MS04: Analysis and applications of PDEs modeling fluids


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 114
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 114
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 114
Session D: Oct.2, 10:30Am-11:50am, Classroom Building 114

10:40am-11:00am (Oct 1)
CLB 114
Xukai Yan,
Oklahoma State University
Stability of homogeneous solutions of stationary incompressible Navier-Stokes equations

In 1944, Landau discovered a three parameter family of explicit (-1)-homogeneous solutions of 3D stationary incompressible NSE with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. Sverak proved that all (-1)-homogeneous solutions that are smooth on the unit sphere must be Landau solutions. Karch and Pilarczyk showed that small Landau solutions are asymptotically stable under any L2-perturbation. In recent joint works with Li Li and Yanyan Li, we studied (-1)-homogeneous solutions of 3D incompressible stationary NSE with finitely many singular rays. In this talk, I will first talk about the existence and asymptotic behavior of such solutions that are axisymmetric with two singular rays passing through the north and south poles. I will then discuss the asymptotic stability of Landau solutions and some of the solutions with singular rays we have obtained.

11:00am-11:20am (Oct 1)
CLB 114

Elaine Cozzi,
Oregon State University
The 2D quasi-geostrophic equation in holder and uniformly local Sobolev spaces
  We show existence of solutions to the 2D quasi-geostrophic equation (SQG) in Holder spaces without placing an integrability assumption on the solution, generalizing a result of Wu. The main challenge in this setting is lack of validity of the SQG constitutive law. To overcome this difficulty, we introduce a modified relationship between the velocity and the temperature. Time permitting, we also discuss an application of our result to existence of solutions to SQG in uniformly local Sobolev spaces. This is joint work with David M. Ambrose, Daniel Erickson, and James P. Kelliher.

11:20am-11:40am (Oct 1)
CLB 114

Anna Mazzucato,
The Pennsylvania State University

On Euler equations with in-flow and out-flow boundary conditions 
  I will discuss recent results concerning the well-posedness and regularity for the incompressible Euler equations when in-flow and out-flow boundary conditions are imposed on parts of the boundary. This is joint work with Gung-Min Gie (U. Louisville, USA) and James Kelliher (UC Riverside, USA).

11:40am-12:00pm (Oct 1)
CLB 114
Kun Zhao,
Tulane University
Existence and Stability of Traveling Waves of Boussinesq-Burgers Equations
  We introduce rigorous mathematical results concerning existence and stability of traveling wave solutions to Cauchy problem of the one-dimensional Boussinesq-Burgers equations modeling propagation of weak tidal bores. Existence of traveling waves is obtained by means of phase plane analysis and geometric singular perturbation. Local stability of traveling waves with arbitrary strength is proven by spatially weighted energy methods. This talk is based on recent joint work with Anita Yang (Chinese University of Hong Kong) and Zhian Wang (Hong Kong Polytechnic University).

3:10pm-3:30pm (Oct 1)
CLB 114

Zhuolun Yang,
Brown University 

Regular solutions of the stationary incompressible Navier-Stokes equations
  In this talk, we will discuss regular solutions of stationary incompressible Navier-Stokes equations. When the base domain is Euclidean space, the existence of such solutions was known for dimensions less than or equal to 5. In a joint work with Yanyan Li, we extended it to dimensions less than or equal to 15.

3:30pm-3:50pm (Oct 1)
CLB 114
Rui Han,
Louisiana State University
Decay of multi-point correlation functions in Zd
  We prove multi-point correlation bounds in Zd for arbitrary d>=1 with symmetrized distances, answering open questions proposed by Sims-Warzel and Aza-Bru-Siqueira Pedra. As applications, we prove multi-point correlation bounds for the Ising model on Zd, and multi-point dynamical localization in expectation for uniformly localized disordered systems, which provides the first examples of this conjectured phenomenon by Bravyi-König.

3:50pm-4:10pm (Oct 1)
CLB 114
Jeaheang Bang,
University of Texas at San Antonio
Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab
  I will present on my recent work with Changfeng Gui, Yun Wang, and Chunjing Xie. Liouville type theorems for the three-dimensional Navier-Stokes system in the entire space is a major open problem. In this work, we investigated Liouville-type theorems in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. For the no-slip boundary conditions, we proved that any bounded solution is trivial if it is axisymmetric or rur is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in L space. For the periodic boundary conditions, we proved that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or rur decays to zero as r tends to infinity. The proofs are based on energy estimates, and the key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions.

4:10pm-4:30pm (Oct 1)
CLB 114
Collin Victor,
University of Nebraska-Lincoln
Data assimilation in turbulent fluids: Movement paradigms for improved convergence rates
  In accurately simulating turbulent flows, two major difficulties arise before the simulation begins; namely the problem of determining the initial state of the flow, and the problem of estimating the parameters. Data assimilation helps to resolve the first problem by eliminating the need for complete knowledge of the initial state. It incorporates incoming observations into the mathematical model to drive the simulation to the correct solution. Recently, a promising new data assimilation algorithm (the AOT algorithm) has been proposed by Azouani, Olson, and Titi, which uses a feedback control term to incorporate observations at the PDE level. In this talk, we examine computationally the effects of observers that move dynamically in time for the 2D incompressible Navier-Stokes equations. We test several movement patterns (which we refer to as "the bleeps, the sweeps, and the creeps") as well as Lagrangian motion and combinations of these patterns, in comparison with static observers. In several cases, order-of-magnitude improvements in terms of the time-to-convergence are observed.

