9:00pracovní seminář NCMM
Na pracovním semináři vystoupí Miroslav Bulíček a Tomáš Bárta.
9:00Josef Štěpán (
MFF UK Praha):
Stochastické reprezentace parciálních diferenciálních rovnic
Abstrakt: pdf
Přednáška bude v češtině.
9:00Mohamed Majdoub (
University Tunis ElManar):
Energy Critical NLS in two space dimensions
We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity
We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case. We also obtain a scattering result in the subcritical regime.
10:15Yutaka Terasawa (Nečas Center and Sapporo University): On Stokes operators with variable viscosity
We consider a genaralization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function bounded from below by a positive constant. This system arises as a linearized system in several fluid mechanics problems. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded $ H_{\infty} $ calculus in $ L^q $ space, which implies the maximal $ L^q $ regularity of the corresponding parabolic evolution equation. The analysis is done for both bounded domains and some class of unbounded domains with boundary of low regularity.
9:00Reinhard Farwig (Technical University Darmstadt): Global Weak Solutions of the Instationary Navier-Stokes System with Non-zero Boundary Values
9:00Bernard Ducomet (Universite Bruyeres, France): Asymptotics for a fluid model of neutron star, after Lattimer
10:15Okihiro Sawada (Waseda University, Tokyo): On the incompressible Navier-Stokes flows around various steady states in the whole space
9:00Jörg Wolf (
University of Magdeburg):
Existence of suitable weak solutions to the Navier-Stokes-Fourier system and partial regularity - Part I
Second lecture of the short course "On the existence and regularity of weak solutions to the Navier-Stokes-Fourier systems under Dirichlet boundary conditions".
Full abstract: pdf
10:15Jörg Wolf (
University of Magdeburg):
Existence of suitable weak solutions to the Navier-Stokes-Fourier system and partial regularity - Part II
Third lecture of the short course "On the existence and regularity of weak solutions to the Navier-Stokes-Fourier systems under Dirichlet boundary conditions".
Full abstract: pdf
9:00Giuseppe Mingione (University of Parma): Non linear aspects of Calderon-Zygmund theory
Calderon-Zygmund theory is a classical tool allowing, in the linear case, to sharply infer the integrability information on solutions to elliptic and parabolic equations starting from that of the given data. I will try to report on some non-linear analogues of such results.
10:15Jörg Wolf (
University of Magdeburg):
Existence of renormalized solutions to the Navier-Stokes-Fourier system and partial regularity
Last lecture of the short course "On the existence and regularity of weak solutions to the Navier-Stokes-Fourier systems under Dirichlet boundary conditions".
Full abstract: pdf
9:00László Székelyhidi (University of Bonn): Convex Integration for Isometric Immersions and the Euler Equations
Since work of Nash and Kuiper in the 1950s the following dichotomy concerning isometric embeddings of $S^2$ into $\R^3$ is well known: whereas the only $C^2$ isometric embedding is the standard embedding modulo rigid motion, there exist many $C^1$ isometric embeddings which can "wrinkle up" the sphere inside arbitrarily small regions in space. It turns out that the same method can be used to construct many examples of weak solutions of the incompressible Euler equations which dissipate energy and are highly oscillatory. In the talk I will show the proof of this result, which is joint work with Camillo De Lellis. I will then address the question of what should be the borderline regularity between rigidity and flexibility for isometric immersions, and the analogous borderline between dissipative and energy-conserving solutions of the Euler equations. These questions are highly relevant in their own right, and seem very much related.
10:15Antonín Novotný (University of Toulon): Singular limits in the thermodynamics of viscous fluids - part III/IV
9:00Andrea Cianchi (University of Firenze): Orlicz-Sobolev spaces and applications to strongly nonlinear PDEs - part I/IV
Basic definitions and properties of Orlicz and Orlicz-Sobolev spaces.
Isoperimetric inequalities.
Symmetrization principles for Orlicz norms of the gradient.
Anisotropic symmetrization and anisotropic Orlicz-Sobolev spaces.
10:15Vuk Milisic (Wolfgang Pauli Institute, Vienna): Multi-scale blood flow modelling for stented arteries: theoretical and numerical results
Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of some realistic boundary conditions at the inlet and outlet parts of the domain. We derive error estimates in terms of the roughness size epsilon either in the a priori norm or in the very veak context (cf Necas). We show some numerical test validating our theoretical results.
11:30Nikolai V. Chemetov (CMAF / University of Lisbon): Strongly Nonlinear Hyperbolic - Elliptic Problem in a Bounded Domain
We investigate a mixed hyperbolic-elliptic type system of PDEs in a given domain. Motivated by physics, we consider nonzero boundary conditions, which describe a flow through the domain. We prove the solvability of this system, using a kinetic formulation of the problem. The system can be used for different physical situations, such as:
a) the motion of superconducting vortices in the superconductor;
b) the Keller-Segel model, describing the collective cell movement;
c) the porous media motion and etc.
