As a toy model for the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equation with singular initial data corresponding to a pair of point vortices with opposite circulations. In the large Reynolds number regime, we construct an approximate solution which takes into account the deformation of the stream lines due to vortex interactions, as well as the corrections to the translation speed of the dipole due to finite size effects. Using energy estimates based on Arnold's variational characterization of equilibria for the Euler equation, we then show that our approximation remains valid over a very long time interval, if the viscosity is sufficiently small. This is a joint work with Michele Dolce (Lausanne), which relies on previous studies in collaboration with Vladimir Sverak (Minneapolis).[-]

The Ocean hosts a combination of slowly evolving balanced mean flows and rapidly evolving inertia-gravity waves (IGWs). While reduced models describing the slow balanced flow are standard textbook material [1], the description of the wave field is an active area of current research. We will first consider near-inertial waves (NIWs) induced by wind blowing over the Ocean, with the goal of determining their spatial reorganization by a background mean flow [2]. After deriving the standard asymptotic model governing NIW mean-flow interaction [3], we will highlight an analogy with the (quantum) dynamics of charged particles in a background electromagnetic field. The analogy offers a shortcut for predicting the spatial organization of the wave field using elementary methods from quantum physics and statistical mechanics. Time-permitting, we will then move on to fully 3D IGWs interacting with a slow background, focusing on the resulting cascade of wave action to small scales [4]. This is an elementary example of a turbulent cascade in a linear wave system, paving the way for subsequent lectures by G. Krstulovic on turbulent cascades in nonlinear wave systems.[-]

The Ocean hosts a combination of slowly evolving balanced mean flows and rapidly evolving inertia-gravity waves (IGWs). While reduced models describing the slow balanced flow are standard textbook material [1], the description of the wave field is an active area of current research. We will first consider near-inertial waves (NIWs) induced by wind blowing over the Ocean, with the goal of determining their spatial reorganization by a background mean flow [2]. After deriving the standard asymptotic model governing NIW mean-flow interaction [3], we will highlight an analogy with the (quantum) dynamics of charged particles in a background electromagnetic field. The analogy offers a shortcut for predicting the spatial organization of the wave field using elementary methods from quantum physics and statistical mechanics. Time-permitting, we will then move on to fully 3D IGWs interacting with a slow background, focusing on the resulting cascade of wave action to small scales [4]. This is an elementary example of a turbulent cascade in a linear wave system, paving the way for subsequent lectures by G. Krstulovic on turbulent cascades in nonlinear wave systems.[-]

Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture is to present an overview of some of the mathematical and numerical difficulties that may be encountered when dealing with fluid–structure interaction problems such as the geometrical nonlinearities or the added mass effect and how one can deal with these difficulties.[-]

Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture is to present an overview of some of the mathematical and numerical difficulties that may be encountered when dealing with fluid–structure interaction problems such as the geometrical nonlinearities or the added mass effect and how one can deal with these difficulties.[-]

Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture is to present an overview of some of the mathematical and numerical difficulties that may be encountered when dealing with fluid–structure interaction problems such as the geometrical nonlinearities or the added mass effect and how one can deal with these difficulties.[-]

Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture is to present an overview of some of the mathematical and numerical difficulties that may be encountered when dealing with fluid–structure interaction problems such as the geometrical nonlinearities or the added mass effect and how one can deal with these difficulties.[-]

Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we develop the $L^q$-$L^r$ decay estimates of the evolution operator $T(t,s)$ as $(t-s)\to\infty$ and then apply them to the Navier-Stokes initial value problem.[-]

The inhomogeneous incompressible Navier-Stokes equations that govern the evolution of viscous incompressible flows with non-constant density have received a lot of attention lately. In this talk, we shall mainly focus on the singular situation where the density is discontinuous, which is in particular relevant for describing the evolution of a mixture of two incompressible and non reacting fluids with constant density, or of a drop of liquid in vacuum. We shall highlight the places where tools in harmonic analysis play a key role, and present a few open problems.[-]