09h25 - 09h30 : Welcome
09h30 - 10h15 : Marco Bravin (Basque Center for Applied Mathematics, Spain) - The vortex filament equation and the well-posedness of the cubic NLS for irregular data - Video
The vortex filament equation (VFE) is a geometric equation that describes the evolution of a curve of vorticity in a three dimensional incompressible inviscid fluid. Through the Hasimoto transformation, the VFE is associated with the cubic NLS equation. In this talk I will focus my attention on vortex filaments that are initially polygonal lines, which correspond to studying the cubic NLS with sum of delta of Dirac as initial data. In particular I will show local and large in time well-posedness for sufficiently small initial data in appropriated spaces. This is a joint work with Luis Vega.
10h15 - 11h00 : Eduardo García-Juárez (Universitat de Barcelona, Spain) - The Muskat and Peskin Problems with Viscosity Contrast - Video
The Muskat problem studies the dynamics of the interface between fluids in a porous medium governed by Darcy’s law. The Peskin problem models the movement of a closed elastic filament immersed in an incompressible fluid. While the former is at the core of petrochemical engineering processes, the latter is a prototypical test problem for biophysical fluid-structure modeling. On the mathematical side, both systems are nonlinear and nonlocal PDEs, of parabolic type, and share the same scaling. We will show how the use of some spaces based on the Wiener algebra turns out to be very convenient to analyze this kind of problems, yielding instant analytic smoothing, global existence, and convergence to the steady states. The techniques allow to consider non too-small initial data, critical regularity in terms of the natural scaling and different viscosities for each fluid.
11h00 - 11h30 : Virtual coffee break
11h30 - 12h15 : Gabriela López Ruiz (Sorbonne Université, France) - Effects of rough coasts on the wind-driven oceanic motion - Video
Surface roughness has been identified as an essential parameter in fluid flow since the nineteenth century, but its effects on fluid dynamics are not fully understood. This talk regards the impact of coastal rough topography on oceanic circulation at the mesoscale. We study a singular perturbation problem from meteorology known as the single-layered quasi-geostrophic model. Assuming the rough coasts do not present a particular structure, the governing boundary layer equations are defined in infinite domains with not-decaying boundary data. Additionally, the eastern boundary layer exhibits convergence issues far from the boundary. In this regime, we establish the well-posedness of the boundary layer profiles in Kato spaces by adding ergodicity properties and using pseudo-differential analysis. We construct an approximate solution to the original problem and show convergence results.
Lunch time
14h30 - 15h15 : Roberta Bianchini (Consiglio Nazionale delle Ricerche, Italy) - Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations - Video
In this talk, we discuss the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for a small initial perturbation of size $\epsilon$ in a suitable Gevrey class. Under the classical Miles-Howard stability criterion on the Richardson number, we show that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. This is a joint work with Jacob Bedrossian, Michele Coti Zelati and Michele Dolce.
15h15 - 16h00 : Chistopher Maulén (Universidad de Chile, Chile) - Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation - Video
In this talk, I shall consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$.I will present in more detail the long-time behavior of zero-speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel, and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.
16h00 - 16h30 : Virtual coffee break
16h30 - 17h15 : Neel Patel (University of Michigan, USA) - Blow-Up for SQG Patches - Video
The two-dimensional surface quasi-geostrophic (SQG) equation is a model for atmospheric or oceanic flows and has strong structural similarity with the 3D Euler equation. Patch solutions represent sharp temperature fronts for the 2D SQG equation, similar to vortex patches for 2D Euler. Interpolating between the 2D Euler equation and the 2D SQG equation, one obtains the one-parameter 0≤ alpha ≤1 family of generalized SQG equations. We will discuss a class of patch solutions that become singular in finite time for a subfamily of these equations in the half-space setting as well as blow-up criteria and well-posedness for patches in the full-space.