5:00pm-5:20pm (Oct 1)
CLB 114
Geng Chen,
University of Kansas
Poiseuille flow of nematic liquid crystals via Ericksen-Leslie model
  In this talk, we will discuss a recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. In this recent progress, we established a new method to study the full model. A singularity formation result will also be discussed, together with the global existence result showing that the solution will in general live in the Holder continuous space. The earlier related result on the stability of variational wave equation using the optimal transport method, and the future work on other wave equations will also be discussed. The talk is on the joint work with Tao Huang, Weishi Liu and Xiang Xu.

5:20pm-5:40pm (Oct 1)
CLB 114
Shuanglin Shao,
University of Kansas
A remark on the two dimensional water wave equation with surface tension
  We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that waves are periodic, we establish a priori energy inequality.

5:40pm-6:00pm (Oct 1)
CLB 114
Adam Larios,
University of Nebraska Lincoln
Reductions of the 2D Kuramoto-Sivashinky equations
  The flame equation, also known as the Kuramoto-Sivashinsky equation (KSE) is a highly chaotic dynamical system that arises in flame fronts, plasmas, crystal growth, and many other phenomena. Due to its lack of a maximum principle, the KSE is often studied as an analogue to the 3D Navier-Stokes equations (NSE) of fluids. We will discuss some of the relationships between these equations of fire and water. Much progress has been made on the 1D KSE since roughly 1984, but for the 2D KSE, even global well-posedness remains a major open question. In analogy with regularizations of the 3D NSE, we present modifications of the 2D KSE which allow for global well-posedness, while still retaining many important features of the 2D KSE. However, as has been demonstrated recently by Kostianko, Titi, and Zelik, standard regularizations, which work well for Navier-Stokes, destabilize the system when applied to even the 1D KSE. Thus, we present entirely new types of modifications for the 2D KSE. This talk will describe key ideas of the analysis, and also show many colorful movies of solutions.

6:00pm-6:20pm (Oct 1)
CLB 114
Matthew Enlow,
University of Nebraska Lincoln
Nonlinear calming for the 2D Kuramoto-Sivashinsky equations
  The Kuramoto-Sivashinsky (KS) equations model chaotic behavior in reaction-diffusion systems and have found many applications in the sciences. Global well-posedness of the equations have been shown in 1D, but this remains an open problem for higher dimensions. This is due in part to a lack of control over the nonlinear term. In this talk we introduce a modification of the 2D KS equations which allow for a better handle on the nonlinearity. We will perform an analysis of these 'calmed' KS equations and observe some computational demonstrations.

10:30am-10:50am (Oct 2)
CLB 114
Kazuo Yamazaki,
Texas Tech University
Non-uniqueness results of stochastic PDEs via probabilistic convex integration
  I will discuss recent developments concerning applications of convex integration to stochastic PDEs such as the Navier-Stokes equations, Boussinesq system, magnetohydrodynamics system, transport-diffusion equation, surface quasi-geostrophic equations, all forced by random noise. The types of noise include additive, linear multiplicative, nonlinearly multiplicative, transport, spatially white noise, as well as space-time white noise. We will also discuss some open problems related to stochastic Yang-Mills equation if time permits.

10:50am-11:10am (Oct 2)
CLB 114
Zachary Bradshaw,
University of Arkansas
Bounds on the separation rate of non-unique 3D Navier-Stokes flows
  There is considerable evidence that non-unique solutions to the 3D Navier-Stokes equations exist within the class of suitable weak solutions. Consequently, it is natural to ask how these hypothetical solutions relate to one another. For example, do these solutions remain relatively close or rapidly separate? In this talk I will present algebraic bounds on the separation rates of suitable weak solutions under appropriate conditions on the initial data. A useful short-time asymptotic expansion will also be discussed, as will related results for non-unique discretely self-similar solutions.

11:10am-11:30am (Oct 2)
CLB 114
Jiahong Wu,
Oklahoma State University
Stabilizing phenomenon for electrically conducting fluids
  Physical experiments and numerical simulations have observed a remarkable phenomenon that a background magnetic field can actually stabilize electrically conducting fluids. Our goal has been to understand the mechanism and establish this observation as a mathematically rigorous fact on MHD systems. This talk presents a stability result on the 3D incompressible MHD system with anisotropic dissipation. The velocity obeys the 3D incompressible Navier-Stokes equation with dissipation in only one direction, and is thus not known to be stable. However, when this Navier-Stokes is coupled with the magnetic field in the MHD system, the solutions near a background magnetic field are always global and stable. The magnetic field stabilizes the fluid. Mathematically, the system governing the perturbations can be converted to a damped wave equation. I will also briefly mention stability results for compressible MHD systems.