• Non-uniqueness of weak solutions to the Euler and Navier-Stokes equations –Tristan Buckmaster

    Abstract: In this course I will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, I will outline the resolution of Onsager’s conjecture as well the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations. I also hope to phrase the results within the context of a broader program of resolving a number of important open problems in the field.

    Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exists weak solutions lying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the phenomena of anomalous dissipation in turbulent cascades, which has been called the zeroth law of turbulence.

    Non-uniqueness of weak solutions to the Navier-Stokes equations has an important consequence for a well known strategy for resolving the famous Millennium Prize problem. Specifically, one may hope to prove existence of global smooth solutions to the Navier-Stokes equations by first constructing global weak solutions and then proving that such solutions are smooth. The recent non-uniqueness result of Vlad Vicol and myself demonstrates that such a strategy is doomed to fail, at least for the class of weak solutions considered. It remains open whether Leray-Hopf weak solutions are non-unique. Non-uniqueness of Leray-Hopf solutions is the subject of a famous conjecture of Ladyženskaja in ’69.

  • Recent developments on water waves — Yu Deng

     

    Abstract: The water wave system describes the motion of water-air interface, like those of lakes and oceans. In the last

    20 years there has been significant progress in the study of these systems. The purpose of this mini-course is
    to give a brief introduction to these recent works. The emphasis will be put on long-time stability results, but
    other important works will be discussed as well.

    Main references:
    [1] D. Lannes. The water waves problem. Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, vol. 188.
    [2] T. Alazard and J. Delort. Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. Éc. Norm. Supér. 48, 1149-1238.
    [3] Y. Deng, A. Ionescu, B. Pausader and F. Pusateri. Global solutions of the gravity-capillary water-wave system in three dimensions. Acta Math. 219, 213-402.
    [4] P. Germain, N. Masmoudi and J. Shatah. Global solutions for the gravity surface water waves equation in dimension 3. Ann. Math. 175, 691-754.
    [5] A. Ionescu and F. Pusateri. Global solutions for the gravity water waves system in 2D. Invent. Math. 199, 653-804.
    [6] S. Wu. Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177, 45-135.
    [7] S. Wu. Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 125-220.

     

  • Recent developments in the theory of relativistic fluids — Marcelo Disconzi

    Abstract: We will discuss several results concerning the mathematical theory of relativistic fluids, with emphasis on geometric aspects and the omnipresence of the dynamic role played by the characteristics of the equations of motion. Starting with perfect fluids, we will outline basic aspects of the Einstein-Euler system. We will then introduce a new formulation of the relativistic Euler equations that exhibits remarkable properties and is particularly well-suited for the study of shock formation. After making a few brief remarks about relativistic magneto-hydrodynamics, we will introduce the problem of studying relativistic fluids with viscosity and discuss some of the mathematical challenges it presents.

    Main references:
    [1] Anile, A. M. Relativistic fluids and magneto-fluids. Cambridge University Press. Chapters 1 and 2.

    [2] Rezzolla, L.; Zanotti, O. Relativistic hydrodynamics. Oxford University Press. Sections 3.1 to 3.4.

    [3] Disconzi, M. M.; Speck, J. The relativistic Euler equations: Remarkable null structures and regularity properties. arXiv:1809.06204 [math.AP]

    [4] Disconzi, M. M. On the existence of solutions and causality for relativistic viscous conformal fluids. Comm. Pure. Appl. Anal., to appear. arXiv:1708.06572 [math.AP].

  • Nonlinear dynamics of the Schrödinger equation with periodic boundary conditions — Emanuele Haus

    Please Click Here for the ABSTRACT