Speaker: Yan Guo, Brown University
Title: Stability of Contact Lines
Abstract: Contact lines (e.g. coffee meets with the coffee cup, boundary of a droplet on table) occurs naturally when a free fluid surface, a capillary surface, meets with a fixed solid boundary. Despite its importance in fluid theory and applications, the dynamical law for contact line has been only studied in relative recent history. We review recent work to establish well-posedness of classical fluid models for describing contact line dynamics.
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Speaker: Andrej Zlatos, UCSD
Title: Stable regime singularity for the Muskat problem
Abstract: The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock. We show that unlike on the whole plane, finite time singularities do arise in the stable regime (lighter fluid above the heavier one) in this setting, including from arbitrarily small smooth initial data. We achieve this by developing a local well-posedness theory for this model as well as obtaining maximum principles for the height, slope, and potential energy of the fluid interface. The former allows the interface to touch the bottom, which applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer, and includes considerably more general fluid interface geometries than even previous whole plane results.
Speaker: Gregory Seregin, University of Oxford
Title: Regularity of axisymmetric suitable weak solutions to the Navier-Stokes equations
Abstract: In the talk, old and new results on regularity of axisymmetric suitable weak solutions to the classical Navier-Stokes equations describing the flow of viscous incompressible fluids will be presented. The absence of the so-called Type I blowups will be shown. Some slightly supercritical sufficient conditions of regularity will be mentioned as well.
Speaker: Alexis Vasseur, University of Texas at Austin
Title: Viscous perturbations to discontinuous solutions of the compressible Euler equation
Abstract: The compressible Euler equation can lead to the emergence of shocks-discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities.The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will present the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent a-contraction with shifts method. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation.
Speaker: Hao Jia, University of Minnesota
Title: Dynamics of linearized 2d Navier Stokes equations in high Reynolds number regime
Abstract: Over the last decade there has been tremendous progresses in the study of fine dynamics of the linearized two dimensional Euler equations around shear flows or vortices, leading to a more or less complete understanding of important physical phenomena such as inviscid damping and vorticity depletion. For the corresponding Navier Stokes equations with a small viscosity, similar results are much harder to prove. On the technical level, this is a singular perturbation problem, and instead of Rayleigh equation (2nd order) one now needs to study, very precisely, the Orr-Sommerfeld equations (4th order) when the spectral parameter is close to the cluster of eigenvalues. In this talk, we will review some recent advances in this direction and outline a unified approach to study uniform-in-viscosity inviscid damping, vorticity depletion and enhanced dissipation. Joint work with Rajendra Beekie (Duke) and Shan Chen (UMN).
Speaker: Susanna Haziot, Brown University
Title: Desingularization and long-term dynamics of solutions to fluid models
Abstract: The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. The Peskin problem describes the flow of a Stokes fluid through the heart valves. In this talk, we investigate the small data critical regularity theory for these two models, and in particular, the desingularization of interfaces with small corners. The first part is joint work with E. Garcia-Juarez, J. Gomez-Serrano and B. Pausader. The second part is joint work with E. Garcia-Juarez.
Speaker: Vlad Vicol, Courant Instiuttute, NYU
Title: The geometry of shock formation and maximal development for multi-D compressible Euler
Abstract: I will describe a new geometric framework for the multidimensional Euler equations, based on Arbitrary-Lagrangian-Eulerian (ALE) coordinates associated to ``fast'' characteristic surfaces. This framework is used to construct the maximal development of smooth Cauchy data for shock formation. The talk is based on a joint work with Steve Shkoller.