We show that weak solutions of degenerate Navier-Sto kes equations converge\nto the strong solutions of the pressureless Euler system with linear drag term\, Newtonian\nrepulsion and quadratic confinem ent. The proof is based on the relative entropy method\nusing the artifici al velocity formulation for the one-dimensional Navier-Stokes system.\n

\n\nThe result is based on the joint work with Jose A. Carrillo and Ew elina Zatorska.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Alexis Vasseur (University of Texas at Austin) DTSTART;VALUE=DATE-TIME:20201123T170000Z DTEND;VALUE=DATE-TIME:20201123T172000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z 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 w orkshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\n Abstract\nIn this talk\, we will discuss the interaction between the st ability\, and the propagation of regularity\, for solutions to the incompr essible 3D Euler equation. It is still unknown whether a solution with smo oth 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 Eule r equation at high frequency. In the axisymmetric case\, we can also study the instability of blow-up profiles. \n

\nThis work was partially s upported by the NSFDMS-1907981. \n

\nThis a joint work with Misha Vi shik and Laurent Lafleche.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Ondřej Kreml (Czech Academy of Sciences) DTSTART;VALUE=DATE-TIME:20201123T172500Z DTEND;VALUE=DATE-TIME:20201123T174500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/7 DESCRIPTION:Title: Non-uniqueness of admissible weak solutions to the comp ressible Euler equations with smooth initial data\nby Ondřej Kreml (Czech Academy of Sciences) as part of BIRS workshop: Multiscale Models for Comp lex Fluids: Modeling and Analysis\n\n\nAbstract\nWe consider the isentr opic Euler equations of gas dynamics in the whole two-dimensional space an d we prove the existence of a $C^\\infty$ initial datum which admits infin itely many bounded admissible weak solutions. Taking advantage of the rela tion between smooth solutions to the Euler system and to the Burgers equat ion we construct a smooth compression wave which collapses into a perturbe d Riemann state at some time instant $T > 0$. In order to continue the sol ution after the formation of the discontinuity\, we adjust and apply the t heory developed by De Lellis and Székelyhidi and we construct infinitely many solutions.\n

\n\nThis is a joint work with Elisabetta Chiodarol i\, V\\'aclav M\\'acha and Sebastian Schwarzacher. \n

\n END:VEVENT BEGIN:VEVENT SUMMARY:Michael Renardy (Virginia Tech) DTSTART;VALUE=DATE-TIME:20201123T175000Z DTEND;VALUE=DATE-TIME:20201123T182000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/8 DESCRIPTION:Title: Pure stress modes for linear viscoelastic flows with va riable coefficients\nby Michael Renardy (Virginia Tech) as part of BIRS wo rkshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nA bstract\nWe consider the equations of a linear Maxwell fluid with spati ally varying coefficients. Pure stress modes are solutions with zero veloc ity but nonzero stresses. We derive equations to characterize such solutio ns. In two dimensions\, we find that under generic hypotheses only certain "trivial" solutions exist. In three dimensions\, on the other hand\, ther e 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. \n\n\n

\n(joint work with Debanjana Mitra and Mythily Ramaswamy)\n

\n END:VEVENT BEGIN:VEVENT SUMMARY:Tomas Barta (Charles University) DTSTART;VALUE=DATE-TIME:20201124T130000Z DTEND;VALUE=DATE-TIME:20201124T132000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/9 DESCRIPTION:Title: Decay of solutions to integrodifferential equations\nby Tomas Barta (Charles University) as part of BIRS workshop: Multiscale Mod els for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe discuss lo ng time behavior of solutions to a non-linear second order integrodifferen tial convolution equation\, in particular we focus on the speed of converg ence to equilibrium. The key assumptions are that the convolution kernel i s small and the non-linear operator satisfies the Lojasiewicz inequality.\ n END:VEVENT BEGIN:VEVENT SUMMARY:Mark Dostalik (Charles University) DTSTART;VALUE=DATE-TIME:20201124T132500Z DTEND;VALUE=DATE-TIME:20201124T134500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/10 DESCRIPTION:Title: Thermodynamically consistent derivation of a micro-macr o model for dilute polymeric fluids\nby Mark Dostalik (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nThe rheology of complex fluids such as polymer ic liquids is highly non-Newtonian in nature and manifests itself as an ex tra stress component in the Cauchy stress tensor. At the purely macroscopi c 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 thermodyna mically based approach to the design of a class of such micro-macro models for dilute polymeric liquids and show that the thermodynamic background o f the model naturally yields stability of the steady state when the fluid occupies an isolated vessel.\n END:VEVENT BEGIN:VEVENT SUMMARY:Piotr Mucha (University of Warsaw) DTSTART;VALUE=DATE-TIME:20201124T135000Z DTEND;VALUE=DATE-TIME:20201124T141000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/11 DESCRIPTION:Title: Flows initiated by ripped density\nby Piotr Mucha (Univ ersity of Warsaw) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nInstead of the abstract\, ple ase see the video on https://youtu. be/l85eQapJ_bA.\n END:VEVENT BEGIN:VEVENT SUMMARY:Dalibor Pražák (Charles University) DTSTART;VALUE=DATE-TIME:20201124T150000Z DTEND;VALUE=DATE-TIME:20201124T152000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/12 DESCRIPTION:Title: A finite-dimensional reduction of dissipative dynamical systems\nby Dalibor Pražák (Charles University) as part of BIRS worksho p: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstra ct\nWe show that under natural regularity assumptions\, an abstract non linear parabolic evolution problem has a finite-dimensional attractor. Mor eover\, the long-time dynamics can be recast as a system of ODEs with expo nentially decaying delay.\n

