Seminar

  • Maja Szlenk
  • (University of Warsaw)
  • Uniqueness of weak solutions for the Stokes system for compressible fluids with general pressure
  • Abstract:
  • We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the transformation to the original Eulerian coordinates. For a nonnegative and bounded initial density, the solution is nonnegative for all $t>0$ as well and belongs to $L^infty([0,infty)timesmathbb{T}^d)$. A key point of our considerations is the uniqueness of such transformation. Since the velocity might not be Lipschitz continuous, we develop a method which relies on the results of Crippa & De Lellis, concerning regular Lagriangian flows. The uniqueness is obtained thanks to the application of a certain weighted flow and detail analysis based on the properties of the $BMO$ space.
  • 30.11.21   09:00

  • Clara Patriarca
  • (Politecnico di Milano)
  • Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel
  • Abstract:
  • In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent the obstacle from going excessively far from the equilibrium position and colliding with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.
  • 23.11.21   09:00

  • Sourav Mitra
  • (University of Würzburg)
  • A control problem of a linear compressible fluid-structure interaction model
  • Abstract:
  • I will talk about a result on the controllability of a compressible FSI model where the structure is located at a part of the fluid boundary. I will first introduce the notion of control and explain the tools to prove the controllability of a linear PDE. In the next part of the talk I will introduce the FSI model under consideration and corresponding linearization. Finally I will speak about the control of the linearized FSI problem and outline a proof.
  • 16.11.21   09:00

  • Srđan Trifunović
  • (University of Novi Sad)
  • On the fluid-structure interaction problem with heat exchange
  • Abstract:
  • Here, I will talk about a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear  thermoelasticity equations and constitutes a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise to a new previously unstudied interaction problem involving heat exchange. The existence of a weak solution for this problem is obtained by combining three approximation techniques - decoupling, penalization and domain extension for fluid.
    This talk is based on a joint work with Václav Mácha, Boris Muha, Šárka Nečasová and Arnab Roy.
  • 09.11.21   09:00

  • Ana Radošević
  • (University of Zagreb)
  • On the regularity of weak solutions to the fluid-rigid body interaction problem
  • Abstract:
  • The fluid-structure interaction (FSI) systems are multi-physics systems that include a fluid and solid component. They are everyday phenomena with a wide range of applications. The simplest model for the structure is a rigid body. We study a nonlinear moving boundary fluid-structure interaction problem where the fluid flow is governed by 3D Navier-Stokes equations, and the structure is a rigid body described by a system of ordinary differential equations called Euler equations for the rigid body. Our goal is to show that a weak solution that additionally satisfy Prodi-Serrin condition is smooth on the interval of its existence, which is a generalization of the well-known regularity result for the Navier-Stokes equations. This is a joint work with Boris Muha and Šárka Nečasová.
  • 02.11.21   09:00

  • Danica Basarić
  • (Institute of Mathematics, CAS)
  • Existence of weak solutions for models of general compressible viscous fluids with linear pressure
  • Abstract:
  • In this talk we will focus on the existence of weak solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. More precisely, we will first prove the existence of dissipative solutions and study under which conditions it is possible to guarantee the existence of weak solutions.
  • 26.10.21   09:00

  • Emil Skříšovský
  • (Charles University)
  • Evolutionary compressible Navier-Stokes-Fourier system in two space dimensions with adiabatic exponent almost one
  • Abstract:
  • In this talk we consider the full Navier-Stokes-Fourier system and present the proof of the existence of a weak solution in two space dimensions for the pressure law given by $p(varrho,theta) sim varrhotheta + varrho log^alpha(1+varrho)+ theta^4$, which can be viewed as a close approximation of the pressure law for ideal gas $p(varrho,theta) sim varrhotheta$. The weak solutions with entropy inequality and total energy balance are considered and the existence of this type of weak solutions without any restriction on the size of the initial conditions or the right-hand sides is shown provided $alpha > frac{17+sqrt{417}}{16}cong 2.34$.
  • 19.10.21   09:00

