Seminar

  • Raphaël Danchin
  • (Université Paris-Est Créteil)
  • A class of global relatively smooth solutions for the Euler-Poisson system
  • Abstract:
  • In this joint work with X. Blanc, B. Ducomet and Š. Nečasová (to appear in JHDE), we construct a class of global solutions to the Cauchy problem for the isentropic Euler equations coupled with the Poisson equation, in the whole space. The initial density is assumed to decay to 0 at infinity and the initial velocity is close to some reference velocity with Jacobian having positive spectrum bounded away from 0. By a suitable adaptation of Grassin-Serre’s work on the `pure’ compressible Euler equations, we obtain a global smooth solution  the large time behavior of which may be described in terms of some solution of the multi-dimensional Burgers equation. The stability of some special spherically symmetric stationary solution is also discussed.
  • 15.12.20   09:00

  • Tong Tang
  • (Hohai University, Nanjing)
  • Global existence of weak solutions to the quantum Navier-Stokes equations
  • Abstract:
  • In this talk, we proved the global existence of weak solutions to the quantum Navier-Stokes equations with non-monotone pressure. Motivated by the work of Antonell-Spirito (2017, Arch. Ration. Mech. Anal., 1161-1199) and Ducomet-Necasova-Vasseur (2010, Z. Angew. Math. Phys., 479-491), we construct the suitable approximate system and obtain the corresponding compactness by B-D entropy estimate and Mellet-Vasseur inequality.
  • 08.12.20   09:00

  • Martin Kalousek
  • (Institute of Mathematics, CAS)
  • Global existence of weak solutions for a magnetic fluid model
  • Abstract:
  • The talk is devoted to the presentation of recent results that concern the global in time existence of weak solutions of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. Presented results are based on the joint work with S. Mitra and A. Schlömerkemper.
  • 01.12.20   09:00

  • Marco Bravin
  • (Basque Center for Applied Mathematics)
  • Interaction of a small rigid body with fluids
  • Abstract:
  • In this talk I will present a recent result in collaboration with Prof Necasova, where we study the interaction between a small rigid body and a compressible viscous fluid modeled by the compressible Navier-Stokes equations.

    In particular I will recall the previous results where the fluids were supposedly incompressible and then I will focus my attention on the improved pressure estimates that are the main novelty in our result. In contrast with the incompressible case the pressure estimates depend on a lower bound of the mass and the inertia matrix of the object as its size tends to zero.
  • 24.11.20   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • A maximal regularity approach to compressible mixtures
  • Abstract:
  • I will present recent results obtained in collaboration with Yoshihiro Shibata and Ewelina Zatorska. We investigate the well posedness of a system describing flow of a mixture of compressible constituents. The system in composed of Navier-Stokes equations coupled with equations describing balance of fractional masses. A crucial property is that the system is non-symmetric and only degenerate parabolic.

    However, it reveals a structure which allows to transform it to a symmetric parabolic problem using appropriate change of unknowns. In order to treat the transformed problem we write it in Lagrangian coordinates and linearize. For the related linear problem we show a Lp-Lq maximal regularity estimate applying the theory of R-bounded solution operators. This estimate allows to show local existence and uniqueness. Next, assuming additionally boundedness of the domain we extend the maximal regularity estimate and show exponential decay  property for the linear problem. This allows us to show global well-posedness of the original problem for small data.
  • 03.11.20   09:00

  • Gianmarco Sperone
  • (Charles University)
  • Explicit bounds for the generation of a lift force exerted by steady-state Navier-Stokes flows over a fixed obstacle
  • Abstract:
  • We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.
    The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).
  • 20.10.20   09:00

  • Ralph Chill
  • (TU Dresden)
  • Degenerate gradient systems: the bidomain problem and Dirichlet-to-Neumann operators
  • Abstract:
  • We identify a common structure behind several parabolic PDE models involving Dirichlet-to-Neumann operators, the bidomain operator, and some more, and we show that they actually have a gradient structure, possibly up to lower order perturbations. This has consequences for wellposedness of these PDEs, regularity of solutions and their asymptotic behaviour.
  • 13.10.20   09:00

  • Marija Galić
  • (University of Zagreb)
  • Existence of a weak solution to a 3d nonlinear, moving boundary FSI problem
  • Abstract:
  • See the attachment.
  • 06.10.20   09:00

  • Paolo Maria Mariano
  • (University of Florence)
  • Varifold-based variational description of crack nucleation: results and perspectives
  • Abstract:
  • In crack nucleation and growth processes, the crack margins may be in contact although not linked by material bonds. In variational descriptions of fractures in elastic-brittle bodies -descriptions all based on energy minimization and De Giorgi's view on minimizing movements - the circumstance can be hardly described just by identifying cracks with the jump set of special bounded variation functions. On the other hand, if we consider both crack paths and deformations as distinct entities - although linked because the deformation jump set is at least included in the crack path - we face the basic problem of controlling minimizing sequences of crack surfaces in three-dimensional environment. The problem can be eluded if we can take - roughly speaking - just sequences of surfaces with bounded curvature, although we should think of the notion of curvature we use because crack surfaces could be extremely rough. To this aim, we can consider crack paths as rectifiable sets supporting vector-valued measures, which admit a generalized notion of curvature: they are the so-called "curvature varifolds". Their introduction suggest a form of the energy for elastic-brittle bodies modifying standard Griffith's one, but  maintaining its polyconvex structure with respect to the deformation gradient. For such an energy it is possible to find existence theorems for minimizers, which are couples deformation/varifold. In this view, deformations can be both weak diffeomorphisms and SBV-based weak diffeomorphisms. In this way, we can describe from a variational view the nucleation of fractures without introducing additional failure criteria. In my talk, I'll describe such a setting with pertinent results, offering some possible research perspectives.
  • 25.02.20   10:15

  • Minsuk Yang
  • (Yonsei University)
  • New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure
  • Abstract:
  • We prove that a suitable weak solutions of the three-dimensional MHD equations are smooth if the negative part of the pressure is suitably controlled.
  •                    09:00

  • Tongkeun Chang
  • (Yonsei University)
  • On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary
  • Abstract:
  • We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's  inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.
  • 18.02.20   09:00

  • Srđan Trifunović
  • (Shanghai Jiao Tong University)
  • On the interaction of compressible fluids and nonlinear thermoelastic plates
  • Abstract:
  • in the attachment
  • 28.01.20   09:00

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