| Date: | January 23, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Yiran Hu, University of Texas at Austin |
| Title: | Global in time solutions to a family of 3D Quasi-Geostrophic Systems |
| Abstract: | Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified atmospheric flows on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+}\times\Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2D$ torus or a bounded domain in $\rt$. In this talk, I will show the existence and some regularity results of global in time solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The main difficulty is handling the degeneration of the background density profile at the boundary. |
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| Date: | February 27, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Claude Bardos, Laboratoire J.-L.Lions |
| Title: | About large medium and shortime behavior of solutions of the collision of the Vlasov equation |
| Abstract: | TBA |
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| Date: | February 27, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Matthias Hieber, Technische Universität Darmstadt |
| Title: | Analysis of Nematic Liquid Crystal Flows: The Ericksen-Leslie and the Q-Tensor Model |
| Abstract: | In this talk we consider two models describing the flow of nematic liquid crystals: the Ericksen-Leslie model and the Q-tensor model. We discuss local as well as global well-posedness results for strong solutions in the incompressible and compressible setting and investigate as well equlibrium sets and the longtime behaviour of solutions. This is joint work with A. Hussein, J. Pruss and M. Wrona. |
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| Date: | March 19, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Marita Thomas, Freie Universitaet - Berlin |
| Title: | Analysis of a model for visco-elastoplastic two-phase flows in geodynamics |
| Abstract: | A model for an incompressible fluid of both viscoelastic and viscoplastic behavior is revisited, which is used in geodynamics, e.g., to describe the evolution of fault systems in the lithosphere on geological time scales. The Cauchy stress of this fluid is composed of a viscoelastic Stokes-like contribution and of an additional internal stress. The model thus couples the momentum balance with the evolution law of this extra stress, which features the Zaremba-Jaumann time-derivative and a non-smooth viscoplastic dissipation mechanism. This model is augmented to the situation of a bi-phasic material that undergoes phase separation according to a Cahn-Hilliard-type evolution law. Suitable concepts of weak solutions are discussed for the coupled
model. This is joint work with Fan Cheng (FU Berlin) and Robert Lasarzik (WIAS and FU Berlin) within project C09 'Dynamics of rock dehydration on multiple scales' of CRC 1114 'Scaling Cascades in Complex Systems' funded by the German Research Foundation.
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| Date: | March 22, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Daniel Boutros, University of Cambridge |
| Title: | On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture |
| Abstract: | Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider an analogue of Onsager's conjecture for the inviscid primitive equations of oceanic and atmospheric dynamics. The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a 'family' of Onsager conjectures for these equations.
Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. Finally, we present a regularity result for the pressure in the Euler equations, which is of relevance to the Onsager conjecture in the presence of physical boundaries. As an essential part of the proof, we introduce a new weaker notion of pressure boundary condition which we show to be necessary by means of an explicit example. These results are joint works with Claude Bardos, Simon Markfelder and Edriss S. Titi. |
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| Date: | March 26, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Dehua Wang, University of Pittsburgh |
| Title: | Hyperbolic and mixed-type problems in gas dynamics and geometry |
| Abstract: | We shall consider the hyperbolic and mixed-type problems arising in gas dynamics and geometry. In particular, the transonic flows past obstacles and in nozzles as well as the isometric embedding in geometry will be discussed. |
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| Date: | April 9, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Yeyu Zhang, Shanghai University of Finance and Economics |
| Title: | Coupled Nonlinear Evolution and Inverse Energy Transfer in Moist Boussinesq Dynamics |
| Abstract: | The interaction between slow and fast components in geophysical fluid dynamics, especially under the influence of phase changes, poses significant analytical challenges. Our study develops a fast-wave averaging framework for the moist Boussinesq system, expanding past dry dynamics to include phase changes between water vapor and liquid water. We examine whether these phase transitions induce coupling between slow and fast waves or if the slow component evolves independently. Numerical simulations with a range of Froude and Rossby numbers reveal that phase changes may disrupt the proportionality of wave influence on the slow component, evidenced by a nonzero time-averaged wave component due to phase transitions. Furthermore, inverse energy transfer to larger scales is investigated in rotating and stratified flows, including water effects and rapid cloud microphysics. The findings could imply that potential vorticity, phase boundaries, and vertical velocity contribute to the formation of coherent structures in strongly rotated and stratified flows, appearing to indicate a revision to the traditional view of energy cascades in geophysical fluids. |
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| Date: | April 9, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Ricardo Alonso, Texas A&M University - Qatar |
| Title: | An energy method for the Boltzmann equation: Higher integrability and boundedness of solutions |
| Abstract: | We cover in detail an argument for proving higher integrability and uniform boundedness for solutions of the homogeneous Boltzmann equation. Techniques are reminiscent of the level set De Giorgi's method for classical elliptic/parabolic PDE. A rough idea of the method's implementation for the spatially inhomogenous problem is discussed at the end.
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| Date: | April 16, 2024 |
| Time: | 04:00am |
| Location: | BLOC 302 |
| Speaker: | Angeliki Menegaki , Imperial College London |
| Title: | Non-equilibrium Steady States in a BGK Model for Dilute Gases |
| Abstract: | We study the BGK equation on the 1D torus coupled to a spatially inhomogeneous thermostat, which models heat transfer in gases and remains out of equilibrium due to the action of the thermostat. We study properties of stationary solutions, also known as non-equilibrium steady states. We will discuss the existence, uniqueness and linear dynamical stability of spatially inhomogeneous steady states. This is based on a joint work with Jo Evans (University of Warwick). |
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| Date: | April 16, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Aseel Farhat, Florida State University |
| Title: | Impact of Rotation on the Regularity and Behavior of Navier-Stokes Solutions |
| Abstract: | In this presentation, we will address into the regularity challenges posed by the three-dimensional (3D) Navier-Stokes equations (NSE) and explore the influence of planetary rotation. Additionally, we will discuss an upper bound on the Hausdorff dimension of the global attractor associated with the 2D Navier-Stokes equations on the beta-plane, which depends on the rotation rate (referred to as the Rossby number). Our findings align with outcomes observed in numerical experiments, suggesting that rotation tends to induce a more zonal solution. |
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| Date: | April 30, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Slim Ibrahim, Univeristy of Victoria |
| Title: | Stable Singularity Formation for the Inviscid Primitive Equations |
| Abstract: | The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this talk, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of non-smooth profiles, and is stable only by smoother perturbations. This is a joint work with C. Collot and Q. Lin. |
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| Date: | April 30, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Quyuan Lin, Clemson University |
| Title: | On the three-dimensional Nernst-Planck-Boussinesq system |
| Abstract: | Electrodiffusion of ions is a phenomenon that takes place in electrolyte solutions when charged
ions are transported in a fluid under the influence of an electric field. It has various real-world applications in neuroscience, semiconductor theory, water purification, desalination, ion separation, etc.
In this talk, I will introduce the Nernst-Planck-Boussinesq (NPB) system, a new ionic electrodiffusion model that incorporates variational temperature and is forced by buoyancy force stemming from temperature and salinity fluctuations. The electromigration term in the NPB system displays a complex nonlinear structure influenced by the reciprocal of the temperature, which distinguishes its mathematical aspects from other electrodiffusion models studied in the literature, such as the Nernst-Planck-Navier-Stokes and the Nernst-Planck-Euler systems. I will discuss the global existence of weak solutions to the 3D NPB system as well as the long-time dynamics of these weak solutions and their exponential decay in time to steady states. |
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