Schedule for: 23w5040 - Fluid Equations, A Paradigm for Complexity: Regularity vs Blow-up, Deterministic vs Stochastic

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Meet and Greet at BIRS Lounge (PDC building, 2nd floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The stationary problem for the free boundary incompressible Navier-Stokes equations lies at the confluence of two distinct lines of inquiry in fluid mechanics. The first views the dynamic problem as an initial value problem. In this context, the stationary problem arises naturally as a special type of global-in-time solution with stationary sources of force and stress. One then expects solutions to the stationary problem to play an essential role in the study of long-time asymptotics or attractors for the dynamic problem. The second line of inquiry, which dates back essentially to the beginning of mathematical fluid mechanics, concerns the search for traveling wave solutions. In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of the speaker and co-authors. For technical reasons, these results were only able to produce traveling solutions with nontrivial wave speed. In this talk we will discuss the well-posedness theory for the stationary problem and show that the solutions thus obtained lie along a one-parameter family of slowly traveling wave solutions. This is joint work with Noah Stevenson.

Fluid-solid interaction problems are widely studied because of their connections with hemodynamics, geophysical and engineering applications. The differential equations governing this type of interactions feature a dissipative component (typically arising from the fluid, through the Navier-Stokes equations) and a conservative component (due to the solid counterpart, through either Euler equations of rigid body dynamics or Navier equations of elasticity). This dissipative-conservative interplay has a fundamental role in questions related to existence, uniqueness and stability of solutions to the governing equations. We will discuss results concerning the existence and stability of solutions to equations characterized by the above-mentioned dissipative-conservative interplay, and describing the dynamics of different mechanical systems featuring fluid-solid interactions.

I will review some recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to C^2 class at all integer times only. Another result is that the \alpha-SQG patches interpolating between Euler and SQG cases are ill-posed in L^p, p \ne 2 or H\"older based spaces. The proofs involve derivation of a new system describing the patch evolution in terms of arc-length and curvature. The talk is based on works joint with Xiaoyutao Luo.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The hyperbolic Navier-Stokes contains an extra double time-derivative term while the hyperbolic MHD differs from the standrad MHD by a double time-derivative term in the magnetic field equation. The appearance of these terms is not an artifact but reflects basic physics laws. Mathematically the global regularity problem on these hyperbolic equations is extremely difficult. In fact, even the L^2-norm of solutions to the 2D equations are not known to be globally bounded in the general case. This talk presents recent results on the regularity, convergence, and the construction of non-unique weak solutions with Kazuo Yamazaki.

In this talk I will present results on singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. These equations describe the motion of a compressible ideal gas, which is characterized by a parameter called the adiabatic constant. Finite time singularities for generic adiabatic constants were found in the recent breakthrough of Merle, Raphaël, Rodnianski and Szeftel. Our results allow us to drop the genericity assumption and construct smooth self-similar profiles for all values of the adiabatic constant. Part of the proof is very delicate and requires a computer-assisted analysis. Joint work with Tristan Buckmaster, Gonzalo Cao-Labora, Jia Shi and Gigliola Staffilani.

We consider the free boundary problem for a 2D and 3D fluid filtered in porous media, which is known as the one-phase Muskat problem. We show that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a unique global Lipschitz strong solution. The proof of the uniqueness relies on a new pointwise $C^{1,\alpha}$ estimate near the boundary for harmonic functions. This is based on recent joint work with Francisco Gancedo (Universidad de Sevilla, Spain) and Huy Q. Nguyen (University of Maryland, USA).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Kolmogorov's K41 theory of fully developed turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids: although smooth solutions of the Euler equations conserve the energy, in a turbulent regime information is transferred to small scales and dissipation can happen even without the effect of viscosity, and it is rather due to the limited regularity of the solutions. In rigorous mathematical terms, however, very little is known. In a recent work in collaboration with M. Colombo and M. Sorella we consider the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. In my talk, I will present the general context and illustrate the main ideas behind our construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. I will also describe how the same techniques provide an example of lack of selection for passive-scalar advection under vanishing diffusivity, and an example of anomalous dissipation for the forced Euler equations in the supercritical Onsager regularity regime (this last result has been obtained in collaboration with E. Brue, M. Colombo, C. De Lellis, and M. Sorella).