  • Milan Pokorný
  • (Charles University)
  • Steady compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions for the temperature
  • Abstract:
  • Based on recent result by Chaudhuri and Feireisl for the evolutionary NSF system we present the proof of existence of weak (and variational entropy) solutions to the steady version with Dirichlet boundary conditions for the temperature. The formulation is based, similarly as in the evolutionary case, on a version of ballistic energy inequality which allows to obtain a priori estimates for the temperature and velocity.
  • 05.10.21   09:00

  • Peter Bella
  • (TU Dortmund)
  • Regularity for degenerate elliptic equations
  • Abstract:
  • I discuss local regularity properties of solutions of linear non-uniformly elliptic equations with non-constant coefficients. Assuming certain integrability conditions on the ellipticity of the coefficient field, we obtain local boundedness of weak solutions and corresponding Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results in the literature (Trudinger). I will also discuss analogous result for the time-independent parabolic equations as well as application to study of the variational integrals with differential (p,q) growth.
  • 08.09.21   10:10

  • Florian Oschmann
  • (TU Dortmund)
  • Inverse of divergence and homogenization of compressible Navier-Stokes equations in randomly perforated domains
  • Abstract:
  • In homogenization of compressible Navier-Stokes equations, an inverse of the divergence operator (called Bogivskiu{i} operator) is crucial to obtain a-priori bounds for the velocity and density independent of the perforation. Such Bogovskiu{i} operators and bounds are well known in the case of periodically arranged holes with fixed diameter, where the mutual distance is of order $varepsilon>0$ and the radii scale like $varepsilon^alpha$ for some $alpha>3$. We generalize these results to the case of randomly distributed holes with random radii and give applications to the homogenization of the Navier-Stokes(-Fourier) equations in such randomly perforated domains.
  •                    09:00

  • Nilasis Chaudhuri
  • (TU Berlin)
  • Convergence of consistent approximations to the complete compressible Euler system
  • Abstract:
  • The aim of the talk is to discuss a result on the weak limit of a `consistent approximation scheme' to the compressible complete Euler system in the full space $ mathbb{R}^d,; d=2,3 $. The main result states that if a weak limit of the consistent approximation scheme is a weak solution of the system, then the approximate solutions converge locally strongly (or at least almost everywhere) in suitable norms under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they do not have to satisfy the minimal principle for entropy.
  • 15.06.21   09:00

  • Ewelina Zatorska
  • (Imperial College London)
  • On the existence of solutions to the two fluids systems
  • Abstract:
  • We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Existence, uniqueness and stability of global weak solutions to this system are obtained with arbitrarily large initial data. Making use of the uniform-in-time bounds for the densities from above and below, exponential decay of weak solution to the unique steady state is obtained without any smallness restriction to the size of the initial data. In particular, our results show that degeneration to single-fluid motion will not occur as long as in the initial distribution both components are present at every point.
    The results are based on joint papers with D. Bresch, P. Mucha, Y. Li and Y. Sun.
  • 25.05.21   10:00

  • Frank Merle
  • (Université de Cergy-Pontoise)
  • On the implosion of a three dimensional compressible fluid
  • Abstract:
  • We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of analysis are the existence of very regular self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).
  • 18.05.21   09:00

  • Yoshihiro Shibata
  • (Waseda University)
  • R-solver, Maximal Regularity, and Mathematical Fluid Dynamics
  • Abstract:
  • Maximal Regularity is an important tool to show the existence of strong solutions of quasi-linear system of parabolic equations, for example free boundary problems for the Navier-Stokes equations. In this lecture, I will talk about a systematic approach for the Lp maximal regularity by using the R-solver. This approach is quite useful to control the high frequency part of solutions to the linearized problem, and so we can prove the local well-posedness for the dynamical equations appearing in the mathematical fluid dynamics. But, to prove the global well-posedness at least for small initial data, we have to control the low frequency part. To do this I use so called Lp-Lq decay estimate for the semigroup group associated with the linearized problem, like Stokes equations. In this talk, I will present how to prove the maximal regularity by using an R-solver and how to control the low frequency part by using Lp-Lq estimates to prove the global wellposedness in some concrete example, like equations describing the compressible viscous fluid flow.
  • 11.05.21   09:00

