# Seminars & Events for Analysis of Fluids and Related Topics

##### On the steady-states of the Navier-Stokes equations in the plane

Despite the seminal work of Jean Leray, the stationary Navier-Stokes equations in two-dimensional unbounded domains are still not completely understood mathematically. More precisely, the behavior at infinity of the weak solutions is an open question. The Stokes paradox states that the linearization of the Navier-Stokes equations have no bounded solutions in general. In this talk, I will explain how the nonlinearity helps to obtain bounded solutions going to zero at infinity as well as their asymptotic behavior for the full nonlinear equations.

##### A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence

A common approach to calculate the solution of a scalar advection-diffusion equation is by a Feynman-Kac representation which averages over stochastic Lagrangian trajectories going backward in time to the initial conditions and boundary data. The trajectories are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing the scalar diffusivity. In this framework, we present an exact formula for scalar dissipation in terms of the variance of the scalar values acquired along each random trajectory. As an important application, we study the connection between anomalous scalar dissipation in turbulent flows for large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle trajectories.

##### Heat Rises: 100 Years of Rayleigh-Bénard Convection

**Please note 5:30 start time. **Buoyancy forces result from density variations, often due to temperature variations, in the presence of gravity. Buoyancy-driven fluid flows shape the weather, ocean and atmosphere dynamics, the climate, and the structure of the earth and stars. In 1916 Lord Rayleigh published a paper entitled "*On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the Under Side*" introducing the minimal mathematical model of buoyancy-driven fluid flow now known as Rayleigh-Bénard convection. For a century this model has served as a primary paradigm of complex nonlinear dynamics displaying spontaneous symmetry breaking and pattern formation, chaos and turbulence.

##### The Boltzmann equation with specular boundary condition in convex domains

We establish the global-wellposedness and stability of the Boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the Maxwellian with or without a small external potential. In particular, we have completely solved the long standing open problem after an announcement of Shizuta and Asano in 1977.

##### Strichartz estimates and local regularity for gravity-capillary water waves

We will consider the water waves problem for 2D and 3D incompressible, irrotational, inviscid fluid flows subject to both gravity and surface tension. The PDE is dispersive, quasilinear and nonlocal. The questions on local existence and uniqueness in the lowest regularity spaces are still in progress. We will discuss our results on a blow-up criterion in terms of the Lipschitz norm of the velocity; nonlinear Strichartz estimates for rough solutions; and their application in estabishing a local well-posedness theory for non-Lipschitz initial velocity.

##### Exponential self-similar mixing by incompressible flows

I will address the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing.

##### Global Stability of Solutions to a Beta-Plane Equation

We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as $\beta$-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions.

##### Blowup for model equations of fluid mechanics

In this talk, I discuss recent progress towards proving a finite time blowup for the 2D inviscid Boussinesq equations, inspired by the hyperbolic flow scenario.I will introduce various model equations for the Boussinesq system that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner. In the second part of my talk, I describe a one-dimensional problem, which has singular solutions converging to a well-defined profile at the time of blowup. This talk is based on joint work with B. Orcan-Ekmecki, M. Radosz, and H. Yang.

##### Transport and mixing by viscous vortex rings

Biomixing is the study of fluid mixing caused by swimming organisms. The swimming of large organisms can lead to mixing by the turbulent flows in their wakes, but the wakes created by small swimming organisms are less turbulent. Instead, the main mechanism of mixing by smaller organisms is the net particle displacement (drift) induced by the swimmer. Several experiments have been performed to examine this drift for small jellyfish; these produce vortex rings that trap and transport a fair amount of fluid. However, since inviscid theory implies infinite particle displacements, the effects of viscosity must be included to understand the damping of real vortex motion. We use a model viscous vortex to compute particle displacements and other relevant quantities, such as the mean kinetic energy fluctuations. This is joint work with Th

##### Global-in-x Steady Prandtl Expansion over a Moving Boundary

I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.

##### The Zakharov-Lvov stochastic model for the wave turbulence

I will discuss the stochastic model of wave turbulence, suggested in 70's by Zakharov-Lvov and present some rigorous results, concerning this theory. Next I will explain the heuristic derivation of the wave kinetic equation for the model and finally present the plan of a rigorous justification of the kinetic equation in a work under preparation with Galina Perelman.

##### Self-similar vortex spirals

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals. Connections to the problem of non-uniqueness for the Euler equations will be discussed.