\n\nAs an application\, we consider a cl ass of non-Newtonian fluids with dynamic boundary conditions.\n

\n END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Woznicki (University of Warsaw) DTSTART;VALUE=DATE-TIME:20201124T152500Z DTEND;VALUE=DATE-TIME:20201124T154500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/13 DESCRIPTION:Title: Mv-strong uniqueness for density dependent\, incompress ible\, non-Newtonian fluids\nby Jakub Woznicki (University of Warsaw) as p art of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and A nalysis\n\n\nAbstract\n\nWe analyse the system of the form\n\\begin{ali gn*}\n {\\partial}_t{\\rho} +{\\rm div \\\,}_x(\\rho u) = 0\\\\\n {\\ partial}_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\, $u$ den otes velocity field\, ${\\mathbb{S}}$ the stress tensor and $p$ is the pre ssure. We are interested in the measure-valued solutions to those equation s and prove the mv-strong uniqueness property. This work bases its assumpt ions on the recent paper by Abbatiello and Feireisl [1]\, but differs from it in density dependency. Surprisingly the solutions are not defined by t he Young measures\, but by the similar tool to the so-called defect measur e.\n

\n\n\n

\nIn m ultiscale models for polymeric fluids\, the evolution of the polymer chain is usually modeled using an entropic force\, computed from the free energ y associated with the end-to-end vector. We will present results which aim at justifying under which circumstances such a dynamics is indeed close t o the original dynamics based on the full-atom chain.\n

\n\n\nF. L egoll\, T. Lelièvre and S. Olla\,

\nF. Legoll\, T. Lelièvre and U. Sharma\,

Classical models describing the motion of Newtonia n fluids\, such as water\, rely on the assumption that the Cauchy stress i s 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 h owever that the framework of classical continuum mechanics\, built upon an explicit constitutive equation for the Cauchy stress\, is too narrow to d escribe inelastic behavior of solid-like materials or viscoelastic propert ies of materials. Our starting point in this work is therefore a generaliz ation of the classical framework of continuum mechanics\, called the impli cit constitutive theory\, which was proposed recently in a series of paper s by K.R. Rajagopal. The underlying principle of implicit constitutive the ory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and\, in particular\, linear) func tion of the symmetric part of the velocity gradient\, one may allow a nonl inear\, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian f luid flow models with implicit and possibly discontinuous power-law-like r heology.\n

\n\nWe develop the analysis of finite element approximati ons of implicit power-law-like models for viscous in-compressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the c lass of models under consideration are related by a\, possibly multi-value d\, 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 paramete r\, measuring the granularity of the finite element triangulation\, tends to zero. A key new technical tool in our analysis is a finite element coun terpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions.\n\n

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

\n END:VEVENT BEGIN:VEVENT SUMMARY:Vít Průša (Charles University) DTSTART;VALUE=DATE-TIME:20201124T175000Z DTEND;VALUE=DATE-TIME:20201124T181000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/17 DESCRIPTION:Title: Thermodynamics of viscoelastic rate-type fluids and its implications for stability analysis\nby Vít Průša (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nAnalysis of finite amplitude stability of flui d flows is a challenging task even if the fluid of interest is described u sing 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.\n\nWe show that the kno wledge of thermodynamical underpinnings of these complex models can be gai nfully exploited in the stability analysis. First we introduce general con cepts that allow one to deal with thermodynamically isolated systems\, and then we proceed to thermodynamically open systems. Next we document the a pplications of these concepts in the case of container flows (thermodynami cally 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 flo w.\n END:VEVENT BEGIN:VEVENT SUMMARY:Anna Abbatiello (Institut für Mathematik\, Technische Universitä t Berlin) DTSTART;VALUE=DATE-TIME:20201125T130000Z DTEND;VALUE=DATE-TIME:20201125T132000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/18 DESCRIPTION:Title: On the motion of a compressible viscous fluid driven by time-periodic inflow/outflow boundary conditions\nby Anna Abbatiello (I nstitut für Mathematik\, Technische Universität Berlin) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\ nAbstract\n

