List of abstracts:
Piotr Biler (University of Wroclaw, Poland): 'New criteria for blowup of solutions in some chemotaxis systems'
Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions
Miroslav Bulicek (Charles University in Prague, Czech Republic): 'Analysis of a viscosity model for concentrated polymers'
The will be concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier--Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic-hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier--Stokes system, includes dependence on the shear-rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We present a result concenrning the existence of global-in-time, large-data weak solutions under fairly general hypotheses.
Rinaldo Colombo (University of Brescia, Italy): 'Modeling and Management of Biological Resources'
Consider a biological resource, i.e., a population characterized by birth, aging and death, grown in order to be sold and produce a profit. The resulting model consists of a part describing the evolution of the resource and of a part describing the costs and the income. The former leads to a structured population model, modified to take into account the selection for reproduction or for the market. The latter is based on suitable cost functions that depend on control parameters describing the selling strategies. We prove the well posedness of the descriptive model and the Gateaux differentiability of the cost with respect to the parameters.
(In collaboration with M.Garavello, University of Milano-Bicocca)
Eduard Feireisl (Academy of Sciences of the Czech Republic): 'Solvability of certain problems concerning inviscid fluids'
We present some examples and counterexamples to well-posedness of problems arising in the mathematical theory of inviscid fluids. We discuss both positive results like weak-strong uniqueness and negative one like the existence of infinitely many solutions for given initial data.
Irena Lasiecka (University of Memphis, US): 'Global stability in a 3-D fluid structure interactions with moving frame.'
Equations of fluid structure interactions are described by Navier Stokes equations coupled to a 3-d dynamic system of elasticity. The coupling is on a free boundary interface between the two regions: the fluid and oscillating structure. The interface is moving with the velocity of the flow. The resulting model is a quasilinear system with parabolic-hyperbolic coupling acting on a moving boundary.
One of the main features and difficulty in handling the problem is a mismatch of regularity between parabolic and hyperbolic dynamics. The existence and uniqueness of smooth local solutions has been established by D. Coutand and S. Shkoller Arch. Rational Mechanics and Analysis in 2005. Global existence of solutions subject to a dissipation placed on the interface has been shown in [1]. We shall show that global well-posedness can be achieved without any interface dissipation. In addition, solutions corresponding to suitably small initial data will be shown to decay to equilibria uniformly. The main hurdle of the problem - the mismatch of the regularity between hyperbolocity and parabolicity - is handled be exploiting recently established sharp regularity of the "Dirichlet-Neuman " map for hyperbolic solvers along with maximal parabolic regularity available for Stokes operators. These two ingredients prevent a typical "loss" of derivatives and allow for the algebraic and topological closure of a suitably constructed fixed point.
This work is joint with M. Ignatova (Princeton), I. Kukavica and A. Tuffaha (USC, Los Angeles)
References:
[1] On wellposedness and small data global existence for an interface damped free boundary fuid-structure interaction model (with. M. Ignatova, I. Kukavica and A. Tuff aha) - Nonlinearity ,vol 27, issue 3, pp 467-499, 2014
Anna Marciniak-Czochra (Heidelberg University, Germany): 'Mass concentration in a structured population model of clonal selection in acute leukemias.'
Recent experimental observations show that although leukemias exhibit clonalheterogeneity, only few clones are detected at diagnosis and at relapses. Thepattern of clone selection may be dierent. To address these questions we pro-pose a mathematical model of dynamics of multiple leukemic clones coupled todynamics of healthy hematopoiesis. Each cell line goes through several dieren-tiation stages and is characterized by parameters describing proliferation ratesand self-renewal potential. Considering a continuum of leukemic clones leads toa structured population model. The model takes a form of a system of integro-dierential equations with a nonlinear and nonlocal coupling, which describesregulatory feedback loops in cell proliferation and dierentiation process. Weshow that such coupling leads to mass concentration in points corresponding tomaximum of the self-renewal potential and the model solutions tend asymptoti-cally to a linear combination of Dirac measures. Furthermore, using a Lyapunovfunction constructed for a nite dimensional counterpart of the model, we provethat the total mass of the solution converges to a globally stable equilibrium.Mathematical analysis suggests which mechanisms of clonal selection predictclonality observed in the course of disease.
Piotr Mucha (University of Warsaw, Poland): 'Two solutions to steady Burgers equation'
Sarka Necasova (Academy of Sciences of the Czech Republic): 'Singular limits in a model of compressible flow.'
We consider relativistic and ”semi-relativistic” models of radiative viscous com-pressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equa-tion extending the classical model introduced in [1] and we study some of its singular limits in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field see [2],[3],[4].
References:
[1] B. Ducomet, E. Feireisl, S. Necasova: On a model of radiation hydrodynamics. Ann. I. H. Poincare-AN 28 (2011) 797–812.
[2] B. Ducomet, S. Necasova: Diffusion limits in a model of radiative flow, to appear in Annali dell Universita di Ferrara, DOI 10.1007/s11565-014-0214-3.
[3] B. Ducomet, S. Necasova: Singular limits in a model of radiative flow, to appear in J. of Math. Fluid Mech.