Workshop Mathematical aspects of fluid mechanics and thermodynamics
In the occassion of 60th birthday of Prof. Jiří Neustupa.
Place: Blue lecture hall, Institute of Mathematics (ground floor of the building beyond the backyard), Žitná 25, Praha 1
Organized by Petr Kučera (Faculty of Civil Engineering, Czech Technical University in Prague) and Šárka Nečasová (Institute of Mathematics, Czech Academy of Sciences).
9:00-9:15Antonín Novotný (University of Toulon): opening
9:20-10:05Patrick Penel (
University of Toulon):
In occasion of sixtieth birthday of Jiří Neustupa: The main originality of three common studies in the theory of Navier-Stokes equations
The mathematical understanding of many qualitative aspects of classical 3D-Navier-Stokes flows is always a fascinating subject of research. Among central issues are the following questions : Nature of eventual singularities developed by the flows ? Influence of boundary conditions complemented the models ? Some related problems have been solved in these directions. The present lecture proposes a brief survey of three common small contributions obtained with Jiri Neustupa based on relatively new approaches. - New in 1996-97, we focused our attention on regularity information in dependance of specific smoothness of one velocity component (expecting the role of the vorticity and this fact that in case of blow'up all velocity components simultaneously loose their regularity).
- New in 2002-03, we weakened the three constraints of Dirichlet boundary conditions and preserved the usual results of the theory generalizing the impermeability properties (thinking that generally speaking the tangential part of the velocity has no reason to be zero on the boundary)
- New in 2008-09, we claimed and proved that the contacts or possible collisions of bodies moving in a fluid can occur with a natural non zero velocity if one consider Navier's boundary condition (a result we cannot expect with the Dirichlet boundary conditions).
10:05-10:50Miloslav Feistauer (Charles University in Prague): Numerical solution of compressible flow in time dependent domains
We are concerned with the simulation of inviscid and viscous compressible flow in time dependent domains. We present an ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow, discretize them in space by the discontinous Galerkin method and introduce a semi-implicit linearized time stepping for the numerical solution of the complete problem. We present some computational results for flow in a channel with a moving wall.
These results were obtained in cooperation with Václav Kučera and Jaroslava Prokopová from Charles University in Prague, Faculty of Mathematics and Physics, and Jaromír Horáček from Institute of Thermomechanics of Academy of Sciences of the Czech Republic.
10:50-11:20break
11:20-12:05Reimund Rautmann (University of Paderborn): On very smooth Navier-Stokes solutions
Recent results on vorticity transport&diffusion [1] suggest the question, wether there are Navier-Stokes solutions taking their values in the domain of the Stokes operator´s power 3/2. We will find such solutions to Navier-Stokes initial-boundary value problems with outer forces which depend on the flow velocity in a special way proposed by V.A. Solonnikov.
[1] R.Rautmann: A direct approach to vorticity transport&diffusion, RIMS Koyuroku Bessatsu B1, Kyoto (2007)305-329.
12:05-12:50Reiner Picard (University of Dresden): On a model in poro-elasticity
A modification of the material law associated with the well-known Biot system as suggested by M. A. Murad and J. H. Cushman in 1996 and first rigorously investigated by R. E. Showalter in 2000 is re-considered in the light of a new approach to a comprehensive class of evolutionary problems and extended to large class of anisotropic inhomogeneous media.
The paper is joint work with Des McGhee (Glasgow, U.K.).
9:00Maurizio Grasselli (Politecnico di Milano): Asymptotic behavior of dissipative evolution equations - Part II/IV
Existence theorems for global attractors, dynamics on the global attractor, Lyapunov functionals and gradient systems, unstable and stable manifolds, global attractors for gradient systems.
10:15Andrea Cianchi (University of Firenze): Orlicz-Sobolev spaces and applications to strongly nonlinear PDEs - part II/IV
Embedding theorems for first-order Orlicz-Sobolev spaces.
Anisotropic Sobolev inequalities.
Embedding theorems for higher-order Orlicz-Sobolev spaces.
9:00Maurizio Grasselli (Politecnico di Milano): Asymptotic behavior of dissipative evolution equations - Part III/IV
Reaction-diffusion equations: well-posedness, bounded absorbing sets, compact absorbing sets, global attractor, further related examples (2D Navier-Stokes equation, Cahn-Hilliard equation, phase-field systems, two-phase fluids).