We consider the barotropic Navier-Stokes system describing t he motion of a compressible viscous fluid confined to a bounded domain dri ven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time-periodic solution in the class of weak solutions sat isfying the energy inequality. \n

\n\nThis is a joint work with Edua rd Feireisl.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Ewelina Zatorska (University College London) DTSTART;VALUE=DATE-TIME:20201125T132500Z DTEND;VALUE=DATE-TIME:20201125T134500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z 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 o f BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analys is\n\n\nAbstract\nIn this talk I will present a derivation of macroscopic model of interacting particles. The population of N particles evolve accor ding to a diffusion process and interacts through a dynamical network. In turn\, the evolution of the network is coupled to the particles' position s. In contrast with the mean-field regime\, in which each particle interac ts 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 particl e interacts\, on average\, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics. The deri vation combines the stochastic averaging (over time-scale parameter) and the many particles ($N\\to \\infty$) limits.\n END:VEVENT BEGIN:VEVENT SUMMARY:Tomasz Dębiec (University of Warsaw) DTSTART;VALUE=DATE-TIME:20201125T135000Z DTEND;VALUE=DATE-TIME:20201125T141000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/20 DESCRIPTION:Title: Incompressible limit for a two-species model with coupl ing through Brinkman’s law.\nby Tomasz Dębiec (University of Warsaw) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe study a two-species model of tissue growth describing dynamics under mechanical pressure and cell growth. The pressur e is incorporated into the common fluid velocity through an elliptic equat ion\, called Brinkman’s law\, which accounts for viscosity effects in th e individual species. \nOur aim is to establish the incompressible limit a s 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.\n

\n\nJoint work with B . Perthame (Sorbonne)\, M. Schmidtchen (TU Dresden) and N. Vauchelet (Pari s 13).

\n END:VEVENT BEGIN:VEVENT SUMMARY:Mária Lukácová-Medvidová (Johannes Gutenberg-Universität Main z) DTSTART;VALUE=DATE-TIME:20201125T150000Z DTEND;VALUE=DATE-TIME:20201125T152000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/21 DESCRIPTION:Title: Viscoelastic phase separation: analysis and numerics\nb y Mária Lukácová-Medvidová (Johannes Gutenberg-Universität Mainz) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nMathematical modelling and numerical simulations o f phase separation becomes much\nmore involved if one component is a macr omolecular compound. In this case\, the large molecular relaxation time\ng ives rise to a dynamic coupling between intra-molecular processes and the unmixing on experimentally relevant time scales\,\nwith interesting new ph enomena\, for which the name “viscoelastic phase separation” has been coined.\n\nOur model of viscoelastic phase separation describes time evolution of the volume fraction of a polymer and the bulk stress\nleadin g to a strongly coupled (possibly degenerate) cross-diffusion system. The evolution of volume fraction is governed\nby the Cahn-Hilliard type equa tion\, while the bulk stress is a parabolic relaxation equation. The syst em is further\ncombined with the Navier-Stokes-Peterlin system\, describi ng time evolution of the velocity and (elastic) conformation tensor.\n

\nUnder some physically relevant assumptions on boundedness of\nmodel par ameters we have proved that global in time weak solutions exist.\nFurther\ , we have derived a suitable notion of the relative energy taking into acc ount the non-convex nature of the energy law\nfor the viscoelastic phase s eparation. This allows us to prove the weak-strong uniqueness principle\na nd consequently the uniqueness of a weak solution in special cases.\n

