WRWDFMII Titles and Abstracts for Main Speakers

Xukai Yan

Anisotropic Caffarelli-Kohn-Nirenberg type inequalities
Weighted norm Sobolev-type inequalities, such as Hardy’s inequalities and their extensions, have also been broadly studied and applied in PDE analysis. Caf-farelli, Kohn and Nirenberg considered such types of interpolation inequalities with isotropic weights given by a power of |x|. These inequalities have many important applications and have been extensively studied from various aspects, including their best constants, extreme functions, and extensions with higher or-der derivative. In this talk, I will discuss some extensions of these inequalities with anisotropic weights. I will first make an overview about the development and related problems on Gagliardo-Nirenberg-Sobolev inequalities and Caffarelli-Kohn-Nirenberg inequalities. I will talk about some recent study on extensions of these inequalities with different types of weights and related problems. I will also present a more general anisotropic version of CKN inequalities, and discuss about an application of them in the study of Navier-Stokes equations.

Aynur Bulut

Title TBA
Abstract TBA

Wojciech Ozanski

Ill-posedness of equations of inviscid fluid mechanics
In the talk we will explore some phenomena of turbulence of fluids, and we will discuss how some recent developments in the mathematical analysis of the 2D and 3D Euler equations, and the Navier-Stokes equations, help understand these phenomena. In particular, we will see that some new results on ill-posedness of the Euler equations provide new analytical methods of studying unique local-in-time solutions for some initial conditions with low regularity. We will discuss how these results demonstrate new mechanisms of growth of solutions of incom­pressible fluids equations and new ways of controlling the errors in the regime well below the Yudovich class.

Jiahong Wu

Wave structures in fluid stability problems
This talk presents examples of a remarkable stabilizing phenomenon in fluid sta­bility problems. The 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor, as in the Oldroyd-B model, small smooth data always lead to global and stable solutions. The 3D Navier-Stokes equation with dissipation in only one direction is not known to always have small global solutions. However, when it is coupled with the magnetic field in the magne­tohydrodynamic system, solutions near a background magnetic field are always global in time. The magnetic field stabilizes the fluid. Mathematically this stabi­lizing phenomenon boils down to wave structures hidden in the systems governing perturbations around physically relevant steady states.

Yannan Shen

Stability of dispersive shock in KdV-Burgers Equation
We study the viscous-dispersive shock profile with infinite oscillations of the Kor­teweg de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock wave, including the rate at which the local extrema converge towards the far field. Then, by exploiting the structural properties of the shock, we show the L2 contraction property of the shock profile under arbitrarily large pertur­bations, up to a time-dependent shift. This result implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coef­ficients. This uniformity yields zero viscosity-dispersion limits.

Ming Chen

Extreme internal waves
Large-amplitude internal waves in stratified fluids give rise to striking nonlinear phenomena, including overturning fronts and gravity currents. In this setting, ”extreme” refers to traveling waves that develop a stagnation point along the interface, allowing the free boundary to lose regularity and form singular features such as vertical tangents. While numerical studies dating back over forty years strongly suggest that such behavior occurs for solutions of the two-layer free-boundary Euler equations, a rigorous mathematical proof has remained open. In this talk, I will describe the construction of a global family of hydrodynamic bores, which are front-type traveling waves that connect distinct asymptotic states, bifurcating from the trivial flat interface. Along the elevation branch, the waves must overturn, and the interface necessarily develops a vertical tangent. This yields the first rigorous proof of overturning obtained through a global bifurcation framework in the fully nonlinear regime where gravity is order one. Along the depression branch, the limiting configuration instead produces a gravity current, a physically fundamental flow in which a denser fluid intrudes beneath a lighter one. I will introduce this phenomenon and explain how the analysis connects to a classical conjecture of von Karman describing the structure of gravity currents near a rigid boundary. This is joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).

Jiajie Chen

Finite time singularities in the Landau equation with very hard potentials
The Landau equation, introduced by Lev Landau in 1936, is one of the central equations in kinetic theory. We consider the Landau equation with very hard potentials γ ∈ (√3,2] which is known to admit global smooth solutions for homogeneous data. Inspired by hydrodynamic limits from kinetic equations to fluid equations, we construct smooth, strictly positive initial data that develop a finite-time singularity by lifting imploding singularities from the compressible Euler equations. In self-similar variables, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data. This is joint work with Jacob Bedrossian (UCLA), Maria Gualdani (UT Austin), Sehyun Ji (UChicago), Vlad Vicol (NYU), and Jincheng Yang (JHU).

Igor Kukavica

Exact boundary controllability for the ideal magneto-hydrodynamic equations
We consider the three-dimensional ideal MHD system on a domain with a con­trollable part of the boundary where we prescribe the boundary data. The basic question of boundary controllability is whether, given two states, one can, by means of boundary control, drive one state to the other. We will review the existing literature on this problem and provide a positive result for domains with only Sobolev regularity. The results are based on works with Matthew Novack, Wojciech Ozanski, and Vlad Vicol.

Leo Rebholz

Enabling fast convergence of nonsolvers for Navier-Stokes using continuous data assimilation
We consider nonlinear solvers for the incompressible, steady Navier-Stokes equa­tions (NSE) in the setting where partial solution data is available, e.g. from physical measurements or sampled solution data from a (too big to send) very high-resolution computation. The measurement data is incorporated/assimilated into the solver through a nudging term addition that penalizes at each iteration the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented at each time step for time dependent dissipative PDEs. For a Picard solver for NSE, we quantify the acceleration provided by the data in terms of the density of the measurement locations and the level of noise in the data and prove the CDA scales the nonlinear solver’s Lipschitz constant by H^1/2 where H is the charac­teristic point spacing of true solution data; this proves that CDA accelerates (or enables) convergence. We also discuss how the technique can be used in more efficient splitting methods that approximate the Picard iteration, and how to use the method to quickly recover an unknown viscosity.