Periodic Reporting for period 2 - CAPA (Global existence and Computer-Assisted Proofs of singularities in incompressible fluids, with Applications)
Reporting period: 2022-01-01 to 2023-06-30
The search for singularities in incompressible flows has become a major challenge in the area of non-linear partial differential equations and is relevant in applied mathematics, physics and engineering. A fluid dynamical understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics.
The goal of this proposal is to go far beyond the state of the art in the study of singularities and global existence for PDE related to fluid mechanics, and to apply some of the techniques to solve fundamental open questions in spectral geometry, or at least provide a totally different perspective on them.
A major theme along this work is the interplay between rigorous computer calculations and traditional mathematics. Interval arithmetics are used as part of a proof whenever needed.
This proposal is divided in three blocks, the first two involving global existence and/or singularities for: the incompressible Euler and Navier-Stokes equations; the surface quasi-geostrophic (SQG), the generalized-SQG equations and related models; and a third one on applications to spectral geometry.
The goal of this proposal is to go far beyond the state of the art in the study of singularities and global existence for PDE related to fluid mechanics, and to apply some of the techniques to solve fundamental open questions in spectral geometry, or at least provide a totally different perspective on them.
A major theme along this work is the interplay between rigorous computer calculations and traditional mathematics. Interval arithmetics are used as part of a proof whenever needed.
This proposal is divided in three blocks, the first two involving global existence and/or singularities for: the incompressible Euler and Navier-Stokes equations; the surface quasi-geostrophic (SQG), the generalized-SQG equations and related models; and a third one on applications to spectral geometry.
During the 2 years of the grant, we have written about 20 papers in fluid mechanics and spectral geometry. About 5 more are in preparation. The most important ones are:
[1] Symmetry in stationary and uniformly-rotating solutions of active scalar equations, J. Gómez-Serrano, J. Park, J. Shi and Y. Yao. Duke Mathematical Journal, 170, no. 13, 2957-3038 (2021).
[2] Smooth imploding solutions for 3D compressible fluids, T. Buckmaster, J. Gómez-Serrano and G. Cao-Labora. Submitted (2022).
[3] Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks, Y. Wang, C. Y. Lai, J. Gómez-Serrano, T. Buckmaster. Submitted (2022).
We have classified stationary and rotating solutions of the 2D incompressible Euler equation and other active scalar equations in [1]. This is an important problem in the context of global existence and singularity formation for equations coming from fluid mechanics and its applicability is very wide. Paper [1] has already been published in Duke Math. Journal, one of the top journals in mathematics. Paper [2] completely solves a central problem in singularity formation for compressible Euler and compressible Navier-Stokes establishing formation of singularities for all values of the adiabatic exponent. Paper [3] develops a new method to find self-similar solutions to fluid equations via deep learning. All of these are or will be landmark papers. Our work has been featured in Quanta Magazine and Quanta Podcast. We have given over 30 seminars/conferences during the last year, and about 15 the year prior to that.
[1] Symmetry in stationary and uniformly-rotating solutions of active scalar equations, J. Gómez-Serrano, J. Park, J. Shi and Y. Yao. Duke Mathematical Journal, 170, no. 13, 2957-3038 (2021).
[2] Smooth imploding solutions for 3D compressible fluids, T. Buckmaster, J. Gómez-Serrano and G. Cao-Labora. Submitted (2022).
[3] Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks, Y. Wang, C. Y. Lai, J. Gómez-Serrano, T. Buckmaster. Submitted (2022).
We have classified stationary and rotating solutions of the 2D incompressible Euler equation and other active scalar equations in [1]. This is an important problem in the context of global existence and singularity formation for equations coming from fluid mechanics and its applicability is very wide. Paper [1] has already been published in Duke Math. Journal, one of the top journals in mathematics. Paper [2] completely solves a central problem in singularity formation for compressible Euler and compressible Navier-Stokes establishing formation of singularities for all values of the adiabatic exponent. Paper [3] develops a new method to find self-similar solutions to fluid equations via deep learning. All of these are or will be landmark papers. Our work has been featured in Quanta Magazine and Quanta Podcast. We have given over 30 seminars/conferences during the last year, and about 15 the year prior to that.
Understanding and classifying stationary solutions for the 2D Euler equations is a major problem that has implications in other fields (e.g. convex integration), other equations (SQG, Navier-Stokes) and would improve the understanding of turbulence.
An important achievement of this project that has gone beyond the state of the art is the set of results concerning rigidity and flexibility of solutions to the 2D Euler and more general active scalar equations. The first part of this result (rigidity) has been published in Duke Math. Journal, one of the leading journals in mathematics and will lead to further developments. The second part (flexibility) is under consideration at a top journal.
We have proved the existence of finite time imploding singularities for 3D compressible Euler for all values of the adiabatic exponent, and for Navier-Stokes, The key idea to go from Euler to Navier-Stokes is that for certain self-similar solutions, under certain conditions in the scaling exponent, the viscosity is a lower order term and can be treated as an error. We believe this has implications beyond the compressible case, in particular in the incompressible one, as well as many other equations (Cordoba-Cordoba-Fontelos model, SQG....).
We have developed a new framework, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations. The solution in addition corresponds to an asymptotic self-similar profile for the 3-dimensional Euler equations in the presence of a cylindrical boundary. In particular, the solution represents a precise description of the Luo-Hou blow-up scenario [G. Luo, T. Hou, Proc. Natl. Acad. Sci. 111(36): 12968-12973, 2014] for 3-dimensional Euler. To the best of the authors' knowledge, the solution is the first truly multi-dimensional smooth backwards self-similar profile found for an equation from fluid mechanics. The new numerical framework is shown to be both robust and readily adaptable to other equations.
We expect the combination of the techniques mentioned in the previous 2 paragraphs to yield very fruitful outcomes in the future, solving longstanding open problems.
On a different topic, we have started to develop tools to tackle hard open questions in spectral geometry (such as the hotspots conjecture from the 1970’s or the Polya-Szego from the 1950’s). We have written two papers. This is more foundational work than results-oriented and sets up a path for the resolution of the aforementioned problems.
An important achievement of this project that has gone beyond the state of the art is the set of results concerning rigidity and flexibility of solutions to the 2D Euler and more general active scalar equations. The first part of this result (rigidity) has been published in Duke Math. Journal, one of the leading journals in mathematics and will lead to further developments. The second part (flexibility) is under consideration at a top journal.
We have proved the existence of finite time imploding singularities for 3D compressible Euler for all values of the adiabatic exponent, and for Navier-Stokes, The key idea to go from Euler to Navier-Stokes is that for certain self-similar solutions, under certain conditions in the scaling exponent, the viscosity is a lower order term and can be treated as an error. We believe this has implications beyond the compressible case, in particular in the incompressible one, as well as many other equations (Cordoba-Cordoba-Fontelos model, SQG....).
We have developed a new framework, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations. The solution in addition corresponds to an asymptotic self-similar profile for the 3-dimensional Euler equations in the presence of a cylindrical boundary. In particular, the solution represents a precise description of the Luo-Hou blow-up scenario [G. Luo, T. Hou, Proc. Natl. Acad. Sci. 111(36): 12968-12973, 2014] for 3-dimensional Euler. To the best of the authors' knowledge, the solution is the first truly multi-dimensional smooth backwards self-similar profile found for an equation from fluid mechanics. The new numerical framework is shown to be both robust and readily adaptable to other equations.
We expect the combination of the techniques mentioned in the previous 2 paragraphs to yield very fruitful outcomes in the future, solving longstanding open problems.
On a different topic, we have started to develop tools to tackle hard open questions in spectral geometry (such as the hotspots conjecture from the 1970’s or the Polya-Szego from the 1950’s). We have written two papers. This is more foundational work than results-oriented and sets up a path for the resolution of the aforementioned problems.