\ nOur extensive numerical simulations confirm robustness of the analysed mo del\nand the convergence of a suitable numerical scheme with respect to th e relative energy.\n END:VEVENT BEGIN:VEVENT SUMMARY:Pierre Degond (Imperial College London) DTSTART;VALUE=DATE-TIME:20201125T152500Z DTEND;VALUE=DATE-TIME:20201125T154500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/22 DESCRIPTION:Title: Topological protection in collective dynamics\nby Pierr e Degond (Imperial College London) as part of BIRS workshop: Multiscale Mo dels for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nStates of ma tter (such as solid\, liquid\, etc) are characterized\nby different types of order associated with local invariances under\ndifferent transformation groups. Recently\, a new notion of topological\norder\, popularized by th e 2016 physics nobel prize awarded to Haldane\,\nKosterlitz and Thouless\, has emerged. It refers to the global rigidity of\nthe system arising in s ome circumstances from topological constraints.\nTopologically ordered sta tes are extremely robust i.e. « topologically\nprotected » against local ized perturbations. Collective dynamics occurs when\na system of self-prop elled particles organizes itself into a coherent\nmotion\, such as a flock \, a vortex\, etc. Recently\, the question of realizing\ntopologically pro tected collective states has been raised. In this work\, we\nconsider a sy stem of self-propelled solid bodies interacting through local\nfull body a lignment up to some noise. In the large-scale limit\, this system\ncan be described by hydrodynamic equations with topologically non-trivial\nexplic it solutions. At the particle level\, these solutions persist for a\ncerta in time but eventually decay towards a uniform flocking state\, due to\nth e stochastic nature of the particle system. We show numerically that the\n persistence time of these topologically non-trivial solutions is far longe r\nthan for topologically trivial ones\, showing a new kind of « topologi cal\nprotection » of a collective state. To our knowledge\, it is the fir st time\nthat a hydrodynamic model guides the design of topologically non- trivial\nstates of a particle system and allows for their quantitative ana lysis and\nunderstanding. In passing\, we will raise fascinating mathemati cal questions\nunderpinning the analysis of collective dynamics systems. \n

\nJoint\nwork with Antoine Diez and Mingye Na (Imperial College Lond on)\n END:VEVENT BEGIN:VEVENT SUMMARY:Barbora Benesova (Charles University) DTSTART;VALUE=DATE-TIME:20201125T155000Z DTEND;VALUE=DATE-TIME:20201125T161000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/23 DESCRIPTION:Title: A variational approach to fluid-structure interaction\n by Barbora Benesova (Charles University) as part of BIRS workshop: Multisc ale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nIn thi s talk we consider the interaction of a Stokes/Navier-Stokes flow with a v iscoelastic body. The elastic body is allowed to undergo large deformation s (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 ext end this approach to the hyperbolic regime including inertia of the solid body. These variational approaches allow us to prove proper energetic esti mates while also controling the geometric restictions posed on the solid b ody and\, eventually\, to establish existence of weak solutions. This is j oint work with Malte Kampschulte and Sebastian Schwarzacher (both Prague). \n END:VEVENT BEGIN:VEVENT SUMMARY:Piotr Gwiazda (Polish Academy of Sciences) DTSTART;VALUE=DATE-TIME:20201125T170000Z DTEND;VALUE=DATE-TIME:20201125T172000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/24 DESCRIPTION:Title: Homogenization of nonlinear elliptic systems in nonrefl exive Musielak-Orlicz spaces\nby Piotr Gwiazda (Polish Academy of Sciences ) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\n

We study the homogenization process for fa milies of strongly nonlinear elliptic systems with the homogeneous Dirichl et boundary conditions. The growth and the coercivity of the elliptic oper ator 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 functi on spaces at each step.\n

\n\n[ 2] Bulíček\, Miroslav\; Gwiazda\, Piotr\; Kalousek\, Martin\; Świerczew ska-Gwiazda\, Agnieszka:

Several notions of weak or 'very weak' solutions have been sugge sted for the incompressible and\ncompressible Euler systems\, motivated by the lack of a satisfactory well-posedness theory for these\nequations in turbulent regimes. Surprisingly\, the speaker and L. Székelyhidi showed i n 2012 that dis-\ntributional and measure-valued solutions are in a sense the same\, although the latter had been expected\nto be a much weaker noti on. In this talk\, we turn to the isentropic compressible Euler system\, w here\nthe situation is fundamentally different. \n

\nJoint work with E. Chiodaroli\, E. Feireisl\, O. Kreml\, and D.\nGallenmüller. \n

\n END:VEVENT BEGIN:VEVENT SUMMARY:Vaclav Macha (Academy of Sciences\, Czech Republic) DTSTART;VALUE=DATE-TIME:20201125T175000Z DTEND;VALUE=DATE-TIME:20201125T181000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/26 DESCRIPTION:Title: On a body with a cavity filled with compressible fluid\ nby Vaclav Macha (Academy of Sciences\, Czech Republic) as part of BIRS wo rkshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nA bstract\nWe 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. \n

