Schedule for: 21w5110 - New Mechanisms for Regularity, Singularity, and Long Time Dynamics in Fluid Equations (Online)

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

I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation in 2 space dimensions with and without advection in the presence of growing modes. The KSE is a model of long-wave instability in dissipative systems.

Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In an effort to provide a rigorous proof of the potential Euler singularity revealed by Luo-Hou's computation, we develop a novel method of analysis and prove that the original De Gregorio model and the Hou-Lou model develop a finite time singularity from smooth initial data. Using this framework and some techniques from Elgindi's recent work on the Euler singularity, we prove the finite time blowup of the 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ initial velocity and boundary. Further, we present some new numerical evidence that the 3D incompressible Euler equations with smooth initial data develop a potential finite time singularity at the origin, which is quite different from the Luo-Hou scenario. Our study also shows that the 3D Navier-Stokes equations develop nearly singular solutions with maximum vorticity increasing by a factor of $10^7$. However, the viscous effect eventually dominates vortex stretching and the 3D Navier-Stokes equations narrowly escape finite-time blowup. Finally, we present strong numerical evidence that the 3D Navier-Stokes equations with slowly decaying viscosity develop a finite time singularity.

We review recent work, joint with Yu Deng and Haitian Yue, about the Gibbs measure for the periodic 2D NLS and 3D Hartree NLS as well as the theory of random tensors, a powerful new framework which allows us to unravel the propagation of randomness under the nonlinear flow beyond the linear evolution of random data. This enables us in particular, to show the existence and uniqueness of solutions to the periodic NLS in an optimal range relative to what we define as the probabilistic scaling.

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We first classify the first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock, as a fractional series expansion with coefficients computed from the data. With this precise pre-shock description, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. We prove that along the slowest surface, all fluid variables except the entropy have $C^{1, {\frac{1}{2}} }$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $C^{1, {\frac{1}{2}} }$ one-sided cusps while the pressure and normal velocity remain $C^2$; as such, we term this surface a weak contact discontinuity.

Contact lines (e.g, where coffee meets the coffee cup or a droplet) appear generically between a free surface and a fixed boundary. Even though the steady contact line and contact angle was studied by people like Gauss and Young, even the modelling of dynamic contact lines has been an active research area in physics. In a joint research program initiated with Ian Tice, global well-posedness and stability of contact lines is established for a recent viscous fluid model in 2D.

We discuss some possible (and still speculative) routes to establishing finite time blowup in fluid equations (and other PDE), focusing in particular on methods based on establishing universality properties for such equations.

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!

I will review some recent advances in the study of the well posedness and singularity formation for the Prandtl system as well as the study of the inviscid limit of the Navier-Stokes system and the Derivation of the Prandtl system.

The wave kinetic equation is a central topic in the theory of wave turbulence, which concerns the thermodynamic limit of interacting wave systems. It can be traced back to the 1920s and has played significant roles in different areas of physics. However, the mathematical justification of the theory has long been open. In this talk we present our recent work, which resolves this problem by providing the rigorous derivation of the wave kinetic equation. This is joint work with Zaher Hani (University of Michigan).

A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on gravitational collapse. The talk is based on joint works with Yan Guo and Mahir Hadzic.

A study of wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed is discussed. For large-amplitude periodic traveling waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded. This is joint work with Walter Strauss (Brown University, USA) and Eugen Varvaruca (University of Iasi, Romania).

The Nernst-Planck-Navier-Stokes equations model the evolution of ions in Newtonian fluids. I will describe results on global existence and stability of smmoth solutions and on asymptotic interior electroneutrality (the vanishing of the charge density away from boundaries, in the limit of zero Debye screening length). The talk is based on recent works with M. Ignatova and with F-N Lee.

Bounded vorticity solutions to the 2D Euler equations on singular domains are typically not close to Lipschitz near boundary singularities, which makes their uniqueness a difficult open problem. I will present a general sufficient condition on the geometry of the domain that guarantees global uniqueness for all solutions initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $\pi$ and, in particular, is satisfied by all convex domains. Its proof is based on showing that fluid particle trajectories for general bounded vorticity solutions cannot reach the boundary in finite time. The condition also turns out to be sharp in the latter sense: there are domains that come arbitrarily close to satisfying it and on which particle trajectories can reach the boundary in finite time.

We discuss the first few terms in the asymptotic expansion of the solutions of $-\Delta u + u\nabla u+\nabla p=f(x)$ at infinity (assuming $f(x)$ is localized and not too large). The first term has been known for some time and is given by Landau solutions. The higher-order terms exhibit interesting behavior. Joint work with Hao Jia.

The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidence suggests that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.

I will discuss solutions to the incompressible Euler equation in two di-mensions with vorticity close to a finite sum of Dirac deltas (vortices). The law of motion of the vortices was known formally for a long time and proved rigorously by Marchioro-Pulvirenti. In collaboration with Juan Davila (U.Bath), Manuel del Pino(U. Bath), and Juncheng Wei (UBC) we have a different point of view, which allows a very precise description of the solution near the vortices. Our construction can be generalized to other situations,such as the construction of leapfrogging vortex rings of the 3D incompressibleEuler equations.

The aim of our talk is to show that axially symmetric suitable weak solutions to the Navier-Stokes equations have no Type I blowups. This can be done by reduction to a Liouville type theorem for a certain governing equation on a scalar function.

We show local regularity of local energy solutions to the Navier-Stokes equations in terms of local scaled integrals of the initial data. It extends previous work of Jia-Sverak, Barker-Prange and ourselves. This refined criterion implies that if a weighted $L^2$ norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982). This is a joint work with Kyungkeun Kang and Hideyuki Miura.

This is a joint work with Y. Guo and K. Widmayer. We consider the Euler equation in 3d with a Coriolis force and we show that small, smooth and localized initial data which are axisymmetric lead to solutions which exist for a long time. The proof uses the dispersive effect induced by the Coriolis term and relies on recent advances for long time estimates for quasilinear dispersive equations.

In this talk, I will discuss characterizations of stationary or uniformly-rotating solutions of 2D Euler and other similar equations. The main question we want to address is whether every stationary/uniformly-rotating solution must be radially symmetric. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

We consider quadratic Klein-Gordon equations with an external potential $V$ in $3+1$ space dimensions. We assume that $V$ is generic and decaying, and that the operator $H = -\Delta + V + m^2$ has an eigenvalue $\lambda^2 < m^2$. This is a so-called ‘internal mode’ and gives rise to time-periodic localized solutions of the linear flow. We address the question of whether such solutions persist under the full nonlinear flow. Our main result shows that all small nonlinear solutions slowly decay as the energy is transferred from the internal mode to the continuous spectrum, provided a natural Fermi golden rule holds. This extends the seminal work of Soffer-Weinstein for cubic nonlinearities to the case of any generic perturbation. Joint work with T. Leger (Princeton University).

I will report some recent results on the existence of finite time blow-up for nematic liquid crystal flows and Landau-Lipschitz-Gilbert equation. The nematic liquid crystal flow is a coupled system of harmonic map flows and Navier-Stokes system while LLG is a standard model in magnetics. I will show how the gluing techniques can be applied to both equations to produce blow-ups.

In this talk, we will discuss the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We will describe how, in such a scenario, the solution becomes unstable as time approaches the blow-up time. The method uses the relation between the vorticity of the solution, and the bi-characteristic amplitude solutions, which describe the evolution of the linearized Euler equation at high frequency. In the axisymmetric case, we can also study the instability of blow-up profiles. This work was partially supported by the NSF DMS-1907981. This a joint work with Misha Vishik and Laurent Lafleche.