Past Events
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.
2022 UMN Summer Workshop on Analysis of PDEs
Monday, July 25, 2022, 9 a.m. through Friday, July 29, 2022, 11:30 a.m.
University of Minnesota
Advisory:
Note that the workshop is intended for graduate students and advanced undergraduates.
Organizers:
- Hao Jia, University of Minnesota, Twin Cities
In this five-day workshop for both graduate students and advanced undergraduate students, there will be a number of lectures given by active and leading experts in several important areas of analysis of partial differential equations, especially those arising from mathematical analysis of fluid dynamics and nonlinear waves. Participants will learn about the physical background, rigorous mathematical formulation, analytic tools, and latest developments in important PDE phenomena including singularity formation, uniqueness and non-uniqueness of weak solutions, stability mechanisms, and soliton resolution. Participants will also have many opportunities to interact with the lecturers in informal settings.
We will provide financial support to facilitate students' participation. To apply, please submit the following documents through the Workshop Application link at the top of the page:
- A brief CV or resume. (A list of publications is not necessary.);
- A reference letter from your advisor or professor.
Supported by NSF CAREER 1945179
Schedule
Subscribe to this event's calendar
Monday, July 25, 2022
9:00 am - 9:30 am | Registration and Coffee | Vincent 502 |
9:30 am - 10:30 am | Instability and non-uniqueness in the Navier-Stokes equations
Dallas Albritton (Princeton University) |
Vincent 570 |
10:30 am - 11:00 am | Break and Discussion | Vincent 502 |
11:00 am - 12:00 pm | Singularity formation in incompressible fluids and related models
Jiajie Chen (California Institute of Technology) |
Vincent 570 |
12:00 pm - 1:30 pm | Lunch | |
1:30 pm - 2:30 pm | 1d scattering theory and its application to nonlinear dispersive equations
Gong Chen (Georgia Institute of Technology) |
Vincent 570 |
2:30 pm - 3:00 pm | Break | Vincent 502 |
3:00 pm - 4:00 pm | Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations
Hao Jia (University of Minnesota, Twin Cities) |
Vincent 570 |
4:00 pm - 4:30 pm | Break and Discussion | Vincent 502 |
Tuesday, July 26, 2022
9:00 am - 9:30 pm | Coffee | Vincent 502 |
9:30 am - 10:30 am | Instability and non-uniqueness in the Navier-Stokes equations
Dallas Albritton (Princeton University) |
Vincent 570 |
10:30 am - 11:00 am | Break and Discussion | Vincent 502 |
11:00 am - 12:00 pm | Singularity formation in incompressible fluids and related models
Jiajie Chen (California Institute of Technology) |
Vincent 570 |
12:00 pm - 1:30 pm | Lunch | |
1:30 pm - 2:30 pm | 1d scattering theory and its application to nonlinear dispersive equations
Gong Chen (Georgia Institute of Technology) |
Vincent 570 |
2:30 pm - 3:00 pm | Break and Discussion | Vincent 502 |
3:00 pm - 4:00 pm | Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations
Hao Jia (University of Minnesota, Twin Cities) |
Vincent 570 |
4:00 pm - 5:00 pm | Outdoor Group Activity |
Wednesday, July 27, 2022
9:00 am - 9:30 am | Coffee | Vincent 502 |
9:30 am - 10:30 am | Instability and non-uniqueness in the Navier-Stokes equations
Dallas Albritton (Princeton University) |
Vincent 570 |
10:30 am - 11:00 am | Break and Discussion | Vincent 502 |
11:00 am - 12:00 pm | Singularity formation in incompressible fluids and related models
Jiajie Chen (California Institute of Technology) |
Vincent 570 |
12:00 pm - 1:30 pm | Lunch | |
1:30 pm - 2:30 pm | 1d scattering theory and its application to nonlinear dispersive equations
Gong Chen (Georgia Institute of Technology) |
Vincent 570 |
2:30 pm - 3:00 pm | Break and Discussion | Vincent 502 |
3:00 pm - 4:00 pm | Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations
Hao Jia (University of Minnesota, Twin Cities) |
Vincent 570 |
4:00 pm - 4:30 pm | Break and Discussion | Vincent 502 |
Thursday, July 28, 2022
9:00 am - 9:30 am | Coffee | Vincent 502 |
9:30 am - 10:30 am | Instability and non-uniqueness in the Navier-Stokes equations
Dallas Albritton (Princeton University) |
Vincent 570 |
10:30 am - 10:45 am | Group Photo | |
10:45 am - 11:00 am | Break and Discussion | Vincent 502 |
11:00 am - 12:00 pm | On the long-term regularity of water waves
Alexandru Ionescu (Princeton University) |
Vincent 570 |
12:00 pm - 1:30 pm | Lunch | |
1:30 pm - 2:30 pm | On the long-term regularity of water waves
Alexandru Ionescu (Princeton University) |
Vincent 570 |
2:30 pm - 3:00 pm | Break and Discussion | Vincent 502 |
3:00 pm - 4:00 pm | Singularity formation in incompressible fluids and related models
Jiajie Chen (California Institute of Technology) |
Vincent 570 |
4:00 pm - 4:30 pm | Break and Discussion | Vincent 502 |
4:30 pm - 5:00 pm | Panel Discussion | Vincent 570 |
Friday, July 29, 2022
8:30 am - 9:00 am | Coffee | Vincent 502 |
9:00 am - 10:00 am | Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations
Hao Jia (University of Minnesota, Twin Cities) |
Vincent 570 |
10:00 am - 10:30 am | Break and Discussion | Vincent 502 |
10:30 am - 11:30 am | On the long-term regularity of water waves
Alexandru Ionescu (Princeton University) |
Vincent 570 |
Participants
Name | Department | Affiliation |
---|---|---|
Dallas Albritton | Institute for Advanced Studies | Princeton University |
Adam Black | Department of Mathematics | Yale University |
Gong Chen | Department of Mathematics | Georgia Institute of Technology |
Jiajie Chen | Department of Applied and Computational Mathematics | California Institute of Technology |
Adriaan de Clercq | School of Mathematics | University of Minnesota, Twin Cities |
Kevin Dembski | Department of Mathematics | Duke University |
Samir Donmazov | Department of Mathematics | University of Kentucky |
Ziyang Gao | Mathematics | University of Minnesota, Twin Cities |
Jialun He | Department of Mathematics | State University of New York - Stonybrook |
Yupei Huang | Department of Mathematics | Duke University |
Alexandru Ionescu | Department of Mathematics | Princeton University |
Hao Jia | School of Mathematics | University of Minnesota, Twin Cities |
Aldis Kurmis | Department of Mathematics | University of Minnesota, Twin Cities |
Noah Lee | Department of Applied and Computational Mathematics | Princeton University |
Kexin Li | Department of Mathematics | University of Michigan |
Zhengjun Liang | Department of Mathematics | University of Michigan |
Jiaqi Liu | Department of mathematics | University of Southern California |
Tal Malinovitch | Department of Mathematics | Yale University |
Frederick Rajasekaran | Department of Mathematics | University of California, San Diego |
Xuanlin Shu | Department of Mathematics | Rutgers, State University of New Jersey |
Yixuan Wang | Department of Applied and Computational Mathematics | California Institute of Technology |
Kin Yau James Wong | Department of Mathematics | University of California, San Diego |
Yantao Wu | Department of Mathematics | Johns Hopkins University |