Math 741. Topics in Partial Differential Equations: Blow up for Euler's equations
- Spring 24 - Hans Lindblad

Although the equations of motion of fluids were derived over 200 years ago it still remains open if they have global smooth solutions in all space in the incompressible case. However, recently for the inviscid case there have been constructions of solutions that blow up, in the nonsmooth case and in the case with a boundary. The proofs involve finding approximate blowup solutions and controlling the errors with linear analysis and numerical calculations. The aim is to cover these results by Elgindi and by Chen and Hou.

We will start with an introduction to incompressible Euler from Chapter 1 and 2 in the textbook [1] (see also [2]). After that we will study the blow up in the nonsmooth case by Elgindi [3] and chapter 4 in the survey article about singularity formulation for incompressible Euler [4] as well as further development in [5]. The aim is to thereafter cover the blow up in the smooth case with a bounday by Chen and Hou [6]. However, it may be helpful to first look at the method in earlier simpler cases in [7] and in [8].

We remark that for the compressible fluids shocks can form and moreover recently Merle, Raphael, Rodnianski and Szeftel [9] showed that a different type of blow up; implosion can occure and the stability analysis is somewhat related. Furthermore [10] gave a somewhat simplier proof of this see overview in [11].

wk  date  Monday  Wednesday
  1  1/22  1.1-1.3.3 in [1] Derivation of Euler's equations.  1.3.4-1.4 Equations for the pressure and vorticity.
  2  1/29  1.4-1.6 in [1] Symmetries and energy conservation  2.2-2.3 [1] Continuation 2.4 [2] vorticity to velocity
  3  2/5  Axial symmetry 2.3.3 in [2] and section 2 in [4]  
  4  2/12    
  5  2/19    
  6  2/26    
  7  3/4    
  8  3/11    
  9  3/18  Spring break  Spring break
10  3/25    
11  4/1    
12  4/8     
13  4/15    
14  4/22    

[1] Bedrossian, Vicol. The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction. AMS (2022)
[2] Majda and Bertozzi Vorticity and incompressible flow
[3] Elgindi Finite-time singularity formation for \(C^{1,\alpha}\!\) solutions to the incompressible Euler equations on \(\Bbb{R}^3\!\) Ann \(\!\)of Math194,647-727(2021\(\!\!\!\))\(\!\!\)
[4] Drivas and Elgindi Singularity formation in the incompressible Euler equation in finite and infinite time Preprint (2022).
[5] Elgindi and Pasqualotto From Instability to Singularity Formation in Incompressible Fluids preprint (2023)
[6] Chen, Hou Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I:Analysis preprint (2023)
[7] Chen and Hou On stability and instability of \(C^{1,\alpha}\) singular solutions of the 3D Euler and 2D Boussinesq equations preprint (2022).
[8] Chen and Hou and D. Huang On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations CPAM (2021).
[9] Merle, Raphael, Rodnianski and Szeftel On the implosion of a three dimensional compressible fluid preprint (2019).
[10] Buckmaster, Cao-Labora and Gomez-Serrano Smooth imploding solutions for 3D compressible fluids preprint (2022).
[11] Buckmaster, Cao-Labora and Gomez-Serrano Smooth self-similar imploding profiles for 3D compressible fluids preprint (2023).
[12] Kiselev, Sverak Small scale creation for solutions of the incompressible two dimensional Euler equation Annals of Math (2014)