17h15 - 18h00 : Annalaura Stingo (University of California Davis, USA) - Almost-global well-posedness for 2d strongly-coupled wave-Klein-Gordon systems - Video
In this talk we discuss the almost-global well-posedness of a wide class of coupled Wave-Klein-Gordon equations in 2+1 space-time dimensions, when initial data are small and localized. The Wave-Klein-Gordon systems arise from several physical models especially related to General Relativity but few results are known at present in lower space-time dimensions. Compared with prior related results, we here consider strong quadratic quasilinear couplings between the wave and the Klein-Gordon equation and no restriction is made on the support of the initial data which are supposed to only have a mild decay at infinity and very limited regularity. Our proof relies on a combination of energy estimates localized to dyadic space-time regions and pointwise interpolation type estimates within the same regions. This is a joint work with M. Ifrim.
09h30 - 10h15 : Pei Su (Université de Bordeaux, France) - Boundary control problem of the water waves system in a tank - Video
Here we are interested in the boundary control problem of the small-amplitude water waves system in a rectangular tank. The model actually we used here is a fully linear and fully dispersive approximation of Zakharov-Craig-Sulem formulation constrained in a rectangle, in particular, with a wave maker. The wave maker acts on one lateral boundary, by imposing the acceleration of the fluid in the horizontal direction, as a scalar input signal. Firstly, we introduce the Dirichlet to Neumann and Neumann to Neumann maps, asscociated to the certain edges of the domain, so that the system reduces to a well-posed linear control system. Then we consider the stabilizability issue on the gravity and gravity-capillary waves. It turns out that, in both cases, there exists a feedback functional, such that the corresponding control system is strongly stable. Finally, we consider the asymptotic behaviour of the above system in shallow water regime, i.e. the horizontal scale of the domain is much larger than the typical water depth. We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a detailed analysis of the Fourier series and the dimensionless version of the evolution operators mentioned above, as well as a scattering semigroup and the Trotter-Kato approximation theorem. This is a joint work with M. Tucsnak (Bordeaux) and G. Weiss (Tel Aviv).
10h15 - 11h00 : Kai Koike (Kyoto University, Japan) - Refined pointwise estimates for 1D viscous compressible flows with application to long-time behavior of a point mass - Video
Think of a point mass moving through a 1D viscous compressible fluid. It's not difficult to imagine that its velocity $V(t)$ would somehow decay to zero as time $t$ goes to infinity. In fact, numerical experiments suggest that it actually decays as $V(t)\sim t^{-3/2}$. In one of my previous works (https://www.sciencedirect.com/science/article/abs/pii/S0022039620304666), I showed an upper bound of the type $V(t)=O(t^{-3/2})$. However, it remained to be answered whether this decay estimate is optimal or not, that is, whether we can prove a corresponding lower bound of the form $C^{-1}t^{-3/2}\leq |V(t)|$. Concerning this problem, I recently obtained a simple necessary and sufficient condition (on the initial data) for the bound $C^{-1}t^{-3/2}\leq |V(t)|$ to hold (https://arxiv.org/abs/2010.06578), hence answering the question of optimality. This result is a corollary to refined pointwise decay estimates of solutions obtained through a very detailed analysis using Green's functions techniques. I shall explain these more in detail in the talk.
11h00 - 11h30 : Virtual coffee break
11h30 - 12h15 : Stefano Scrobogna (Universidad de Sevilla, Spain) - On the effect of viscosity in surface gravity waves - Video
The motion of water waves is a classical research topic that has attracted a lot of attention from many different researchers in Mathematics, Physics and Engineering and it is classically modeled by the free-boudary irrotational Euler equations. Usually, these assumptions are enough to describe the main part of the dynamics of real water waves, however, discrepancies between experimental experiences and computer simulations show that sometimes viscosity needs to be taken into account. In this setting the Euler equations should be replaced by the Navier-Stokes equations and the irrotationality hypothesis has to be dropped. It is known however, since the works of Boussinesq (1895) and Lamb (1932) that the vorticity plays a role only close to the free boundary, thus, it would be desirable to add dissipative effects to the water waves equations without going all the way to the Navier-Stokes equations and the subsequent removal of the irrotationality assumption. This problem has been addressed by a number of people starting with Boussinesq and Lamb, in this talk we will investigate a a model proposed by Dias, Dyachenko & Zakharov (Physics Letters A 2008). Joint work with R. Granero-Belinchón.