# The Abel Symposium: scheduled activities

Tuesday (morning) – Friday (afternoon)

June 13–16, 2023 at **Bårdshaug Herregård**

The Abel Symposium 2023 will host world-leading speakers in the field of partial differential equations, who over a period of four days will present their work in the traditional setting of Bårdshaug mansion.

**The event is by invitation only.**

# Schedule, presentations and videos

We apologise for the sound quality in the first six videos.

## Tuesday

**Title:** On continuous time bubbling for the harmonic map heat flow in two dimensions.

**Abstract:** I will describe recent work with Jacek Jendrej (CNRS, Paris Nord) and Andrew Lawrie (MIT) on harmonic maps of finite energy from the plane to the two sphere, without making any symmetry assumptions. While it has been known since the 1990s that bubbling occurs along a carefully chosen sequence of times via an elliptic Palais-Smale mechanism, we show that this continues to hold continuously in time. The key notion is that of the “minimal collision energy” which appears in the soliton resolution result by Jendrej and Lawrie on critical equivariant wave maps.

**Title:** Paralinearization of free boundary problems in fluid dynamics

**Abstract:** A classical topic in the mathematical theory of hydrodynamics is to study the evolution of the free surface separating air from an incompressible perfect fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. These equations are different in nature, dispersive or parabolic, but we will see that they can be studied using related tools. In particular, I will discuss a paradifferential approach to these problems.

**Title:** Nonlinear forward-backward problems

**Abstract:** This talk is devoted to the study of the equation \(u u_x - u_{yy}=f\) in the domain \((x_0,x_1)\times (-1,1)\), in the vicinity of the shear flow profile \(u(x,y)=y\). This equation serves as a toy model for more complicated fluid equations such as the Prandtl system.

The difficulty lies in the fact that we are interested in changing sign solutions. Hence the equation is forward parabolic in the region where \(u>0\), and backward parabolic in the region \(u<0\). The line \(u=0\) is a free boundary and an unknown of the problem.

Unexpectedly, we prove that even when the data (i.e. the source term \(f\) or the boundary data) are smooth, existence of strong solutions of the equation fails in general. This phenomenon is already present at the linear level, and linked to the existence of singular profiles for the homogeneous linearized equation. In fact, we prove that strong solutions exist (both for the linearized and for the nonlinear system) if and only if the data satisfy a finite number of orthogonality conditions, whose purpose is to avoid the presence of singular profiles in the solution.

A key difficulty of our work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow.

This is a joint work with Frédéric Marbach and Jean Rax.

**Title:** Analysis of models of living tissues and free boundary problems

**Abstract:** Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.

We will give an overview of the modeling aspects and focuss on the links between those mathematical models. Then, we will focus on the 'compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.

The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, uniform \(L^4\) estimate on the pressure gradient and emergence of instabilities.

## Wednesday

**Title:** A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

**Abstract:** In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.

**Title:** Area Variations under Lagrangian and Legendrian constraints

**Title:** Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

**Abstract:** We discuss stationary solutions of Euler's equation on a rotating sphere and their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system. We present some rigidity results (showing that certain stationary solutions must be either zonal or rotated zonal solutions) and some stability results of low-mode Rossby-Haurwitz stationary solutions. This is joint work with Pierre Germain (Courant Institute, New York).

## Thursday

**Title:** Recent Advances in Wave Turbulence Theory

**Abstract:** Wave Turbulence Theory is widely used across several areas of physics to describe the statistical behavior of dispersive waves. It was formulated throughout the twentieth century by Peierls, Hasselman and Zakharov. Central to this theory is the derivation of the Wave Kinetic Equation which describes energy cascades between waves. In recent years a rigorous mathematical proof of the derivation of the Wave Kinetic Equation and its range of applicability started to emerge, a feat that has taken a long time since the theory was first proposed. In this expository talk I will be discussing rigorous results and mathematical techniques that lead to proving the validity of the kinetic description of waves. I will also discuss the range of parameters where the kinetic description is valid.

**Title:** Some recent developments in wave turbulence theory

**Abstract:** In this talk I will present two different approaches in the study of wave turbulence theory. The first, introduced by Bourgain, consists in analyzing the long time behavior of high Sobolev norms for the defocusing, cubic NLS equation on 2D tori (periodic solutions). In this context I will emphasize how the rationality or irrationality of the torus affects the analysis. The second approach deals with the rigorous derivation of the 3-wave kinetic equation from a weakly nonlinear multidimensional KdV type equation.

**Title:** Turbulence in completely integrable PDEs: The Calogero-Moser derivative NLS

**Abstract:** I will discuss a new type of a derivative nonlinear Schrödinger equation, which can be seen as a continuum version of completely integrable Calogero-Moser many-body systems in classical mechanics. The resulting NLS exhibits many intriguing features such as a Lax pair structure on Hardy spaces, \(L^2\)-criticality, and turbulent solutions. In this talk, I will focus on dynamics of multi-soliton solutions, which exhibit an unbounded growth of Sobolev norms (turbulence). This is based on joint work with Patrick Gérard (Orsay).

**Title:** Low regularity and long time solutions in quasilinear dispersive flows

**Abstract:** Both the study of low regularity solutions and the study of the long time dynamics for quasilinear pde's are challenging fundamental questions in dispersive flows. Usually these are considered separately, but recent research shows that it is instead natural to study them together. The talk will provide an overview of several new ideas an results in this direction, ranging from water waves and nonlinear wave equation to nonlinear Schroedinger flows. This is joint work with Mihaela Ifrim, and also in part with Albert Ai.

**Title:** The quartic integrability and long time existence of steep water waves in 2d

**Abstract:** It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle.

We construct a sequence of energy functionals \(\mathfrak E_j(t)\), directly in the physical space, that involves material derivatives of order \(j\) of the solutions for the 2d water wave equation, so that \(\frac d{dt} \mathfrak E_j(t)\) is quintic or higher order.

We show that if some scaling invariant norm, and a norm involving one spatial derivative above the scaling of the initial data are of size no more than \(\varepsilon\), then the lifespan of the solution for the 2d water wave equation is at least of order \(O(\varepsilon^{-3})\), and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size \(\varepsilon\), then the lifespan of the solution is at least of order \(O(\varepsilon^{-5/2})\).

Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.

## Friday

**Title:** Explicit formula and zero-dispersion limit for the Benjamin-Ono equation

**Abstract:** I will show how the Lax pair structure of the Benjamin-Ono equation on the line leads to an explicit formula for general initial data in the Sobolev space \(H^1\). Applications to the zero-dispersion limit will be discussed.

**Title:** Generic alignment conjecture and the problem of emergence of collective behavior

**Abstract:** Emergence is a phenomenon of formation of collective outcomes in systems where communication between agents has local range. For a wide range of applications, such as swarming behavior of animals or exchange of opinions between individuals, such outcomes result in a globally aligned state or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions is a challenging mathematical problem. In this talk we will overview three programs of research directed on understanding the emergent phenomena: statistical approach to generic alignment for agent-based systems, kinetic approach based on relaxation and hypocoercivity, and hydrodynamic models incorporating a novel way of interaction based on topological communication.

**Title:** A variational regularity theory for Optimal Transportation, and its application to matching

**Abstract:** A couple of years ago, with M. Goldman we devised a new approach to the regularity theory for Optimal Transportation that mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-dimensional smallness condition, the displacement is close to the gradient of a harmonic function.

Probably the main advantage of this variational regularity theory over the one based on maximum principle — and attached to the name of Caffarelli — is that it does not require any regularity of the involved measures. Hence it can be applied to the popular matching problem, where it provides regularity on large scales.