  • Paolo Maremonti
  • (Universitŕ della Campania Luigi Vanvitelli)
  • On the uniqueness of a suitable weak solution to the Navier-Stokes Cauchy problem
  • Abstract:
  • We are dealing with the Navier-Stokes Cauchy problem. We investigate some results of regularity and uniqueness related to suitable weak solutions. The suitable weak solution notion is meant in the sense introduced by Caffarelli-Kohn-Nirenberg. In paper [1], we recognize that a suitable weak solution enjoys more regularity than Leray-Hopf weak solutions, that allows us to furnish new uniqueness results for the solutions. Actually, we realize two results. The first one is a new sufficient condition on the initial datum u0 for uniqueness. We work on existing suitable weak solution, that is, we do not construct a more regular weak solution corresponding to our initial datum. The second result employs a weaker condition with respect to previous ones (almost $u_0 in  L^2$), but, just for one of the two compared weak solutions, we need a “special” Prodi-Serrin condition. It is “special” as it is local in space.

    References: [1] Crispo F. and Maremonti P., On the uniqueness of a suitable weak solution to the Navier-Stokes Cauchy problem, SN Partial Differential Equations and Applications, to appear.
  • 04.05.21   09:00

  • Franco Flandoli
  • (Scuola Normale Superiore, Pisa)
  • Mixing and dissipation properties of transport noise
  • Abstract:
  • This talk is based on recent works with Dejun Luo and Lucio Galeati devoted to the investigation of a suitable scaling limit of several different PDE models subject to transport noise, when the noise is extremized in a suitable limit sense. Among the consequences there are certain forms of mixing, enhanced dissipation, delayed blow-up due to noise; these results hold for several classes of equations including Euler and Navier-Stokes equations, Keller-Siegel and reaction diffusion equations; and also rigorous results on eddy viscosity and eddy dissipation in turbulent fluids have been proved. Along with arguments of stochastic model reduction, developed with Umberto Pappalettera, a picture arises of the potential effects of small scale fluctuations on large scale properties of turbulent fluids.
  • 27.04.21   09:00

  • Mythily Ramaswamy
  • (TIFR Centre for Applicable Mathematics, Bangalore)
  • Local stabilization of time periodic flows
  • Abstract:
  • Fluid flows have been studied for a long  time, with a view to understand better the models like channel flow, blood flow, air flow in the lungs etc. Here we focus on  time periodic fluid flow models. Local stabilization here concerns the decay of the perturbation in the flow near a periodic trajectory. The main motivating example is the incompressible Navier-Stokes system. I will discuss the general framework to study periodic solutions and then indicate some results in this direction.
  • 20.04.21   09:00

  • Francesco Fanelli
  • (University of Lyon)
  • Statistical solutions to the barotropic Navier-Stokes equations
  • Abstract:
  • In this talk we are concerned with the notion of statistical solutions to some models of fluid mechanics. We focus on the barotropic Navier-Stokes equations, supplemented with non-homogeneous boundary data.
    In the first part of the talk, we study dynamical properties of statistical solutions. Our approach, different from the classical one of Foia?-Prodi and Vishik-Fursikov for the incompressible system, is based on a semiflow selection procedure. This allows us to define statistical solution as the push-forward measure of the initial probability distribution on the space of data of the Navier-Stokes system. We then investigate questions like existence and stability of
    statistical solutions.
    In the second part of the talk, we focus on the special class of stationary statistical solutions. In particular, we explore their role in the investigation of the validity of the so-called ergodic hypothesis in the context of the barotropic Navier-Stokes equations.

    This talk is based on joint works with Eduard Feireisl (Czech Academy of Sciences) and Martina Hofmanová (Universität Bielefeld).
  • 13.04.21   09:00

  • Martina Hofmanová
  • (University of Bielefeld)
  • Non-uniqueness in law of stochastic 3D Navier-Stokes equations
  • Abstract:
  • We consider the stochastic Navier-Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on two iconic examples of a stochastic perturbation: either an additive or a linear multiplicative noise driven by a Wiener process. In both cases, we develop a stochastic counterpart of the convex integration method  introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,infty). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval [0,T], T>0.
  • 06.04.21   09:00

  • Antonín Novotný
  • (University of Toulon)
  • Compressible fluids with nonhomogeneous boundary data II
  • Abstract:
  • We shall discuss several problems in the mathematical analysis of  viscous compressible fluids under the action of non zero inflow-outflow boundary conditions.

    The lecture will be delivered on Zoom (only):
    Meeting ID: 944 2924 8803
    Passcode: 697514

    NEW: The record of the lecture
  • 30.03.21   09:00

  • Antonín Novotný
  • (University of Toulon)
  • Compressible fluids with nonhomogeneous boundary data I
  • Abstract:
  • We shall discuss several problems in the mathematical analysis of  viscous compressible fluids under the action of non zero inflow-outflow boundary conditions.

    The lecture will be delivered on Zoom (only):
    Meeting ID: 944 2924 8803
    Passcode: 697514

    NEW: The record of the lecture
  • 23.03.21   08:30

  • Michele Coti Zelati
  • (Imperial College London)
  • Stationary Euler flows near the Kolmogorov and Poiseuille flows
  • Abstract:
  • We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This is in contrast with both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel, for which we can show that the only stationary states near them must be shears. This has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes.

    NEW: The slides of the lecture
  • 16.03.21   09:00

  • Václav Mácha
  • (Institute of Mathematics, CAS)
  • Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations
  • Abstract:
  • We show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem. This work was done in collaboration with M. Kalousek and Š. Nečasová.
  • 09.03.21   09:00

  • David Lannes
  • (Université de Bordeaux)
  • Some problems arising in wave-structure interactions
  • Abstract:
  • There are different formulations of the water waves problem. One of them is to formulate it as a system of equations coupling two quantities, e.g. the free surface elevation $zeta$ and the horizontal discharge $Q$. Actually, one can understand the water waves problem as a system on three quantities, $zeta$, $Q$ and the surface pressure $P_s$ under the constraint that $P_s$ is constant (and therefore disappears from the equations).
    When we consider in addition a floating body then, under the body, we still have a system of equations on the same three quantities, but this time the constraint is not on the pressure but on the surface of the water, that must coincide with the bottom of the floating object.
    Wave-structure interactions can be understood as the coupling of these two different constrained problems. We shall briefly analyse this coupling and show among other things how it dictates the evolution of the contact line between the surface of the water and the surface of the floating body, and how to transform it into transmission problems that raise many mathematical issues such as fully nonlineary hyperbolic initial boundary value problems, dispersive boundary layers, initial boundary value problems for nonlocal equations, etc.
  • 02.03.21   09:00

  • Piotr B. Mucha
  • (University of Warsaw)
  • Flows initiated by ripped densities
  • Abstract:
  • We address the question: Are solutions to the equations of viscous flows that are initiated by a density function given by a characteristic function of a set regular and unique?  The positive answer is possible for the compressible Navier-Stokes equations if the bulk/volume viscosity is large. The limit case of the homogeneous incompressible NSEs will be discussed too.

    The talk will be based on results with Raphael Danchin:
    RD, PBM: Compressible NSEs with ripped density, arXiv;
    RD, PBM: The incompressible NSEs in vacuum, CPAM2019.
  • 05.01.21   09:00

prof. RNDr. Eduard Feireisl, DrSc.
Šárka Nečasová, Milan Pokorný
chairmen