# Past Events

## Singularity formation in incompressible fluids and related models

**Wednesday, July 27, 2022, 11 a.m.** through **Wednesday, July 27, 2022, Noon**

Vincent 570

Jiajie Chen (California Institute of Technology)

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In these lectures, we will describe some recent progress on singularity formation in incompressible fluids and related models. We will begin with some properties of the 3D Euler equations useful for studying singularity formation and the dynamic rescaling formulation of the 3D Euler equations. Then we will discuss some ideas to overcome some difficulties in singularity formation and study finite time blowup based on the stability of an approximate blowup profile. We will compare this stability and another notion of stability of blowup. Lastly, we will discuss some ideas for constructing finite time blowup from smooth initial data, particularly in 1D models of the Euler equations, which can be helpful in studying the singularity formation of 3D Euler with smooth data.

## Instability and non-uniqueness in the Navier-Stokes equations

**Wednesday, July 27, 2022, 9:30 a.m.** through **Wednesday, July 27, 2022, 10:30 a.m.**

Vincent 570

Dallas Albritton (Princeton University)

It is not yet known whether Navier-Stokes solutions develop singularities in finite time. If they do, then the solutions can be continued beyond the singularities as Leray-Hopf solutions. It is therefore a fundamental question, Are Leray-Hopf solutions unique? The goal of this course is to present very recent developments in our understanding of this question.

We will begin by quickly reviewing the Navier-Stokes basics: dimensional analysis, the energy balance, pressure, weak solutions, perturbation theory, and weak-strong uniqueness. To save time, we will not present the complete proofs.

Next, we will explain aspects of the Jia, Sverak, and Guillod program (Jia-Sverak, Inventiones 2014, JFA 2015; Guillod-Sverak, arXiv 2017) and, in particular, how instability in self-similarity variables can generate non-uniqueness. We will briefly discuss bifurcations, stable and unstable manifolds, and, time permitting, a short proof of the existence of large self-similar solutions.

Finally, we will present the recent work (A.-Brue-Colombo, Ann. Math. 2022) which rigorously established non-uniqueness of Leray-Hopf solutions with forcing. We will present the main idea but focus on the spectral perturbation arguments.

No prior knowledge of the Navier-Stokes equations is required, though it might be beneficial to preview background on weak and mild solutions in, for example, Chapters 4 and 11 in Robinson-Rodrigo-Sadowski or Chapters 3 and 5 in Tsai.

## Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations

**Tuesday, July 26, 2022, 3 p.m.** through **Tuesday, July 26, 2022, 4 p.m.**

Vincent 570

Hao Jia (University of Minnesota, Twin Cities)

In this sequence of lectures, we will introduce the classical stability problem for incompressible Euler and Navier Stokes equations. We will focus on the specific setting of the perturbative regime near a spectrally stable monotonic shear flow, and explain various dynamical phenomena, such as inviscid damping for the Euler equation and enhanced dissipation for the Navier Stokes equations in the high Reynolds number regime. Recent advances and further open problems will also be discussed if time permits.

## 1d scattering theory and its application to nonlinear dispersive equations

**Tuesday, July 26, 2022, 1:30 p.m.** through **Tuesday, July 26, 2022, 2:30 p.m.**

Vincent 570

Gong Chen (Georgia Institute of Technology)

In these lectures, I will introduce the spectral theory and the scattering theory associated with the 1d Schr\”odinger operator. Then I will illustrate how these tools can be used to understand and compute the large-time behaviors of nonlinear dispersive equations.

## Singularity formation in incompressible fluids and related models

**Tuesday, July 26, 2022, 11 a.m.** through **Tuesday, July 26, 2022, Noon**

Vincent 570

Jiajie Chen (California Institute of Technology)

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In these lectures, we will describe some recent progress on singularity formation in incompressible fluids and related models. We will begin with some properties of the 3D Euler equations useful for studying singularity formation and the dynamic rescaling formulation of the 3D Euler equations. Then we will discuss some ideas to overcome some difficulties in singularity formation and study finite time blowup based on the stability of an approximate blowup profile. We will compare this stability and another notion of stability of blowup. Lastly, we will discuss some ideas for constructing finite time blowup from smooth initial data, particularly in 1D models of the Euler equations, which can be helpful in studying the singularity formation of 3D Euler with smooth data.

## Instability and non-uniqueness in the Navier-Stokes equations

**Tuesday, July 26, 2022, 9:30 a.m.** through **Tuesday, July 26, 2022, 10:30 a.m.**

Vincent 570

Dallas Albritton (Princeton University)

It is not yet known whether Navier-Stokes solutions develop singularities in finite time. If they do, then the solutions can be continued beyond the singularities as Leray-Hopf solutions. It is therefore a fundamental question, Are Leray-Hopf solutions unique? The goal of this course is to present very recent developments in our understanding of this question.

We will begin by quickly reviewing the Navier-Stokes basics: dimensional analysis, the energy balance, pressure, weak solutions, perturbation theory, and weak-strong uniqueness. To save time, we will not present the complete proofs.

Next, we will explain aspects of the Jia, Sverak, and Guillod program (Jia-Sverak, Inventiones 2014, JFA 2015; Guillod-Sverak, arXiv 2017) and, in particular, how instability in self-similarity variables can generate non-uniqueness. We will briefly discuss bifurcations, stable and unstable manifolds, and, time permitting, a short proof of the existence of large self-similar solutions.

Finally, we will present the recent work (A.-Brue-Colombo, Ann. Math. 2022) which rigorously established non-uniqueness of Leray-Hopf solutions with forcing. We will present the main idea but focus on the spectral perturbation arguments.

No prior knowledge of the Navier-Stokes equations is required, though it might be beneficial to preview background on weak and mild solutions in, for example, Chapters 4 and 11 in Robinson-Rodrigo-Sadowski or Chapters 3 and 5 in Tsai.

## Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations

**Monday, July 25, 2022, 3 p.m.** through **Monday, July 25, 2022, 4 p.m.**

Vincent 570

Hao Jia (University of Minnesota, Twin Cities)

In this sequence of lectures, we will introduce the classical stability problem for incompressible Euler and Navier Stokes equations. We will focus on the specific setting of the perturbative regime near a spectrally stable monotonic shear flow, and explain various dynamical phenomena, such as inviscid damping for the Euler equation and enhanced dissipation for the Navier Stokes equations in the high Reynolds number regime. Recent advances and further open problems will also be discussed if time permits.

## 1d scattering theory and its application to nonlinear dispersive equations

**Monday, July 25, 2022, 1:30 p.m.** through **Monday, July 25, 2022, 2:30 p.m.**

Vincent 570

Gong Chen (Georgia Institute of Technology)

In these lectures, I will introduce the spectral theory and the scattering theory associated with the 1d Schr\”odinger operator. Then I will illustrate how these tools can be used to understand and compute the large-time behaviors of nonlinear dispersive equations.

## Singularity formation in incompressible fluids and related models

**Monday, July 25, 2022, 11 a.m.** through **Monday, July 25, 2022, Noon**

Vincent 570

Jiajie Chen (California Institute of Technology)

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In these lectures, we will describe some recent progress on singularity formation in incompressible fluids and related models. We will begin with some properties of the 3D Euler equations useful for studying singularity formation and the dynamic rescaling formulation of the 3D Euler equations. Then we will discuss some ideas to overcome some difficulties in singularity formation and study finite time blowup based on the stability of an approximate blowup profile. We will compare this stability and another notion of stability of blowup. Lastly, we will discuss some ideas for constructing finite time blowup from smooth initial data, particularly in 1D models of the Euler equations, which can be helpful in studying the singularity formation of 3D Euler with smooth data.

## Instability and non-uniqueness in the Navier-Stokes equations

**Monday, July 25, 2022, 9:30 a.m.** through **Monday, July 25, 2022, 10:30 a.m.**

Vincent 570

Dallas Albritton (Princeton University)

It is not yet known whether Navier-Stokes solutions develop singularities in finite time. If they do, then the solutions can be continued beyond the singularities as Leray-Hopf solutions. It is therefore a fundamental question, Are Leray-Hopf solutions unique? The goal of this course is to present very recent developments in our understanding of this question.

We will begin by quickly reviewing the Navier-Stokes basics: dimensional analysis, the energy balance, pressure, weak solutions, perturbation theory, and weak-strong uniqueness. To save time, we will not present the complete proofs.

Next, we will explain aspects of the Jia, Sverak, and Guillod program (Jia-Sverak, Inventiones 2014, JFA 2015; Guillod-Sverak, arXiv 2017) and, in particular, how instability in self-similarity variables can generate non-uniqueness. We will briefly discuss bifurcations, stable and unstable manifolds, and, time permitting, a short proof of the existence of large self-similar solutions.

Finally, we will present the recent work (A.-Brue-Colombo, Ann. Math. 2022) which rigorously established non-uniqueness of Leray-Hopf solutions with forcing. We will present the main idea but focus on the spectral perturbation arguments.

No prior knowledge of the Navier-Stokes equations is required, though it might be beneficial to preview background on weak and mild solutions in, for example, Chapters 4 and 11 in Robinson-Rodrigo-Sadowski or Chapters 3 and 5 in Tsai.