Schedule for: 18w5057 - Regularity and Blow-up of Navier-Stokes Type PDEs using Harmonic and Stochastic Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

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

I will describe results regarding the surface quasi-geostrophic equation (SQG) in bounded domains. The results concern global interior Lipschitz bounds for large data for the critical SQG in bounded domains. In order to obtain these, we establish nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. As an application global existence of weak solutions of SQG were obtained.

In this talk we will introduce a new class of flocking models that feature emergence of global alignment via only local communication. Such models have been sought for since the introduction of Cucker-Smale dynamics which showed global unconditional alignment for models with substantially strong non-local interaction kernels. We introduce a set of new structural components into the communication protocol, including singular kernel, and topological adaptive diffusion, that enhance alignment mechanisms with purely local interactions. We highlight some challenges that arise in the problem of global well-posendess and stability of flocks.

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy.

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 the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

We discuss `scaling' and `saturation' methods on general state spaces which can be used to solve low mode control problems for certain nonlinear partial differential equations. These methods naturally generalize ideas of Jurdjevic and Kupka in the finite-dimensional setting of ODEs. The methods will be highlighted by applying them to specific equations, including the KvD, 3d Euler equations and the 2d Boussinesq equations. Applications to support properties and ergodicity of randomly-forced versions of these equations will be noted.

In his 1991 paper, J.-M. Delort proved existence of solutions to the 2D Euler equations with $H^{-1}$-valued nonnegative vorticity; this includes the case of initial nonnegative vorticity concentrated on a line (vortex sheet). Here we prove the analogue result for the stochastic case, with transport noise on the vorticity. Namely, we consider 2D stochastic Euler equations (in vorticity form) $\partial_{t} \xi + u\cdot\nabla \xi + \sum_{k} \sigma_{k} \cdot \nabla \xi \circ \dot{W}^{k} = 0$, $\xi =$ const $+$ curl $u$, where $\sigma_{k}$ are given (divergence-free) regular vector fields and $W^{k}$ are independent Brownian motions. Our main result is existence of a weak (in the probabilistic sense) $H^{-1}$-valued nonnegative solution $\xi$. This is a joint work with Zdzislaw Brzezniak.

Intermittent flows, possessing more intense energy flux, exhibit deviations from Kolmogorov’s scaling laws, which can be measured in numerical simulations and experiments. I will rigorously define the spectrum of intermittency dimensions (as a function of the Holder exponent) and discuss how it affects regularity properties of solutions to the Navier-Stokes equations (NSE) and their ability to satisfy the energy equality. In particular, I will present new Onsager's spaces for the NSE and compare intermittent flows used for recent non-uniqueness constructions for the NSE in various dimensions.

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

We shall deal in this talk with non uniform rotating vortices for planar Euler equations. We propose to give a general approach to construct some of them near radial solutions. We provide a complete study for the truncated quadratic profile and explore the rarefaction of the bifurcating curves with respect to the parameters of the profile. This is a joint work with Claudia Garcia and Juan Soler.

The scenario of Type I singularity is a natural generalization of the self-similar or discretely self-similar singularities. In this talk we present recent progresses on the study of the Type I blow-up in the 3D incompressible Euler equations. By applying the blow-up method and the other local analysis tools we could make some progresses in our study of the Type I blow-up in relation to the concentration phenomena of the energy. The is a joint work with J. Wolf.

I will discuss some recent results on global existence for the Muskat equation and on stationary solutions with a splash singularity for the two-fluid Euler equations.

Many equations that model fluid behavior are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting PDEs often involve multiple non-dimensional parameters. Frequently some of these parameters are very small and they enter into the analysis in different ways. We will discuss one such system which has been proposed as a model for magnetostrophic turbulence and describe results that can be obtained in several different small parameter limits. In this talk we will concentrate on a forced drift-diffusion equation for the temperature where the fluid viscosity enters via the drift velocity. We examine the convergence of solutions in the limit as the viscosity goes to zero. We introduce a natural notion of ``vanishing viscosity'' weak solutions and prove the existence of a compact global attractor for the critical drift-diffusion equation. This is joint work with Anthony Suen.

For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382-–386 (2018).

We consider time dependent hypersurfaces immersed in Euclidean space and compute the evolution of geometric quantities such as the first and second fundamental form, curvatures, area and volume enclosed. We contrast geometric and hydrodynamic evolution. We give two recent results: a proof of an old conjecture regarding slender jet breakup, and a rigorous proof of finite or infinite time pinchoff in a model of Hele-Shaw.

In this talk we will use a basic example of shear flow to demonstrate some of the recent advances in the three-dimensional Euler equations. Specifically, this example was introduced by DiPerna and Majda to show that weak limit of classical solutions of Euler equations may, in some cases, fail to be a weak solution of Euler equations. We use this shear flow example to show the immediate loss of smoothness and ill-posedness of solutions of the 3D Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we also show the existence of weak solutions for the 3D Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface (vortex sheet). This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz (Birkhoff-Rott) problem where a minimal regularity implies the real analyticity of the interface. Furthermore, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. Eventually, we will discuss the recent remarkable work of De Lellis and Sz\'{e}kelyhidi concerning the wild weak solutions of Euler equations and their non-uniqueness. In particular, we propose the following ruling out criterion for non-physical weak solutions of Euler equations: ``In the absence of physical boundaries any weak solution of Euler equations which is not a vanishing viscosity limit of Leray-Hopf weak solutions of the Navier-Stokes equations should be ruled out". We will use this shear flow, and other solutions of Euler equations with certain spatial symmetry, to provide nontrivial examples for the use of this ruling out criterion. If time allows we will also discuss (i) recent progress concerning the Onsager conjecture in bounded domains; (ii) the nonuniqueness of weak solutions to the 3D Navier-Stokes equations with Hyper-viscosity $(-\Delta)^\theta$, for $\theta < 5/4$, demonstrating the sharpness of the J.-L. Lions result. This is a joint work with Claude Bardos.

TBA

The 3D Quasi-geostropic equation is a model used in climatology to model the evolution of the atmosphere for small Rossby numbers. It can be derived from the primitive equation. The surface quasi-geostrophic equation (SQG) is a special case where the atmosphere above the earth is at rest. The evolution then depends only on the boundary condition, and can be reduced to a 2D model. In this talk, we will show how we can derive the physical lateral boundary conditions for the inviscid 3D QG, and construct global in time weak solutions. Finally, we will discuss the global regularity of solutions to the QG equation with Ekman pumping.

We consider the Navier-Stokes equations modified with a hyperviscosity $(-\Delta)^p$ instead of the usual Laplacian and perturbed by a random forcing term, which comes from a Wiener process. For these equations we prove global existence and uniqueness as in the deterministic setting. Moreover, we investigate how large has to be the hyperviscosity parameter $p$ in order to apply Girsanov transform to analyze this hyperviscous Navier-Stokes equation. This is done by taking as a reference equation either the linear Stokes equation associated with the equation for the velocity or a suitable equation related to the equation for the vorticity.

I will report some recent results about both stationary and non-stationary Stokes systems with variable coefficients. Applications to the Navier-Stokes equations and the construction of Green's functions will also be presented. Based on joint work with Doyoon Kim (Korea University), Jongkeun Choi (Brown University), and Tuoc Phan (U of Tennessee).

The first part of the presentation is a short review of the statistical mechanics theory for points vortex models for the 2D Euler equations. In the second part we outline a recent result obtained in collaboration with F. Grotto (Scuola Normale, Pisa) about the fluctuations of the mean field limit for point vortices. In the last part we outline an extension of the theory to a slightly more general class of models with singular interaction. This is a work in progress with C. Geldhauser (Chebychev Laboratory, St. Petersburg).

Scale-invariance for the statistical Navier-Stokes equations is contrasted with their deterministic counterpart. The use of the vector potential (or, the stream function) achieves scale-invariance for the deterministic Navier-Stokes equations, where the dissipative term is given by the Ornstein-Uhlenbeck operator. We consider scale-invariance for the statistical Navier-Stokes equations in d-dimensions on the basis of the Hopf-Foias equation. It is achieved by choosing the d-th derivative of the vector potential as a relevant variable so that the dissipative term is given by the Fokker-Planck operator. On the basis of the latter, as an example we give a decaying profile with forward self-similarity explicitly for the Burgers equations in d-dimensions. For the 3D Navier-Stokes equations, a preliminary result is presented on the self-similar decaying profile using the vorticity gradient. Time permitting, a brief remark is made on the role played by the composition operator (known as Koopman operator) in the statistical theory of the Navier-Stokes equations.

We consider existence and uniqueness results for stochastic pde driven by a fractional Brownian motion. A fractional Brownian motion is a Gauss process with a particular parameter $H$ (the Hurst parameter). We have to differ two cases when the Hurst parameter $H\in (1/3,1/2]$ and $H\in (1/2,1)$. The case $H=1/2$ gives us a Brownian motion. In the other case this noise is not a martingale such that such that classical techniques of Ito integration cannot be applied. We show the existence of a random dynamical system generated by (s)pde driven by this noise and discuss some stability results.

We consider some models (compressible and incompressible) for three-dimensional fluid flow on the torus that incorporate the ambient temperature of the fluid. If the kinematic viscosity grows with the local temperature and the temperature is fed by the local dissipation of kinetic energy, then the fluid displays a novel self-regularizing effect in turbulent regions. This manifests as a strong a priori bound on a thermally-weighted enstrophy norm.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.