastic phase separation: analysis and numerics\nby Mária Lukácová-Medvidová (Johannes Gutenberg-Universität Mainz) as pa rt of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and An alysis\n\n\nAbstract\nMathematical modelling and numerical simulations of phase separation becomes much\nmore involved if one component is a macrom olecular compound. In this case\, the large molecular relaxation time\ngiv es rise to a dynamic coupling between intra-molecular processes and the un mixing on experimentally relevant time scales\,\nwith interesting new phen omena\, for which the name “viscoelastic phase separation” has been c oined.\nWe study the homogenization process for fami lies of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operat or is assumed to be indicated by a general inhomogeneous anisotropic N−f unction\, which may be possibly also dependent on the spatial variable\, i .e.\, the homogenization process will change the characteristic function s paces at each step.\n
\nSeveral notions of weak or 'very weak' solutions have been suggest ed for the incompressible and\ncompressible Euler systems\, motivated by t he lack of a satisfactory well-posedness theory for these\nequations in tu rbulent regimes. Surprisingly\, the speaker and L. Székelyhidi showed in 2012 that dis-\ntributional and measure-valued solutions are in a sense th e same\, although the latter had been expected\nto be a much weaker notion . In this talk\, we turn to the isentropic compressible Euler system\, whe re\nthe situation is fundamentally different. \n
\nJoint work with E . Chiodaroli\, E. Feireisl\, O. Kreml\, and D.\nGallenmüller. \n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Vaclav Macha (Academy of Sciences\, Czech Republic) DTSTART:20201125T175000Z DTEND:20201125T181000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/26 DESCRIPTION:Title: On a body with a cavity filled with compressible fluid\nb y Vaclav Macha (Academy of Sciences\, Czech Republic) as part of BIRS work shop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbs tract\nWe concern the system consisting of a moving body filled with a compressible fluid. We present several existence proofs\, however\, our ma in aim is to deal with the long-time behavior of the whole system. \n
\ nResults presented during this work were done in collaboration with G. P. Galdi\, S. Nečasová and B. She.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Yong Lyu (Nanjing University) DTSTART:20201126T130000Z DTEND:20201126T132000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/27 DESCRIPTION:Title: Homogenization of stationary Navier–Stokes–Fourier system in domains with tiny holes\nby Yong Lyu (Nanjing University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analy sis\n\n\nAbstract\nWe study the homogenization of stationary compressible Navier–Stokes–Fourier system in a bounded three dimensional domain per forated 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.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Pierre-Emmanuel Jabin (University of Maryland) DTSTART:20201126T132500Z DTEND:20201126T134500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/28 DESCRIPTION:Title: Compressible Navier-Stokes equations with heterogeneous press ure laws\nby Pierre-Emmanuel Jabin (University of Maryland) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysi s\n\n\nAbstract\nWe prove the existence of global weak solutions à la Ler ay for compressible Navier-Stokes equations with a pressure law which depe nds on the density and on time and space variables t and x. The assumption s on the pressure contain only locally Lipschitz assumption with respect t o 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-condu cting Navier-Stokes equations with physical laws such as the truncated vir ial assumption. The paper focuses on the construction of approximate solut ions through a new regularized and fixed point procedure and on the weak s tability 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.In the talk\, we discuss a resp onse of the fluid on the boundary\, which acts as a delayed slip due to ma terial properties. In the moment when the slip changes rapidly\, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent re laxation. When these effects become significant\, the so-called dynamic sl ip phenomenon occurs. We develop a mathematical analysis of Navier-Stokes- like problems with dynamic slip boundary condition\, which requires a prop er generalisation of the Gelfand triplet and the corresponding function sp aces setting. \n
\n\nIt is a joint work with Anna Abbatiello and Mir oslav Bulíček.\n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Marie Doumic (INRIA) DTSTART:20201126T152500Z DTEND:20201126T154500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/31 DESCRIPTION:Title: Estimating the division of amyloid fibrils\nby Marie Doum ic (INRIA) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\n\nAmyloid fibrils are important b iological structures associated with devastating human diseases such as Al zheimer disease\, as well as have vital biological functions such as adhes ion and biofilm formation. The division of amyloid protein fibrils is requ ired for the propagation of the amyloid state and is an important contribu tor to their stability\, pathogenicity\, and normal function. \nWe apply a symptotic results on the fragmentation equation to develop an inverse pro blem approach\, allowing us to compare the division stability of amyloid f ibrils and estimate their division features (fragmentation rate and kernel ).\n