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

\n END:VEVENT BEGIN:VEVENT SUMMARY:Yong Lyu (Nanjing University) DTSTART;VALUE=DATE-TIME:20201126T130000Z DTEND;VALUE=DATE-TIME:20201126T132000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/27 DESCRIPTION:Title: Homogenization of stationary Navier–Stokes–Fourier system in domains with tiny holes\nby Yong Lyu (Nanjing University) as par t of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Ana lysis\n\n\nAbstract\nWe study the homogenization of stationary compressibl e Navier–Stokes–Fourier system in a bounded three dimensional domain p erforated with a large number of very tiny holes. Under suitable assumptio ns imposed on the smallness and distribution of the holes\, we show that t he homogenized limit system remains the same in the domain without holes.\ n END:VEVENT BEGIN:VEVENT SUMMARY:Pierre-Emmanuel Jabin (University of Maryland) DTSTART;VALUE=DATE-TIME:20201126T132500Z DTEND;VALUE=DATE-TIME:20201126T134500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/28 DESCRIPTION:Title: Compressible Navier-Stokes equations with heterogeneous pressure laws\nby Pierre-Emmanuel Jabin (University of Maryland) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analy sis\n\n\nAbstract\nWe prove the existence of global weak solutions à la L eray for compressible Navier-Stokes equations with a pressure law which de pends on the density and on time and space variables t and x. The assumpti ons on the pressure contain only locally Lipschitz assumption with respect to the density variable and some hypothesis with respect to the extra tim e and space variables. It may be seen as a first step to consider heat-con ducting Navier-Stokes equations with physical laws such as the truncated v irial assumption. The paper focuses on the construction of approximate sol utions through a new regularized and fixed point procedure and on the weak stability process taking advantage of the new method introduced by the tw o first authors with a careful study of an appropriate regularized quantit y linked to the pressure.\nThis is a joint work with D. Bresch and F. Wang.\n END:VEVENT BEGIN:VEVENT SUMMARY:Ansgar Juengel (TU Wien) DTSTART;VALUE=DATE-TIME:20201126T135000Z DTEND;VALUE=DATE-TIME:20201126T141000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/29 DESCRIPTION:Title: Analysis of degenerate cross-diffusion systems for heat -conducting fluid mixtures\nby Ansgar Juengel (TU Wien) as part of BIRS wo rkshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nA bstract\nWe present global-in-time existence results for two cross-diffusi on systems modeling\nheat-conducting fluid mixtures. Both models consist o f the balance equations for the\nmass densities and temperature. The key d ifficulty is the nonstandard degeneracy in the \ndiffusion (Onsager) matri ces\, i.e.\, ellipticity is lost when the fluid density or \ntemperature v anishes. This problem is overcome in the first model by exploiting the \nv olume-filling property of the mixture\, leading to gradient estimates for the square \nroot of the partial densities\, and in the second model by co mpensated compactness\nand renormalization techniques from mathematical fl uid dynamics. \n

\nThe first model is\njoint work with C. Helmer\, the second one with G. Favre\, C. Schmeiser\, and N. Zamponi.\n END:VEVENT BEGIN:VEVENT SUMMARY:Erika Maringová (TU Wien) DTSTART;VALUE=DATE-TIME:20201126T150000Z DTEND;VALUE=DATE-TIME:20201126T152000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/30 DESCRIPTION:Title: On the dynamic slip boundary condition\nby Erika Maring ová (TU Wien) as part of BIRS workshop: Multiscale Models for Complex Flu ids: Modeling and Analysis\n\n\nAbstract\n

In the talk\, we discuss a re sponse 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 wal l 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-Stoke s-like problems with dynamic slip boundary condition\, which requires a pr oper generalisation of the Gelfand triplet and the corresponding function spaces setting. \n

\n\nIt is a joint work with Anna Abbatiello and M iroslav Bulíček.\n

\n END:VEVENT BEGIN:VEVENT SUMMARY:Marie Doumic (INRIA) DTSTART;VALUE=DATE-TIME:20201126T152500Z DTEND;VALUE=DATE-TIME:20201126T154500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/31 DESCRIPTION:Title: Estimating the division of amyloid fibrils\nby Marie Do umic (INRIA) as part of BIRS workshop: Multiscale Models for Complex Fluid s: Modeling and Analysis\n\n\nAbstract\n\nAmyloid fibrils are important biological structures associated with devastating human diseases such as Alzheimer disease\, as well as have vital biological functions such as adh esion and biofilm formation. The division of amyloid protein fibrils is re quired for the propagation of the amyloid state and is an important contri butor to their stability\, pathogenicity\, and normal function. \nWe apply asymptotic results on the fragmentation equation to develop an inverse p roblem approach\, allowing us to compare the division stability of amyloid fibrils and estimate their division features (fragmentation rate and kern el).\n

