slideshow 3

Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

On some two-dimensional incompressible inhomogeneous viscous fluid flows
Zihui He
University of Bielefeld
Wednesday, 18. May 2022 - 9:00 to 10:00
In this talk, we will present some existence, uniqueness and regularity results for the motion of two-dimensional incompressible inhomogeneous viscous fluid flows in presence of a density-/temperature-dependent viscosity coefficient.

Firstly, we will discuss the boundary value problem for the stationary Navier-Stokes equation, where the viscosity coefficient is density-dependent. We will give some explicit solutions with piecewise constant viscosity coefficients, where some regularity and irregularity results will be considered.

We will also discuss the initial value problem for the evolutionary Boussinesq equation, which is a nonlinear coupling between a heat equation and a Navier-Stokes type of equation. In this case, the viscosity coefficient is temperature-dependent.

This talk is based on joint work with Xian Liao (KIT).

Parabolic boundary-value problems in generalized Sobolev spaces
Aleksandr Murach
NAS of Ukraine, Institute of Mathematics
Tuesday, 3. May 2022 - 9:00 to 10:00
See the attached file.
Murach_abstract.pdf

Global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions
Huanyao Wen
South China University of Technology
Tuesday, 26. April 2022 - 9:00 to 10:00
We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluids and the momentum equations for the mixture. This model can be derived from a generic compressible two-fluid model with equal velocities and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes equations. Under some assumptions on the initial data which can be discontinuous, unbounded and large, we show existence of global weak solutions with finite energy. The main difference compared with previous works on the same model, is that transition to each single-phase flow is allowed without any domination conditions of densities.

Reference: H. Wen, On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions. Calc. Var. (2021) 60:158.

Results for a bilinear control problem associated to a repulsive chemotaxis model
María Ángeles Rodríguez-Bellido
University of Sevilla
Tuesday, 19. April 2022 - 9:00 to 10:00
Chemotaxis is understood as the biological process of the movement of living organisms in response to a chemical stimulus which can be given towards a higher (attractive) or lower (repulsive) concentration of a chemical substance. At the same time, the presence of living organisms can produce or consume chemical substance.
In this talk, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a 2D and 3D models. The existence of a global optimal solution with bilinear control is analyzed. First-order optimality conditions for local optimal solutions are derived by using a Lagrange multiplier theorem.

References:

[1] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M. A.
Optimal bilinear control problem related to a chemo-repulsion system in 2D domains.
ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp.

[2] Guillen-Gonzalez, F.; Mallea-Zepeda, E.;... more

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