\n\nThis is a joint work with Magali Tournus\, Miguel Escobedo and Wei-Feng Xue.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Petr Kaplicky (Charles University) DTSTART:20201126T155000Z DTEND:20201126T161000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/32 DESCRIPTION:Title: Uniqueness and regularity of flows of non-Newtonian fluids wi th critical power-law growth\nby Petr Kaplicky (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet bound ary condition. Under the natural monotonicity\, coercivity and growth cond ition 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 all ows to improve regularity of any weak solution. Improving further the regu larity of weak solutions along a regularity ladder allows to show that act ually solution belongs to a uniqueness class provided data of the problem are sufficiently smooth.\n
\n\nWe also briefly discuss if the simila r technique is applicable to critical Convective Brinkman-Forchheimer equa tion.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Michal Bathory (University of Vienna) DTSTART:20201126T170000Z DTEND:20201126T172000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/33 DESCRIPTION:Title: Analysis of an unsteady flow of an incompressible heat-conduc tive rate-type viscoelastic fluid with stress diffusion\nby Michal Bat hory (University of Vienna) as part of BIRS workshop: Multiscale Models fo r Complex Fluids: Modeling and Analysis\n\n\nAbstract\nViscoelastic fluids often exhibit high sensitivity of material properties on temperature chan ges. Nevertheless\, the available mathematical theory for these fluids con cerns only models that are isothermal or that are simplified in other ways . For example\, one can find existence theories in 2D\, for small data\, w ith only the corotational derivative\, with only the spherical part of the elasticity tensor etc. In the talk\, we introduce an existence theory wit hout any of these assumptions and treat a rather general class of Johnson- Segalman-like models including full thermal evolution. To avoid the well-k nown ill-posedness of the corresponding PDE system\, we modify the ``elast ic part'' of the dissipation of the fluid far from the equilibrium\, while preserving thermodynamic compatibility of the model. This way\, we are ab le to prove the existence of a global-in-time weak solution for any initia l datum with finite total energy and entropy.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Paige Davis (Charles University) DTSTART:20201126T172500Z DTEND:20201126T174500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/34 DESCRIPTION:Title: Absolute Instabilities of Travelling Waves Solutions in a Kel lerSegel Model\nby Paige Davis (Charles University) as part of BIRS wo rkshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nA bstract\nThe Keller-Segel model for bacterial chemotaxis supports travelli ng wave solutions which have been described in the literature as both line arly stable and unstable and in the case of linear consumption (conditiona lly) nonlinearly stable. We reconcile this apparent contradiction by loca ting 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 t o absolute instability. Furthermore\, we show that the absolute spectrum d eforms as the consumption is changed\, illustrating a connection between t he constant\, sublinear and linear cases.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicola Fusco (Universita di Napoli) DTSTART:20201126T175000Z DTEND:20201126T181000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/35 DESCRIPTION:Title: Stability results for the nonlocal Mullins-Sekerka flow\n by Nicola Fusco (Universita di Napoli) as part of BIRS workshop: Multiscal e Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nThe nonl ocal 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 wil l show that three-dimensional periodic configurations that are strictly st able with respect to this energy are exponentially stable also for the non local Mullins-Sekerka flow. This result is contained in a joint paper with E. Acerbi\, M. Morini and V. Julin\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Dongjuan Niu (Capital Normal University) DTSTART:20201127T132500Z DTEND:20201127T134500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/36 DESCRIPTION:Title: Vanishing porosity limit of the coupled Stokes-Brinkman syste m\nby Dongjuan Niu (Capital Normal University) as part of BIRS worksho p: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstra ct\nIn this talk\, I will discuss with the small porosity asymptotic be havior of the coupled Stokes-Brinkman system in the presence of a curved i nterface between the Stokes region and the Brinkman region. In particular\ , we derive a set of approximate solutions\, validated via rigorous analys is\, to the coupled Stokes-Brinkman system. Of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-J ones interface condition (1.9) with the constant of proportionality indepe ndent of the curvature of the interface. \n
\nIt is a joint work wit h Mingwen Fei and Xiaoming Wang.
