MS08: PDEs and dynamical systems


  • Ning Ju, Oklahoma State university

Session A: Oct.1, 10:40am-12:00pm, Classroom Building 101
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 101

10:40am-11:00am (Oct 1)
CLB 101

Igor Kukavica,
University of Southern California

A free boundary inviscid model of flow-structure interaction


We address a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by a fourth-order hyperbolic PDE. We provide a priori estimates for the existence of solutions with a sharp regularity for the Euler initial data and construct solutions with the regularity construct solutions with the fluid initial data in Hr, where r >= 3. The result is joint with Amjad Tuffaha.

11:00am-11:20am (Oct 1)
CLB 101
Yanqiu Guo,
Florida International University
Inertial manifolds for regularized Navier-Stokes equations

For a large class of dissipative evolution equations, long-time behavior of solutions possesses a resemblance of the behavior of finite-dimensional systems. In order to capture such phenomena, Foias, Sell, and Temam introduced the concept of inertial manifolds. An inertial manifold of an evolution equation is a finite-dimensional Lipschitz invariant manifold attracting exponentially all the trajectories of the dynamical system. The existence of an inertial manifold for an infinite-dimensional evolution equation represents the best analytical form of reduction of an infinite system to a finite-dimensional one. A number of dissipative PDEs possess inertial manifolds, such as the nonlinear reaction-diffusion equation, the Kuramoto-Sivashinsky equation, and the Cahn-Hilliard equation. But, whether the Navier-Stokes equations possess an inertial manifold is unknown. In this presentation, I will talk about the existence of inertial manifolds for some regularized Navier-Stokes equations, and discuss its connection with large gaps between sums of squares in number theory.

11:20am-11:40am (Oct 1)
CLB 101
Vincent R. Martinez,
City University of New York

Relaxation-based parameter recovery from partial observations for hydrodynamic systems
  This talk will discuss a dynamical approach for recovering unknown parameters of hydrodynamic equations such as damping coefficients or external forcing from partial observations of the underlying system. The method is based on a feedback-control scheme which incorporates observations as an exogenous term and drives the system to synchronize with the observations. This can be used to devise algorithms that systematically filters the model errors resulting from the unknown parameters. Under suitable assumptions on the observational density and tuning parameters of the algorithm, convergence of the algorithm to the true value of the parameters can be guaranteed.

11:40am-12:00pm (Oct 1)
CLB 101
Quyuan Lin,
University of California, Santa Barbra

Error estimates for the physical informed neural networks (PINN) approximating the primitive equations
  Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is in general a hard task. Neural network has been shown to be a promising machine learning tool to tackle this challenge. In this talk, I will introduce the physical informed neural networks (PINNs) that approximate the solutions to the PEs with either full viscosity and diffusivity or only the horizontal ones, and establish the error estimates. In particular, I will show the existence of two layer tanh PINNs such that the corresponding training error is arbitrarily small by taking the width of PINNs to be wide enough, and the error between the true solution and the approximation can be made arbitrarily small provided that the training error is small enough and the sample set is large enough. Furthermore, the results are on the higher order (in spatial Sobolev norm) error estimates, which improves some previously known results for PINNs concerning only the L2 error.

3:10pm-3:30pm (Oct 1)
CLB 101
Aseel Farhat,
Florida State University

Intermittency in turbulence and the 3D NSE regularity problem
  We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present a proof that the hyper-dissipative 3D Navier-Stokes are regular within an appropriate functional setting incorporating the intermittency in turbulent regimes, with any power of the Laplacian greater than 1.

3:30pm-3:50pm (Oct 1)
CLB 101
Krutika Tawri,
University of California, Berkeley

On stochastic partial differential equations with a Ladyzenskaya-Smagorinsky type nonlinearity
  The theory of monotone operators plays a central role in many areas of nonlinear analysis. Monotone operators often appear in fluid dynamics, for example the p-Laplacian appears in a non-Newtonian variant of the Navier- Stokes equations modeled by Ladyzenskaya or in the Smagorinsky model of turbulence. In this talk, we will discuss global existence results of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general L\'evy noise. The classical approach based on using directly the Galerkin approximation is not valid. In this talk we will discuss how one can approximate a monotone operator by a family of monotone operators acting in a Hilbert space, so as to recover certain useful properties of the orthogonal projectors and overcome the challenges faced while applying the Galerkin scheme.

3:50pm-4:10pm (Oct 1)
CLB 101
Ning Ju,
Oklahoma State university

Particle trajectories of large scale oceanic flow
  I will present my recent research on global existence and uniqueness of particle trajectories for vector field of viscous Primitive Equations for large scale oceanic flow.

4:10pm-4:30pm (Oct 1)
CLB 101
Paolo Piersanti,
Indiana University

Ice sheets melting as an obstacle problem
  In this talk, which is the result of a joint work of the speaker with Roger Temam (IU), we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative, rendering the problem under consideration an obstacle problem.
A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.