10:15Enrich Fernandez-Cara (University of Sevilla, Spain): Some recent results and open questions concerning the control of nonlinear PDEs from fluid mechanics
This talk concerns the formulation of control problems for some partial differential systems with origin in fluid mechanics: the Navier-Stokes and Boussinesq equations, quasi-geostrophic models, the Maxwell and Oldroyd viscoelastic models, etc. The most interesting results are connected to the local exact controllability to the trajectories. The particular properties of some of these systems lead to challenging open problems.
11:30Andrea Cianchi (University of Firenze): Orlicz-Sobolev spaces and applications to strongly nonlinear PDEs - part III/IV
Continuity and differentiability properties of Orlicz-Sobolev functions. Boundedness of solutions to variational problems under general growth conditions.
9:00Maurizio Grasselli (Politecnico di Milano): Asymptotic behavior of dissipative evolution equations - Part IV/IV
Damped semilinear wave equations: the 3D case with critical growth, the global attractor and its smoothness.
10:15Andrea Cianchi (University of Firenze): Orlicz-Sobolev spaces and applications to strongly nonlinear PDEs - part IV/IV
Local boundedness of solutions to fully anisotropic elliptic equations in divergence form. Higher-integrability properties of the gradient of solutions to variational problems under general growth conditions.
9:00Roger Lewandowski (University Rennes 1): Main mathematical tools for the study of turbulent models
We show first the Boccardo-Gallouet interpolation inequality. We next prove existence results for the coupled k-u system in case of bounded eddy viscosity terms, in 2D and 3D cases.
10:15Roger Lewandowski (University Rennes 1): Renormalized solutions to the scalar k-u system for unbounded eddy viscosties
We turn to the 3D scalar k-u system with homogeneous boundary conditions. We define the notion of « renormalized solutions » and we prove the existence of a renormalized solution to the system in case of unbounded eddy viscosities.
9:00Victor Isakov (Department of Mathematics and Statistics, Wichita State University): The inverse doping profile problem for semiconductors
We consider a standard drift-diffusion model(system of 3 PDE) of a semiconductor device and its simplifications suitable for studying the problem of identifying the doping profile (source term in the first equation of the system) from some practically available data. We introduce a dual problem, calculate the first term of an asymptotic expension with respect to conductivity in various parts of a semiconductor and obtain the first uniqueness result in an important unipolar case. We outline possible future research.
9:00Roman Shvydkoy (University of Illinois, Chicago): On ill-posedness of basic equations of fluid dynamics in Besov spaces
In this talk we will discuss several results concerning ill-posedness and conditional regularity of Navier-Stokes and Euler equations in various critical Besov spaces. In particular we present a construction of a vector field $u_0 \in B^{-1}_{\infty,\infty}\cap H^s$ for all $s<1/2$ in 3D such that no Leray-Hopf solution to the NSE with this initial condition has a continuous trajectory in the metric of $B^{-1}_{\infty,\infty}$. The construction exploits the basic mechanism of the forward energy cascade. On the other hand, if we know that a weak Leray-Hopf solution is continuous in $B^{-1}_{\infty,\infty}$, then it is automatically regular. This illustrates the criticality of Besov space $B^{-1}_{\infty,\infty}$ for the well-posedness theory for NSE. Similar examples will presented for the Euler equation to prove ill-posedness in a range of Besov spaces including $B^{5/2}_{2,\infy}$. We recall that H^{5/2} = B^{5/2}_{2,2}$ is the critical space for applicability of the energy method, and the best known local well-posedness result in only known for $B^{5/2}_{2,1}$. The case of $H^{5/2}$ itself remains an open question. The talk is based on a joint work with A. Cheskidov.
10:15Christian Komo (TU Darmstadt): Regularity of weak solution to the Navier-Stokes equations in exterior domains
11:30Karolina Goetze (TU Darmstadt): Strong L^p solutions for a problem of fluid-rigid body interaction
14:00Eiji Yanagida (Tohoku University, Sendai, Japan): Solutions with moving singularities for a semilinear parabolic equation
For a semilinear parabolic equation with power nonlinearity, it is known that there exists a singular steady state that is radially symmetric with respect to the singular point. In this talk, we consider time-dependent singular solutions whose singularity resembles that of the singular steady state. In some parameter range, given suitable initial data, we establish the local existence and uniqueness of a solution with moving singularities. We also show the global existence of a solution whose singular point traces any prescribed smooth curve.
9:00Yuming Qin (Donghua University, Shanghai): Attractors for Autonomous Dynamical Systems I.
10:15Yuming Qin (Donghua University, Shanghai): Attractors for Autonomous Dynamical Systems II.