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Sébastien Boyaval (Ecole des Ponts ParisTech & Inria Paris) DTSTART:20201127T135000Z DTEND:20201127T141000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/37 DESCRIPTION:Title: Viscoelastic motions of Maxwell fluids with finite propagatio n speed\nby Sébastien Boyaval (Ecole des Ponts ParisTech & Inria Pari s) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modelin g and Analysis\n\n\nAbstract\nIn continuum models for non-perfect fluids\, viscoelastic stresses have often been introduced as extra-stresses of pur ely-dissipative (entropic) nature\, \nsimilarly to viscous stresses that i nduce motions of infinite propagation speed.\nA priori\, it requires only one to couple an evolution equation for the (extra-)stress with the moment um balance.\nIn many cases\, the apparently-closed resulting system is oft en not clearly well-posed\, even locally in time.\nThe procedure also rais es questions about how to encompass transition toward alastic solids.\n\nA noticeable exception is K-BZK theory where one starts with a purely elast ic fluids.\nViscoelasticity then results from dissipative (entropic) stres ses due to the relaxation of the fluids'"memory".\nThat K-BKZ approach is physically appealing\, but mathematically quite difficult because integral s are introduced to avoid material ('natural') configurations.\n\nWe propo se to introduce viscoelastic stress starting with hyperelastic fluids (lik e K-BKZ) and evolving material configurations (unlike K-BKZ).\nAt the pric e of an enlarged system with an additional material-metric variable\,\none can define well-posed (compressible) motions with finite propagation spee d\nthrough a system of conservation laws endowed with a "contingent entrop y" (like in standard polyconvex elastodynamics).\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Milan Pokorný (Charles University) DTSTART:20201127T150000Z DTEND:20201127T152000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/38 DESCRIPTION:Title: Existence analysis of a stationary compressible fluid model f or heat-conducting and chemically reacting mixtures\nby Milan Pokorný (Charles University) as part of BIRS workshop: Multiscale Models for Comp lex Fluids: Modeling and Analysis\n\n\nAbstract\nWe present large-data exi stence result for weak solutions to a steady compressible\nNavier-Stokes-F ourier system for chemically reacting fluid mixtures.\nGeneral free energi es satisfying some structural assumptions are considered\,\nwith a pressur e containing a $\\gamma$-power law.\nThe model is thermodynamically consis tent and contains the Maxwell-Stefan\ncross-diffusion equations in the Fic k-Onsager form\nas a special case. Compared to previous works\, a very gen eral model class is\nanalyzed\, including cross-diffusion effects\, temper ature gradients\,\ncompressible fluids\, and different molar masses.\nA pr iori estimates are derived from the entropy balance and the total\nenergy balance. The compactness for the total mass density follows from\nan estim ate for the density in $L^{\\gamma}$ with $\\gamma>3/2$\,\nthe effective viscous\nflux identity\, and uniform bounds related to Feireisl's oscillat ions defect measure.\nThese bounds rely heavily on the convexity of the fr ee energy and the strong convergence\nof the relative chemical potentials. \n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Tomas Los (Charles University) DTSTART:20201127T152500Z DTEND:20201127T154500Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/39 DESCRIPTION:Title: On planar flows of viscoelastic fluids of the Burgers type\nby Tomas Los (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nRate-type fluid models involving the stress and its observer-invariant time derivati ves of higher order are used to describe a large class of viscoelastic mix tures - geomaterials like asphalt\, biomaterials such as vitreous in the e ye\, synthetic rubbers such as SBR. A standard model that belongs to the c ategory 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 m odels of the first order. This viewpoint allows one to develop the whole h ierarchy of generalized models of a Burgers type. We study one such genera lization. Carrying on the study by \nMasmoudi (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 o f two Giesekus models in two spatial dimensions.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Skrzeczkowski (University of Warsaw) DTSTART:20201127T155000Z DTEND:20201127T161000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/40 DESCRIPTION:Title: Fast reaction limit with nonmonotone reaction function\nb y Jakub Skrzeczkowski (University of Warsaw) as part of BIRS workshop: Mul tiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\n\nWe analyse fast reaction limit in the reaction-diffusion system\n\\begi n{align*}\n\\partial_t u^{\\varepsilon} &= \\frac{v^{\\varepsilon} - F(u^ {\\varepsilon})}{\\varepsilon}\, \\\\\n\\partial_t v^{\\varepsilon} &= \\D elta v^{\\varepsilon} + \\frac{F(u^{\\varepsilon}) - v^{\\varepsilon}}{\\v arepsilon}\,\n\\end{align*}\nwith nonmonotone reaction function $F$. As sp eed of reaction tends to infinity\, the concentration of non-diffusing com ponent $u^{\\varepsilon}$ exhibits fast oscillations. We identify precisel y 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 ne w insights\, relaxes assumptions on model functions and provides a weak fo rmulation for the evolution of the Young measure.\n
\n\n\nThis is a joint work with Beno\\^\\i t Perthame (Paris) [1]\n
\n\n\n[1] B. Per
thame\, J. Skrzeczkowski. Fast reaction limit with nonmonotone reaction
function.\narXiv: 2008.11086\, submitted.