\n\nThis is a joint work with Magali Tournus\, Miguel Escobe do and Wei-Feng Xue.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Petr Kaplicky (Charles University) DTSTART;VALUE=DATE-TIME:20201126T155000Z DTEND;VALUE=DATE-TIME:20201126T161000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/32 DESCRIPTION:Title: Uniqueness and regularity of flows of non-Newtonian flu ids with critical power-law growth\nby Petr Kaplicky (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling a nd Analysis\n\n\nAbstract\nWe deal with the flows of non-Newtonian flui ds in three dimensional setting subjected to the homogeneous Dirichlet bou ndary condition. Under the natural monotonicity\, coercivity and growth co ndition 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 a llows to improve regularity of any weak solution. Improving further the re gularity of weak solutions along a regularity ladder allows to show that a ctually solution belongs to a uniqueness class provided data of the proble m are sufficiently smooth.\n

\n\nWe also briefly discuss if the simi lar technique is applicable to critical Convective Brinkman-Forchheimer eq uation.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Michal Bathory (University of Vienna) DTSTART;VALUE=DATE-TIME:20201126T170000Z DTEND;VALUE=DATE-TIME:20201126T172000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/33 DESCRIPTION:Title: Analysis of an unsteady flow of an incompressible heat- conductive rate-type viscoelastic fluid with stress diffusion\nby Michal B athory (University of Vienna) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nViscoelastic flui ds often exhibit high sensitivity of material properties on temperature ch anges. Nevertheless\, the available mathematical theory for these fluids c oncerns only models that are isothermal or that are simplified in other wa ys. For example\, one can find existence theories in 2D\, for small data\, with only the corotational derivative\, with only the spherical part of t he elasticity tensor etc. In the talk\, we introduce an existence theory w ithout any of these assumptions and treat a rather general class of Johnso n-Segalman-like models including full thermal evolution. To avoid the well -known ill-posedness of the corresponding PDE system\, we modify the ``ela stic part'' of the dissipation of the fluid far from the equilibrium\, whi le preserving thermodynamic compatibility of the model. This way\, we are able to prove the existence of a global-in-time weak solution for any init ial datum with finite total energy and entropy.\n END:VEVENT BEGIN:VEVENT SUMMARY:Paige Davis (Charles University) DTSTART;VALUE=DATE-TIME:20201126T172500Z DTEND;VALUE=DATE-TIME:20201126T174500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/34 DESCRIPTION:Title: Absolute Instabilities of Travelling Waves Solutions in a KellerSegel Model\nby Paige Davis (Charles University) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\ nAbstract\nThe Keller-Segel model for bacterial chemotaxis supports travel ling wave solutions which have been described in the literature as both li nearly stable and unstable and in the case of linear consumption (conditio nally) nonlinearly stable. We reconcile this apparent contradiction by lo cating the essential spectrum\, absolute spectrum and point spectrum of th e linear operators associated with the travelling wave solutions. We deriv e conditions for the spectral (in)stability of the travelling wave solutio ns 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.\n END:VEVENT BEGIN:VEVENT SUMMARY:Nicola Fusco (Universita di Napoli) DTSTART;VALUE=DATE-TIME:20201126T175000Z DTEND;VALUE=DATE-TIME:20201126T181000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/35 DESCRIPTION:Title: Stability results for the nonlocal Mullins-Sekerka flow \nby Nicola Fusco (Universita di Napoli) as part of BIRS workshop: Multisc ale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nThe no nlocal Mullins-Sekerka flow can be seen as the $H^{-\\frac12}$-gradient fl ow of the so called sharp-interface Ohta-Kawaski energy. In this talk we w ill show that three-dimensional periodic configurations that are strictly stable with respect to this energy are exponentially stable also for the n onlocal Mullins-Sekerka flow. This result is contained in a joint paper wi th E. Acerbi\, M. Morini and V. Julin\n END:VEVENT BEGIN:VEVENT SUMMARY:Dongjuan Niu (Capital Normal University) DTSTART;VALUE=DATE-TIME:20201127T132500Z DTEND;VALUE=DATE-TIME:20201127T134500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/36 DESCRIPTION:Title: Vanishing porosity limit of the coupled Stokes-Brinkman system\nby Dongjuan Niu (Capital Normal University) as part of BIRS works hop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbst ract\nIn 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 particula r\, we derive a set of approximate solutions\, validated via rigorous anal ysis\, to the coupled Stokes-Brinkman system. Of particular interest is th at the approximate solution satisfies a generalized Beavers-Joseph-Saffman -Jones interface condition (1.9) with the constant of proportionality inde pendent of the curvature of the interface. \n

\nIt is a joint work w ith Mingwen Fei and Xiaoming Wang.

