Background and purpose:

Finite time singularity formation for Nonlinear Evolution PDEs, and small scales instabilities in fluid dynamics, have seen a great progress recently. The Cauchy problem for numerous equations has been solved, and researchers are now aiming at understanding precisely these phenomena.

Over the last thirty years, parabolic equations have been particularly well-understood. Removing little by little conditions on the nonlinearity (sub-super-critical), and symmetry assumptions, mathematicians were able to prove a wide range of behaviors in different regimes, even for the simplest models. C. Collot, V. T. Nguyen and collaborators have brought recently a unified spectral and modulational approach that applies to various settings, with many applications yet to be developed.

Progresses were also made in the study of the in/stability of boundary layers in fluid dynamics. C. Collot and collaborators analyzed recently solutions displaying a separation for the Prandtl’s system, with a new approach on singularities of nonlinear hyperbolic transport equations. At the same time, singular solutions were constructed or understood precisely for the first time for the incompressible and compressible Euler equations, and the compressible Navier-Stokes equations.

The course will detail the recent analysis made for these problems, focusing on two classes of equations: nonlinear parabolic problems such as the semilinear heat equation and the Keller-Segel system, and models of fluid mechanics such as Burgers and Prandtl’s equations. It is an introduction to the problems of the construction and stability of particular dynamics, and of the classification of behaviors. The audience will be provided up-to-date analysis techniques for nonlinear PDEs. An overview of the recent developments of the field will be presented. Open problems and directions of research will be discussed.

 The course is accessible to graduate students who have basic knowledge in PDEs and functional analysis. It is also of interest for any researcher in the general field of analysis.

 

• Selected bibliography of the authors:

- Collot, C., Ghoul, T. E., Masmoudi, N., & Nguyen, V. T. (2020) Refined description and stability for singular solutions of the 2D Keller-Segel system. Communications on Pure and Applied Mathematics (to appear) [arXiv :1912 :00721]

- Collot, C., Merle, F., & Raphaël, P. (2020). Strongly anisotropic type II blowup at an isolated point. Journal of the American Mathematical Society, 33(2), 527-607.

- Collot, C., Ghoul, T. E., & Masmoudi, N. (2019). Unsteady separation for the inviscid two-dimensional Prandtl’s system. [arXiv:1903.08244].

- Collot, C., Ghoul, T. E., Ibrahim, S., & Masmoudi, N. (2018). On singularity formation for the two-dimensional unsteady Prandtl’s system. [arXiv:1808.05967].

- Ghoul, T. E., Nguyen, V.T. & Zaag, H. (2018) Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Annales de l’Institut Henri Poincare I Analyse Non Lineaire, no. 6, 1577-1630.

- Nguyen, V.T. & Zaag, H. (2017) Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation. Ann. Scient. Éc. Norm. Supér (4). 50:5, 1241-1282.

- Nguyen, V.T. & Zaag, H. (2016). Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method. Analysis and PDE, no. 1, 229-257.

 

Outline and Speakers:

Week 1: Singularities in nonlinear parabolic equations (Van Tien Nguyen)

• Part 1: The semilinear heat equation 

Lecture 1: Introduction to the semilinear heat equation

- Local well-posedness in Lebesgue spaces

- Standard energy identity for blowup criteria

- Similarity transformation, self-similar solutions and asymptotic self-similarity

Lecture 2: Existence and stability of Type I blowup solutions to the semilinear heat equation (Analysis from [Ghoul-Nguyen-Zaag, AIHP 2018])

- Hermite operator: spectrum, Hermite semigroup, parabolic regularity 

- Formal derivation of blowup profile

- Formulation of the linearized problem

- Setting up the bootstrap regime

Lecture 3a: (Continued) Existence and stability of Type I blowup solutions to the semilinear heat equation (Analysis from [Ghoul-Nguyen-Zaag, AIHP 2018])

- Control of the radiation in the bootstrap regime (reduction to a finite-dimensional problem)

- Contradiction argument to control the finite-dimensional problem 

- Conclusion of the stability result

• Part 2: The 2D Keller-Segel system

Lecture 3b: Introduction to the 2D Keger-Segel system:

- Local well-posedness results

- Energy identity for blowup criterion: the mass threshold 8

- Non-existence of type I blowup

Lecture 4: Existence and stability of Type 2 blowup solutions to the 2D Keller-Segel system (Analysis from [Collot-Ghoul-Masmoudi-Nguyen, CPAM 2020])

- Self-similar variables, partial mass setting

- Properties of the linearized operator: 

o Eigenproblem in the partial mass setting

o Coercivity estimate

- Formal derivation of blowup rate

o Spectral analysis

o Via matching expansions (Velazquez)

Lecture 5: (Continued) Existence and stability of Type 2 blowup solutions to the 2D Keller-Segel system (Analysis from [Collot-Ghoul-Masmoudi-Nguyen, CPAM 2020])

- Formulation of the linearized problem

- Setting up the bootstrap regime

- Derivation of modulation equations driving the blowup law

- Control of the radiation in the bootstrap regime

- Conclusion to the stability result.

Discussing open problems and related parabolic problems

Week 2: Singularities in fluid mechanics (Charles Collot)

Lecture 1: Classification of singularities for analytic solutions to Burgers equation (Analysis from [Collot-Ghoul-Masmoudi]). Global in time continuation after the singularity by a weak solution (viscosity solutions, Oleinik, Hopf). Comment on uniqueness of entropy solutions. 

Lecture 2: Inviscid limit for Navier-Stokes equation and appearance of a boundary layer. Kato criterion. Prandtl’s expansion. Grenier’s instability result. Comment on Kukavica-Vicol and Sammartino-Caflisch for local well posedness and stability of the boundary layer in the analytical setting.

Lecture 3: Separation for the inviscid Prandtl’s boundary layer (Collot-Ghoul-Masmoudi). Incompressible characteristics, self-similar profiles, geometric approach to the genericity of blow-up solutions. Simpler case of the absence of Eulerian flow at infinity.

Lecture 4: Separation for the viscous Prandtl’s boundary layer (Collot-Ghoul-Ibrahim-Masmoudi). Blow-up profile, Modulation equations, exterior Lyapunov functionals, applications of techniques from the previous week.

Lecture 5: Selected topics on singularity formation and instability in fluid dynamics (comment on work of Gerard-Varet Dormy, Buckmaster-Shkoller-Vicol, Merle-Raphael-Rodnianski-Szeftel, Elgindi Jeong)