\n[2] P. I. Plotnikov. <
i>Passage to the limit with respect to viscosity in an equation with a var
iable direction of parabolicity. Differ. Uravn.\, 30:4 (1994)\, 665--6
74\; Differ. Equ.\, 30:4 (1994)\, 614--622.\n
We consider the Kelvin-Voigt model for viscoelastic ity and prove propagation of $H^1$-regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored en ergy satisfies the Andrews-Ball condition\, in particular allowing for a n on-monotone stress. By contrast\, a counterexample indicates that for non- monotone stress-strain relations (even in 1-d) initial oscillations\nof th e strain lead to solutions with sustained oscllations. In addition\, in tw o space dimensions\, we prove that the weak solutions with deformation gra dient in $H^1$ are in fact unique\, providing a striking analogy to the 2D Euler equations with bounded vorticity. \n
\n\n(joint work with K. Koumatos (U. of Sussex)\, C. Lattanzio and S. Spirito (U. of L’Aquila)). \n
\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Agnieszka Świerczewska-Gwiazda (University of Warsaw) DTSTART:20201127T175000Z DTEND:20201127T181000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/43 DESCRIPTION:Title: Dissipative measure-valued solutions for the Euler-Poisson eq uation\nby Agnieszka Świerczewska-Gwiazda (University of Warsaw) as p art of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and A nalysis\n\n\nAbstract\nWe consider pressureless compressible Euler equati ons driven by nonlocal repulusion-attraction and alignment forces. Our att ention is directed to measure-valued solutions\, i.e.\, very weak solutio ns described by a\nclassical Young measure together with appropriate conce ntration defects. We investigate the evolution of a relative energy funct ional to compare\na measure-valued solution to a regular solution emanatin g from the same initial datum. This leads to a weak-strong uniqueness prin ciple.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Victoria Patel (University of Oxford) DTSTART:20201127T130000Z DTEND:20201127T132000Z DTSTAMP:20260404T060944Z UID:BIRS_20w5188/44 DESCRIPTION:Title: Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body\nby Victoria Patel (University of Oxford) as part of BIRS workshop: Multiscale Models for Complex Fluids: Mo deling and Analysis\n\n\nAbstract\nWe will consider a system of evolutiona ry PDEs that describe a model of\nviscoelastic bodies exhibiting a certain strain-limiting property. \nNamely\, working in the small strain setting\ , we ask that a sum of the linearised \nstrain and the strain rate is give n by some function $F$ acting on the Cauchy\nstress tensor\, where this fu nction $F$ is nonlinear and bounded. These \nmodels come from the much lar ger class of implicit constitutive\nrelations. We will show the existence and uniqueness of global-in-time\nlarge-data weak solutions to this strain -limiting problem by first\nproving the existence of solutions to a broade r class of models. This \nbroader class replaces the bounded function $F$ on the stress by one that\nexperiences some level of polynomial growth. Us ing a suitable approximation of the\nstrain-limiting problem by these prob lems with growth\, we are able to deduce\nsuitable a priori bounds that al low us to prove the existence of a \nsolution to our original problem. The main issue is that the stress tensor\, and\nthus approximations of the st ress\, are initially seen to be bounded a priori \nonly in $L^1$. However\ , we are able to circumvent such an issue without introducing \nany notion of measure-valued solutions\, and in particular\, we obtain a satisfactor y \nexistence theory for these problems under some suitable assumptions on the data.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5188/44/ END:VEVENT END:VCALENDAR