\n END:VEVENT BEGIN:VEVENT SUMMARY:Sébastien Boyaval (Ecole des Ponts ParisTech & Inria Paris) DTSTART;VALUE=DATE-TIME:20201127T135000Z DTEND;VALUE=DATE-TIME:20201127T141000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/37 DESCRIPTION:Title: Viscoelastic motions of Maxwell fluids with finite prop agation speed\nby Sébastien Boyaval (Ecole des Ponts ParisTech & Inria Pa ris) as part of BIRS workshop: Multiscale Models for Complex Fluids: Model ing and Analysis\n\n\nAbstract\nIn continuum models for non-perfect fluids \, viscoelastic stresses have often been introduced as extra-stresses of p urely-dissipative (entropic) nature\, \nsimilarly to viscous stresses that induce motions of infinite propagation speed.\nA priori\, it requires onl y one to couple an evolution equation for the (extra-)stress with the mome ntum balance.\nIn many cases\, the apparently-closed resulting system is o ften not clearly well-posed\, even locally in time.\nThe procedure also ra ises questions about how to encompass transition toward alastic solids.\n\ nA noticeable exception is K-BZK theory where one starts with a purely ela stic fluids.\nViscoelasticity then results from dissipative (entropic) str esses due to the relaxation of the fluids'"memory".\nThat K-BKZ approach i s physically appealing\, but mathematically quite difficult because integr als are introduced to avoid material ('natural') configurations.\n\nWe pro pose to introduce viscoelastic stress starting with hyperelastic fluids (l ike K-BKZ) and evolving material configurations (unlike K-BKZ).\nAt the pr ice of an enlarged system with an additional material-metric variable\,\no ne can define well-posed (compressible) motions with finite propagation sp eed\nthrough a system of conservation laws endowed with a "contingent entr opy" (like in standard polyconvex elastodynamics).\n END:VEVENT BEGIN:VEVENT SUMMARY:Milan Pokorný (Charles University) DTSTART;VALUE=DATE-TIME:20201127T150000Z DTEND;VALUE=DATE-TIME:20201127T152000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/38 DESCRIPTION:Title: Existence analysis of a stationary compressible fluid m odel for heat-conducting and chemically reacting mixtures\nby Milan Pokorn ý (Charles University) as part of BIRS workshop: Multiscale Models for Co mplex Fluids: Modeling and Analysis\n\n\nAbstract\nWe present large-data e xistence result for weak solutions to a steady compressible\nNavier-Stokes -Fourier system for chemically reacting fluid mixtures.\nGeneral free ener gies satisfying some structural assumptions are considered\,\nwith a press ure containing a $\\gamma$-power law.\nThe model is thermodynamically cons istent and contains the Maxwell-Stefan\ncross-diffusion equations in the F ick-Onsager form\nas a special case. Compared to previous works\, a very g eneral model class is\nanalyzed\, including cross-diffusion effects\, temp erature gradients\,\ncompressible fluids\, and different molar masses.\nA priori estimates are derived from the entropy balance and the total\nenerg y balance. The compactness for the total mass density follows from\nan est imate for the density in $L^{\\gamma}$ with $\\gamma>3/2$\,\nthe effectiv e viscous\nflux identity\, and uniform bounds related to Feireisl's oscill ations defect measure.\nThese bounds rely heavily on the convexity of the free energy and the strong convergence\nof the relative chemical potential s.\n END:VEVENT BEGIN:VEVENT SUMMARY:Tomas Los (Charles University) DTSTART;VALUE=DATE-TIME:20201127T152500Z DTEND;VALUE=DATE-TIME:20201127T154500Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/39 DESCRIPTION:Title: On planar flows of viscoelastic fluids of the Burgers t ype\nby Tomas Los (Charles University) as part of BIRS workshop: Multiscal e Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nRate-typ e fluid models involving the stress and its observer-invariant time deriva tives of higher order are used to describe a large class of viscoelastic m ixtures - 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 th e 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 gene ralization. Carrying on the study by \nMasmoudi (2011)\, who briefly prove d 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.\n END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Skrzeczkowski (University of Warsaw) DTSTART;VALUE=DATE-TIME:20201127T155000Z DTEND;VALUE=DATE-TIME:20201127T161000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/40 DESCRIPTION:Title: Fast reaction limit with nonmonotone reaction function\ nby Jakub Skrzeczkowski (University of Warsaw) as part of BIRS workshop: M ultiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\n\nWe analyse fast reaction limit in the reaction-diffusion system\n\\be gin{align*}\n\\partial_t u^{\\varepsilon} &= \\frac{v^{\\varepsilon} - F( u^{\\varepsilon})}{\\varepsilon}\, \\\\\n\\partial_t v^{\\varepsilon} &= \ \Delta v^{\\varepsilon} + \\frac{F(u^{\\varepsilon}) - v^{\\varepsilon}}{\ \varepsilon}\,\n\\end{align*}\nwith nonmonotone reaction function $F$. As speed of reaction tends to infinity\, the concentration of non-diffusing c omponent $u^{\\varepsilon}$ exhibits fast oscillations. We identify precis ely its Young measure which\, as a by-product\, proves strong convergence of the diffusing component $v^{\\varepsilon}$\, a result that is not obvio us from a priori estimates. Our work is based on analysis of regularizatio n for forward-backward parabolic equations by Plotnikov [2]. We rewrite hi s 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.\n

