# Fluid turbulence and Singularities of the Euler/ Navier Stokes equations

The Workshop on Fluid turbulence and Singularities of the Euler/ Navier Stokes equations will take place on March 13-15, 2019. This is the first of two workshop organized by Michael Brenner, Shmuel Rubinstein, and Tom Hou. The second, Machine Learning for Multiscale Model Reduction, will take place on March 27-29, 2019. Both workshops will be held in room

List of registrants

### Schedule:

Wednesday, March 13

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Thursday, March 14

 Time Speaker Title/Abstract 9:00 – 9:45am Sijue Wu, University of Michigan Title: On the motion of water waves with angled crests             Abstract: A common phenomena in the ocean is waves with angled crests. In this talk, I will discuss some recent progress on the understanding of the motion of water waves that allows for angled crested type singularities in the interface. 9:50 – 10:35am Vladimir Sverak, University of Minnesota Title: Singularities and their stability in various models.             Abstract:  We will discuss models (mostly 1d with non-zero viscosity) with both stable and unstable singularities, together with effects of the singularities on uniqueness. Issues concerning the full Navier-Stokes equations will also be mentioned. 10:40 – 11:10am Coffee Break 11:10 – 11:55am Alexander Kiselev, Duke University Title: Small scale creation in solutions to equations of fluid dynamics             Abstract: Small scale creation in fluid motion is ubiquitous. There remains much to understand about mathematical mechanisms driving it. In particular, the question of global regularity vs finite time singularity formation stands open for 3D Euler and Navier-Stokes equations, and for the SQG equation with smooth initial data. In this talk, I plan to review some recent results on simplified models that have been designed to understand this process, in particular inspired by the Hou-Luo scenario for singularity formation in solutions to 3D Euler equation 12:00 – 2:00pm Lunch Break 2:00 – 2:45pm Edriss Titi, TAMU Title: Remarks on the Navier-Stokes and Euler Equations             Abstract: In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness and the effect of physical boundaries on the potential formation of singularity. In addition, I will present a blow-up criterion based on a class of inviscid regularization for these equations. 2:50 – 3:35pm Vlad Vicol, Courant Institute Title: Convex integration for the Navier-Stokes equations             Abstract: We consider applications of the method of convex integration, to the Navier-Stokes and related equations. Topics discussed include the nonuniqueness of finite energy distributional solutions for the system. 3:40 – 4:10pm Coffee Break 4:10 – 4:55pm Andrej Zlatos, UCSD Title: Euler Equations in Domains with Corners             Abstract: When the velocity is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the 2D Euler equations has been established only in some special cases, and in all these the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular.  Assuming that the latter holds initially, the challenge is to propagate this property along the Euler dynamic via an appropriate control of the Lagrangian trajectories. In domains with corners whose angles are all less than $\pi$, and sufficiently smooth elsewhere, the velocity fails to be almost Lipschitz only near any obtuse corners.  We show for these domains that if the vorticity is initially constant near the whole boundary, then it remains such forever (and global weak solutions are unique). We also show that this may fail for domains with corners that have angles greater than $\pi$.

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Friday, March 15

 Time Speaker Title/Abstract 9:00 – 9:45am Tarek Elgindi, UCSD Title: Singularity formation for solutions to the 3d Euler equation             Abstract: We will briefly review recent results on singularity formation for strong solutions to the incompressible Euler and Boussinesq systems obtained with In-Jee Jeong. Then we discuss a more recent construction of finite time singularity for more regular solutions. 9:50 – 10:35am Jiajie Chen, Caltech Title: On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equation             Abstract: De Gregorio introduced the model ωt + uωx = uxω, a variant of the Constantin-Lax-Majda (CLM) model ωt = ωHω, to study the convection and vortex stretch effect of the 3D Euler equation. In this talk, we will show that the De Gregorio model on the real line develops a finite time self-similar singularity from smooth compactly supported initial data using a dynamical rescaling approach. We also study the modified De Gregorio’s model ωt + auωx = uxω for a > 0 and show that the C α self-similar solutions of the CLM model is stable in some suitable functional space. As a result, we can construct compactly supported C α initial data that have finite energy, i.e. ||u||2 < +∞, and develops a self-similar singularity in finite time. It improves the result recently obtained by Elgindi and Jeong. Due to the stability result and that the singularity develops near the origin, we can prove that such initial data also lead to finite time singularity of modified De Gregorio model on the circle. Finally, we will talk about the viscous De Gregorio model and present the blowup result for a close to ½ and a criterion for global well posedness.This is joint work with Thomas Hou and De Huang. 10:40 – 11:10am Coffee Break 11:10 – 11:55am Tom Hou, Caltech Title: Computer-Assisted Analysis of 3D Euler Singularity             Abstract: Whether the 3D incompressible Euler equation can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. In a recent joint work with Dr. Guo Luo, we provided convincing numerical evidence that the 3D Euler equation develops finite time singularities. Inspired by this finding, we have recently developed an integrated analysis and computation strategy to analyze the finite time singularity of a regularized 3D Euler equation. We first transform the regularized 3D Euler equation into an equivalent dynamic rescaling formulation. We then study the stability of an approximate self-similar solution. By designing an appropriate functional space and decomposing the solution into a low frequency part and a high frequency part, we prove nonlinear stability of the dynamic rescaling equation around the approximate self-similar solution, which implies the existence of the finite time blow-up of the regularized 3D Euler equation. This is a joint work with Jiajie Chen, De Huang, and Dr. Pengfei Liu.

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