Semester Theme: PDE of Fluid Dynamics

Working Seminar

The working seminar meets on Wednesdays 1:45pm-3:15pm at Physics 128. Here is the schedule of the talks. 

• 9/1 Organizational meeting
• 9/22  Siming He: Hou-Luo scenario for singularity formation in the 3D Euler equation, based on: G. Luo and T. Hou, Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation, Multiscale Model. Simul. 12 (2014) 1722–1776 
•   9/29 Kevin Hu and Logan Stokols: Criticality of the axi-symmetric Navier-Stokes equation, based on: Z. Lei and Q. Zhang, Criticality of the axially symmetric Navier-Stokes equations. Pacific J. Math. 289 (2017), no. 1, 169–187 and D. Wei, Regularity criterion to the axially symmetric Navier-Stokes equations. J. Math. Anal. Appl. 435 (2016), no. 1, 402–413
• 10/6 Ayman Rimah Said: Small scale creation in solutions of the 2D Euler equation, based on A. Kiselev and V. Sverak, Small scale creation for solutions of the incompressible two dimensional Euler equation, Annals of Math. 180 (2014), 1205–1220
• 10/13 Kevin Dembski and Omar Melikechi: Global regularity in the De Gregorio model,  based on J. Chen, On the regularity of the De Gregorio model for the 3D Euler equations, preprint arXiv:2107.07772
• 10/20 Karim Rida Moh Shikh Khalil and Yupei Huang: Stability of solutions to Euler equation, based on Z. Lin, Some stability and instability criteria for ideal plane flows. Comm. Math. Phys. 246 (2004), no. 1, 87–112 and Z. Lin, Instability of some ideal plane flows. SIAM J. Math. Anal. 35 (2003), no. 2, 318–356 
• 10/27 No meeting
• 11/3 Federico Pasqualotto: Nonlinear instability in Euler equation, based on E. Grenier, On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091
• 11/10 Federico Pasqualotto: Nonlinear instability in Euler equation, based on E. Grenier, On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091, continued
• 11/17 This week we have one of the Igor Kukavica’s minicourse lectures (see below). 
• 11/24 No meeting 
• 12/1  Federico Pasqualotto and Ayman Rimah Said: Finite time singularity for 3D Euler equation, based on T. M. Elgindi, Finite-Time Singularity Formation for C^{1,\alpha} Solutions to the Incompressible Euler Equations on R^3, preprint, arXiv:1904.04795
• 12/8 Federico Pasqualotto and Ayman Rimah Said: Finite time singularity for 3D Euler equation, based on T. M. Elgindi, Finite-Time Singularity Formation for C^{1,\alpha} Solutions to the Incompressible Euler Equations on R^3, preprint, continued

Minicourses

This semester, we also have two minicourses.

Ryan Murray (NC-State). 

11/1 at 2pm, 11/2 at 4:30pm and 11/4 at 4:30pm, all in Physics 119.

COURSE TITLE:  Self-similar solutions for Euler’s equations: in search of non-uniqueness 
Abstract: This series of talks will consider the incompressible Euler equation in two dimensions. These equations are fundamental for understanding basic phenomena in fluid dynamics, but many fundamental aspects of these equations are still not fully understood. In particular, the question of well-posedness and uniqueness within physically relevant classes of solutions is not completely resolved. These talks will comprise of three lectures: the first will discuss the broader picture of non-uniqueness for incompressible Euler, with a special focus on long standing, physically-motivated examples which only possess point singularities, but which lack rigorous analysis. The second and third lectures will focus on an ongoing project, in collaboration with Alberto Bressan and Wen Shen, seeking to construct two distinct self-similar solutions to Euler’s equations with the same initial data, which only exhibit a finite number of point singularities of the vorticity. The resolution of the equations near these singularities depends upon previous analysis by Elling, and these talks will describe some of the most important points of the construction.

Igor Kukavica (University of Southern California).

11/15 at 4:30pm, 11/16 at 3:15pm (both at Physics 119), and 11/17 at 1:45pm in Physics 128.

COURSE TITLE: On the inviscid limit problem for the Navier-Stokes equations

Abstract: Whether the solution of the Navier-Stokes equation converges to the solution of Euler equation as the vorticity vanishes is one of the fundamental problems in fluid dynamics. In the three lectures, we will provide a gentle introduction to this problem and also discuss recent results. First. we will prove that the convergence holds in strong Sobolev norms when domain is the whole space. Then we concentrate on the domains with a boundary. First, we will prove Kato’s necessary and sufficient condition for the inviscid limit to hold by evaluating the dissipation of the solution inside boundary layer. Then we will discuss the role of analyticity and Prandtl expansions when solving the inviscid limit in a half-plane. The aim is to show the main ideas in the proof of a recent result with V. Vicol and F. Wang on the inviscid problem with an initial datum which is analytic only close to the boundary and has finite Sobolev regularity in the interior.