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

\n\n\n[1] B. P
erthame\, J. Skrzeczkowski. *Fast reaction limit with nonmonotone reacti
on function*.\narXiv: 2008.11086\, submitted.

\n[2] P. I. Plotnikov.
*Passage to the limit with respect to viscosity in an equation with a v
ariable direction of parabolicity.* Differ. Uravn.\, 30:4 (1994)\, 665-
-674\; Differ. Equ.\, 30:4 (1994)\, 614--622.\n

We consider the Kelvin-Voigt model for viscoelast icity and prove propagation of $H^1$-regularity for the deformation gradie nt 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 no n-monotone stress-strain relations (even in 1-d) initial oscillations\nof the strain lead to solutions with sustained oscllations. In addition\, in two space dimensions\, we prove that the weak solutions with deformation g radient 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 END:VEVENT BEGIN:VEVENT SUMMARY:Agnieszka Świerczewska-Gwiazda (University of Warsaw) DTSTART;VALUE=DATE-TIME:20201127T175000Z DTEND;VALUE=DATE-TIME:20201127T181000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/43 DESCRIPTION:Title: Dissipative measure-valued solutions for the Euler-Pois son equation\nby Agnieszka Świerczewska-Gwiazda (University of Warsaw) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe consider pressureless compressible Euler equa tions driven by nonlocal repulusion-attraction and alignment forces. Our a ttention is directed to measure-valued solutions\, i.e.\, very weak solut ions described by a\nclassical Young measure together with appropriate con centration defects. We investigate the evolution of a relative energy fun ctional to compare\na measure-valued solution to a regular solution emanat ing from the same initial datum. This leads to a weak-strong uniqueness pr inciple.\n END:VEVENT BEGIN:VEVENT SUMMARY:Victoria Patel (University of Oxford) DTSTART;VALUE=DATE-TIME:20201127T130000Z DTEND;VALUE=DATE-TIME:20201127T132000Z DTSTAMP;VALUE=DATE-TIME:20201204T170636Z UID:BIRS_20w5188/44 DESCRIPTION:Title: Existence of large-data global weak solutions to a mode l of a strain-limiting viscoelastic body\nby Victoria Patel (University o f Oxford) as part of BIRS workshop: Multiscale Models for Complex Fluids: Modeling and Analysis\n\n\nAbstract\nWe will consider a system of evolutio nary PDEs that describe a model of\nviscoelastic bodies exhibiting a certa in strain-limiting property. \nNamely\, working in the small strain settin g\, we ask that a sum of the linearised \nstrain and the strain rate is gi ven by some function $F$ acting on the Cauchy\nstress tensor\, where this function $F$ is nonlinear and bounded. These \nmodels come from the much l arger class of implicit constitutive\nrelations. We will show the existenc e and uniqueness of global-in-time\nlarge-data weak solutions to this stra in-limiting problem by first\nproving the existence of solutions to a broa der class of models. This \nbroader class replaces the bounded function $F $ on the stress by one that\nexperiences some level of polynomial growth. Using a suitable approximation of the\nstrain-limiting problem by these pr oblems with growth\, we are able to deduce\nsuitable a priori bounds that allow us to prove the existence of a \nsolution to our original problem. T he main issue is that the stress tensor\, and\nthus approximations of the stress\, are initially seen to be bounded a priori \nonly in $L^1$. Howeve r\, we are able to circumvent such an issue without introducing \nany noti on of measure-valued solutions\, and in particular\, we obtain a satisfact ory \nexistence theory for these problems under some suitable assumptions on the data.\n END:VEVENT END:VCALENDAR