[4] B. Ducomet, S. Necasova: Non equilibrium diffusion limit in a barotropic ra-diative flow Submitted to Volume Contemporary Mathematics Series of the American Mathematical Society; editors: Vicentiu Radulescu, Adelia Se-queira, Vsevolod A. Solonnikov.
Benoit Perthame (Universit Pierre et Marie Curie): 'The Hele-Shaw free-boundary asymptotics for fluid models of tumor growth'
The growth of solid tumors can be described at a number of different scales from the cell to the organ scales. For a large number of cells, the 'fluid mechanical' approach has been advocated recently by many authors in mathematics or biophysics. Several levels of mathematical descriptions are commonly used, including only elastic effects, nutrients, active movement, surrounding tissue, vasculature remodeling and several other features.
We will focus on the links between two types of mathematical models. The `microscopic' description is at he cell population density level and a more macroscopic, description is based on a free boundary problem close to the classical Hele-Shaw equation. Asymptotic analysis is a tool to derive these Hele-Shaw free boundary problems from cell density systems in the stiff pressure limit. This modeling also opens other questions as circumstances in which instabilities develop.
This work is a collaboration with F. Quiros and J.-L. Vazquez (Universidad Autonoma Madrid), M. Tang (SJTU) and N. Vauchelet (LJLL).
Milan Pokorny (Charles University in Prague, Czech Republic): 'Heat-conducting, compressible mixtures with multicomponent diffusion'
We study a model for heat conducting compressible chemically reacting gaseous mixture, based on the coupling between the compressible Navier–Stokes–Fourier system and the full Maxwell-Stefan equations. The viscosity coefficients are density-dependent functions vanishing on vacuum and the internal pressure depends on species concentrations. We consider the question of existence of a solution to this system and based on several levels of approximations we construct a weak solution without any restriction on the size of the data. We also consider steady version of the system. The presentation is based on joint papers with P.B. Mucha (University of Warsaw), E. Zatorska (IMPAN Warsaw) and V. Giovangigli (Ecole Polytechnique, Palaiseau).
Ryszard Rudnicki (Polish Academy of Sciences): 'Asymptotic properties of some phenotypic evolution models'
We present an individual based model of phenotypic evolution which includes random and assortative mating process of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of distribution densities of phenotypic trait. In the case of random mating we show that this equation has one-dimensional attractor. In the case of assortative mating we expect convergence of the phenotype profile to multimodal limit distributions. This result suggests that assortative mating can lead to polymorphic population and adaptive speciation.
The talk is based on a manuscript: R. Rudnicki and P. Zwolenski, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol. 70 (2015), 1295-1321.
Wojciech Zajączkowski (Polish Academy of Sciences): 'Stability results for the Navier-Stokes equations'
We present results on existence of regular global special solutions to the Navier-Stokes equations. Next we present results on stability of these solutions. Finally we show ideas of proofs on stability. We are interested in existence of nonvanishing in time solutions. Hence we consider problems with nonvanishing in time external forces. We consider incompressible and compressible cases.
Short communication
Jan Burczak (Polish Academy of Science): 'Keller-Segel meets Burgers on a circle group.'
A disproof of a blowup conjecture for the critical fractional Keller-Segel system in one space dimension will be presented.
Filip Klawe (University of Warsaw, Poland): 'Thermo-visco-elasticity for Norton-Hoff-type models with Cosserat effects'
We consider the quasi-static evolution of thermo-visco-elastic material. The main goal of this talk is to present how taking into account the additional effects may improve the result of solutions' existence. We added a micropolarity effect to thermo-visco-elastic model regarding Norton-Hoff-type constitutive function. This additional phenomenon improves the regularity of solution.
Piotr Minakowski (University of Warsaw, Poland): 'A New Eulerian Approach to Crystal Plasticity'
Looking at severe plastic deformationexperiments, it seems that crystalline materials at yield behave as a special kind of anisotropic, highly viscous fluids flowing through an adjustable crystal lattice space. High viscosity provides a possibility to describe the flow as a quasi-static process, where inertial and other body forces can be neglected. The flow through the lattice space is restricted to preferred crystallographic planes and directions causing anisotropy. In the deformation process the lattice is strained and rotated.
We present derivation of a model that is based on the rate form of the decomposition rule: the velocity gradient consists of the lattice velocity gradient and the sum of the velocity gradients corresponding to the slip rates of individual slip systems. We employ the Gibbs potential to obtain rate of stress--strain response.
Aneta Wróblewska-Kamińska (Polish Academy of Science): 'Heat-conducting non-Newtonian fluids with nonstandard rheology'
Our purpose is to show existence of weak solutions to unsteady flow of non-Newtonian incompressible nonhomogeneous, heat-conducting fluids with a general form of the stress tensor. Being motivated by implications arising in nonstandard (nonlinear) rheology we focus on more general than polynomial growth condition. We use x-dependent convex functions which indicate suitable anisotropic Musielak-Orlicz spaces.
Ewelina Zatorska (University of Warsaw, Poland): 'Mathematical theory of the multicomponent flows'
I will present the results obtained for the mathematical model of multicomponent, reactive, compressible flow. I will start from analysis of the simple 2-species isothermal flow and explain how these basic ideas can be used in the case of multicomponent diffusion. I will aslo discuss some open problems in this field and our recent contributions concerning the systems with degenerate viscosity coefficients.