The very basic question of well-posedness remains open for many fluid equations. We will discuss some recent progresses in the effort to understand the classical problem by exploring ill-posedness behavior of solutions. The emphasis is on the construction of pathological solutions which either indicate non-uniqueness or develop finite time singularity.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Convex integration has proven to be a successful technique for modeling instabilities in fluid dynamics (Kelvin-Helmholtz, Rayleigh-Taylor, or Saffman-Taylor instabilities). A prime example is the unstable Muskat problem, which is the mathematical treatment of a two-phase, incompressible fluid evolving through porous media. In this talk, I will review the existence theory and demonstrate how maximizing potential energy dissipation reconciles Otto's minimizing scheme on the Wasserstein space with the convex integration subsolution, leading to a unique equation for macroscopic evolution. Furthermore, I will explain that such an equation admits entropy solutions. This later work is a collaboration with Ángel Castro (ICMAT Madrid) and Björn Gebhard (UAM Madrid).

In ideal MHD (magnetohydrodynamics), smooth solutions conserve total energy, cross helicity and magnetic helicity. Nevertheless, in view of numerical evidence, ideal MHD should possess weak solutions that 1) arise at the ideal (inviscid, non-resistive) limit, 2) conserve magnetic helicity but 3) dissipate total energy and do not conserve cross helicity. I will discuss a result directly on ideal MHD, in collaboration with Daniel Faraco and László Székelyhidi Jr. (to appear in CPAM): there exist bounded solutions of ideal MHD with prescribed total energy and cross helicity profiles and magnetic helicity. The proof uses a new convex integration scheme on two-forms consistent with the conservation of magnetic helicity. I will also discuss another result from the same paper (which proves a conjecture of Buckmaster and Vicol): L^3_{t,x} is the L^p-threshold for magnetic helicity conservation in ideal MHD.

We prove strong illposedness for the generalized SQG equations for smooth data, when roughly speaking the stream function is more singular than the advected scalar. The mechanism is due to degenerate dispersion near a quadratic shear flow. The case when the stream function is logarithmically singular is of particular interest ("Ohkitani model"); in this case, there is a wellposedness in a scale of Sobolev spaces with time-decreasing exponents, and this result has immediate applications to long-time dynamics of inviscid and viscous alpha-SQG models. Joint work with D. Chae, Jungkyoung Na and S.-J. Oh.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We provide a new boundary estimate on the vorticity for the incompressible Navier-Stokes equation endowed with no-slip boundary condition. The estimate is rescalable through the inviscid limit. It provides a control on the layer separation at the inviscid Kato double limit, which is consistent with the Layer separation predictions via convex integration. This is a joint work with Jincheng Yang.

In 2017 Chemin, Zhang and Zhang posed the question of whether initial data with small third-component (with respect to a scale-invariant norm) implies global regularity of the associated 3D Navier-Stokes solution. Motivated by this open question, in this talk I will discuss growth properties of solutions for certain initial data with zero third-component. This talk is based on joint work with Christophe Prange (Cergy) and Jin Tan (Cergy).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will discuss recent results on the blow-up problem of classical solutions for the incompressible Euler equations with finite energy. We will construct solutions that has instantaneous gap loss of Sobolev regularity in the plane and finite time singularities in the whole space. Joint works with Luis Martinez-Zoroa, Wojciech Ożański and Fan Zheng.

I will start by presenting the lake equations which can be considered as a generalization of the 3D axisymmetric Euler equations without swirl. This 2D model differs from the well-known 2D Euler equations due to an anelastic constraint in the div-curl problem. I will explain how this new constraint implies a very different behavior of concentrated vortices: the point vortex moves under its own influence according to a binormal curvature law. This work is done in collaboration with Lars Eric Hientzsch and Evelyne Miot.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A common method used to sample from a distribution with density proportional to $p = e^{-V/\kappa}$ is to run Monte Carlo simulations on an overdamped Langevin equation whose stationary distribution is also proportional to $p$. When the potential $V$ is not convex and the temperature $\kappa$ is small, this can take an exponentially large (i.e. of order $e^{C/\kappa}$) amount of time to generate good results. I will talk about a method that introduces a "mixing drift" into this system, which allows us to rigorously prove convergence in polynomial time (i.e. a polynomial in $1/\kappa$). This is joint work with Alex Christie, Yuanyuan Feng and Alexei Novikov.

We prove global in time well-posedness for perturbations of the 2D Navier-Stokes equations driven by a perturbation of additive space-time white noise.The proof relies on a dynamic high-low frequency decomposition, tools from paracontrolled calculus and an L2 energy estimate for low frequencies. Our argument requires the solution to the linear equation to be a log-correlated field. We do not rely on (or have) explicit knowledge of the invariant measure: the perturbation is not restricted to the Cameron-Martin space of the noise. Our approach allows for anticipative and critical (L2) initial data. Time permitting, we will discuss some further developments. Joint work with Martin Hairer.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.