9:00Dieter Bothe (TU Darmstadt): The Maxwell-Stefan approach to multicomponent diffusion
From classical experiments on diffusion in mixtures of chemical components it is well-known that anomalous effects like up-hill diffusion occur and, hence, Fickian diffusion does not apply. The Maxwell-Stefan approach considers force balances to account for cross-effects between different species as well as thermodynamic driving forces to include effects in non-ideal solutions. We recall the derivation of the Maxwell-Stefan equations and provide a first result on the wellposedness of the resulting reaction-diffusion equations.
10:15Yuming Qin (Donghua University, Shanghai): Attractors for Autonomous Dynamical Systems III.
11:30Yuming Qin (Donghua University, Shanghai): Attractors for Autonomous Dynamical Systems - Applications in PDEs
9:00Joachim Naumann (
Humboldt University, Berlin):
On the limit case p → 1 of the power law model (stationary motion)
Abstract: pdf
10:15Jörg Wolf (
Otto-von-Guericke University, Magdeburg):
On the partial regularity of suitable weak solutions to the generalized Navier-Stokes equations
Abstract: pdf
9:00Nicola Fusco (
University of Naples):
Equilibrium configurations of epitaxially strained elastic films: qualitative properties of solutions
Abstract: pdf
9:00Milan Pokorný (
Mathematical Institute of Charles University, Praha):
Regularity criteria for incompressible Navier-Stokes equations involving one velocity component
10:15Ondřej Kreml (Faculty of Mathematics and Physics, Charles University, Praha): Steady flow of a second grade fluid past an obstacle
Seminář zrušen - seminar cancelled
9:00Jan Stebel (Institute of Mathematics, AS CR, Praha): Shape stability of incompressible fluids subject to Navier’s slip
10:15Václav Mácha (Faculty of Mathematics and Physics, Charles University, Praha): Existence and uniqueness of solutions to generalized Stokes problem
9:00Pavel Krejčí (
Institute of Mathematics, AS CR, Praha):
Regularity and uniqueness of solutions to quasilinear parabolic systems
Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on an $L^p$ variant of the Gronwall lemma (joint work with Lucia Panizzi).
Seminář zrušen - seminar cancelled
9:00Radek Erban (
University of Oxford):
On some partial differential equations arising in mathematical modelling of biological systems
The material to be covered will follow on from my talks "Stochastic and Multiscale Modelling in Biology I and II" given on Monday, 23rd November 2009, as part of the Necas Seminar Series on Continuum Mechanics. However, this talk will be understandable without having attended the first two lectures. I will focus on two problems in which partial differential equations (PDEs) play a significant role. The first one is the analysis of the behaviour of stochastic chemical systems which is based on the chemical Fokker-Planck equation. The second area will be concerned with hyperbolic and parabolic PDEs arising in modelling of chemotaxis of bacteria and amoeboid cells.
10:15Yulia Namlyeyeva (Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk): Isolated singularities of solutions of nonlinear anisotropic elliptic and parabolic equations
It is considered wide classes of quasilinear anisotropic elliptic equations and doubly nonlinear anisotropic parabolic equations, whose solutions have singularity at a point. The sharp point-wise conditions for removable isolated singularities of solutions are established.
9:00Miroslav Bulíček (Faculty of Mathematics and Physics, Charles University, Praha): Regularity results for unsteady incompressible Stokes-Fourier and Navier-Stokes-Fourier systems
We consider unsteady flow of homogeneous incompressible non-Newtonian or/and Newtonian heat conducting fluid in spatially periodic setting. For non-Newtonian fluids we identify a class of viscosities for which it is relatively easy to get estimates on the second gradient of the velocity in 2D case. Using then regularity methods developed for two-dimensional flows of non-heat conducting fluids we are able to show the existence of a classical solution to 2D Navier-Stokes-Fourier system. On the other hand the case of Newtonian fluids is more delicate and one cannot directly obtain the estimates on the second velocity gradient even for Stokes-Fourier system, i.e., the system without convective terms. Although this fact, we are still able to identify a class of physically relevant viscosities for which the desired estimate can be derived and then consequently the full regularity of the solution in 2D setting can be established.
10:15Libor Pavlíček (Faculty of Mathematics and Physics, Charles University, Praha): Lower monotone maps
New class of k-monotone maps will be introduced and some of their basic properties will be discussed. Some natural, till now, unanswered questions will be formulated.
9:00Dalibor Pražák (Faculty of Mathematics and Physics, Charles University, Praha): Attractors for nonlinear parabolic problems in unbounded domains
We study the long time behavior of solutions to nonlinear parabolic equations in unbounded domains. We propose to show the existence of a global attractor and estimate its Kolmogorov entropy, using the method of trajectories and suitable estimates using Bochner spaces with spatial weights.
Everyone is warmly welcome.