##### On the relativistic Boltzmann equation without angular cutoff

We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of spatially periodic box. We consider the general conditions on the collision kernel from Dudy´nski and Ekiel-Je´zewska (Commun Math Phys 115(4):607–629, 1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized relativistic Boltzmann collision operator in terms of a geometric fractional Sobolev norm; this shows a spectral gap exists and this behavior is similar to that of non-relativistic case as shown by Gressman and Strain(Journal of AMS 24(3), 771–847, 2011).

##### Scattering of the Gross-Pitaevskii equation in the 3D radial energy space

**PLEASE NOTE CHANGE IN TIME (5:00).** We consider the 3D Gross-Pitaevskii (GP) equation, or the nonlinear Schrodinger (NLS) equation with non-vanishing constant amplitude at spatial infinity. We are interested in the asymptotic stability of the plane wave solutions. This was first answered positively by Gustafson, Nakanishi, and Tsai assuming that the perturbation belongs to a weighted Sobolev space. Such a control is not provided by the conserved energy of the system, so the natural question is whether this stability holds for perturbations in the energy space. Such a result is crucial to addressing the large-data theory. We give a positive answer under the assumption of radial symmetry (or angular regularity), and prove small-data scattering to appropriate free states.

##### On a Slightly Compressible Water Wave

In this talk, I would like to go over some recent results on a compressible water wave. We generalize the apriori energy estimates for the compressible Euler equations established in Lindblad-Luo to when the fluid domain is unbounded. In addition, we establish weighted elliptic estimates that allow us to find initial data in some weighted Sobolev spaces with weight $w(x)=(1+|x|^2)^{\mu}, \mu \geq 2$, and we show this propagates within short time; in other words, we are able to prove weighted energy estimates for compressible water waves. These results serve as good preparation for proving long time existence also for compressible water waves.

##### Recent results on 2D density patches for inhomogeneous Navier-Stokes

This talk is about the dynamics of a patch given by two fluids of different constant densities, evolving by the inhomogeneous Navier-Stokes equations. The main question to address is whether the regularity of the boundary of the initial patch is preserved in time. Using classical Sobolev spaces for the velocity, we establish the propagation of $C^{1+\gamma}$, $W^{2,\infty} and $C^{2+\gamma}$ regularities with $0<\gamma<1$. The results are based on new cancelations found for time dependent singular integrals given by the linear nonhomogeneous heat kernel acting on quadratic terms.

##### An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem

**Please note special time (5:30). **It is shown--within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components--that the ever-resisting `scaling gap' in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor; all preexisting improvements have been logarithmic in nature, regardless of the functional set up utilized. The mathematics was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows. This is a joint work with A. Farhat and Z. Bradshaw.

##### Mixing in Compressible Hydrodynamics as Diffusivities Approach Zero

The Kelvin-Helmholtz (KH) instability is a prototypical hydrodynamic mixing process driven by velocity shear. I will present simulations of the KH instability in compressible hydrodynamics. Compressibility introduces baroclinic instabilities which can further enhance mixing. I compare simulations run at specific Reynolds numbers to ``implicit large eddy simulations'' (ILES) in which numerical errors play the role of a sub-grid scale diffusivity parameterization. Many of the simulations were run using Dedalus, an open-source spectral code which can solve nearly arbitrary PDEs. I will then discuss extrapolating our simulations to the limit of Re->infinity, extrapolating the ILES to the limit of resolution approaching infinity, and whether or not these two limits are the same.

##### Brief survey of computer assisted proofs for partial differential equations

I will present a brief survey of computer assisted methods of studying partial differential equations that I have worked on. The methods I am going to discuss allow for obtaining proofs of the existence of particular solutions of a certain class of PDEs in a prescribed range of parameters. I will discuss opportunities and limitations of the presented approach. In particular most of the presented results have not been obtained using known techniques of 'classical analysis'.

##### Stochastic homogenization for reaction-diffusion equations

We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses. That is, the medium becomes effectively homogeneous in the large-scale limit of the dynamics of solutions to the PDE. Hypotheses that guarantee this include fairly general stationary ergodic KPP reactions, as well as homogeneous ignition reactions in up to three dimensions perturbed by radially symmetric impurities distributed according to a Poisson point process. In contrast to the original (second-order) reaction-diffusion equations, the limiting "homogenized" PDE for this model are (first-order) Hamilton-Jacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense.