Search Research Symposium Abstracts

The viscosity operator on the Riemannian manifolds

There are inequivalent formulations of the Navier-Stokes equations on the Riemannian manifolds. This is due to the different possibilities for the Laplacian operator acting on vector fields. In this talk, we discuss some of the possible choices, and present a recent progress in the search for the "correct" viscosity operator in the setting of Riemannian [Read More...]

Presenter: Magdalena Czubak, University of Colorado Boulder
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 10:15 am

Almost sure well-posedness for Hall MHD*

We consider the magnetohydrodynamics system with Hall effect accompanied with initial data in supercritical Sobolev space. Via an appropriate randomization of the supercritical initial data, both local and small data global well-posedness for the system are obtained almost surely in critical Sobolev [Read More...]

Presenter: Mimi Dai, Institute for Advanced Study / University of Illinois at Chicago
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 10:40 am

Spatial integral of the solution to SPDEs with time-independent noise

In this talk, we review some recent results regarding the asymptotic behavior of the spatial integral of the solution to the hyperbolic/parabolic Anderson model, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a spatially homogeneous Gaussian noise. The noise does not depend on time, which means that Itô's [Read More...]

Presenter: Raluca Balan, University of Ottawa
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 11:05 am

Recent developments on non-uniqueness of stochastic PDEs via convex integration

In this talk I will review some recent developments concerning non-uniqueness of solutions to stochastic PDEs via the technique of convex integration adapted to probabilistic settings. Examples of stochastic PDEs include Navier-Stokes equations, Euler equations, Boussinesq system, and magnetohydrodynamics system, some of which have diffusion in the form of [Read More...]

Presenter: Kazuo Yamazaki, Texas Tech University
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 11:30 am

Long-time behavior of nonlinearly coupled measure-valued partial differential equations

The mean dynamics of many stochastic processing networks in which jobs with general service distributions are routed in a network of multiple servers are often described by nonlinearly coupled systems of measure-valued partial differential equations. These have some similarities with partial differential equations that describe age-structured population [Read More...]

Presenter: Kavita Ramanan, Brown University
Authors: Rami Atar, Weining Kang, Haya Kaspi, Kavita Ramanan,
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 3:30 pm

A Nonautonomous Free Boundary Model for Tumor Growth

A non-autonomous free boundary model for tumor growth will be introduced. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equations describing the evolution of the tumor size. Global existence and uniqueness of solutions, existence and uniqueness [Read More...]

Presenter: Xiaoying (Maggie) Han, Auburn University
Authors: Wenlong Sun, Tomas Caraballo, Xiaoying Han, Peter Kloeden
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 3:55 pm

Mixing and weak convergence of numerical approximations of SPDEs

We consider a general framework for obtaining uniform-in-time rates of convergence for numerical approximations of SPDEs in suitable Wasserstein distances. The framework is based on two general results under an appropriate set of assumptions: a Wasserstein contraction result for a given Markov semigroup; and a uniform-in-time weak convergence result for a [Read More...]

Presenter: Cecilia Mondaini, Drexel University
Authors: Cecilia Mondaini and Nathan Glatt-Holtz
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 4:20 pm

Various numerical schemes for stochastic 2D Navier-Stokes equations

We introduce various space-time numerical schemes for Stochastic 2D Navier-Stokes equations and show rate of [Read More...]

Presenter: Hakima Bessaih, Florida International University
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 17, 2022; 4:45 pm

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an [Read More...]

Presenter: Kaitlin Hill, Wake Forest University
Authors: Kaitlin Hill, Jessica Zanetell, and John Gemmer
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 18, 2022; 10:15 am

Local well-posedness in the bi-harmonic NLS with low power nonlinearities

In this work we consider one of the variant of the famous Schrödinger equation, where the potential term can be expressed as a power nonlinearity (with any positive power) and the dispersion operator has the higher order, i.e., instead of the Laplacian (or two derivatives in one dimension) we consider a double Laplacian (with four derivatives). This [Read More...]

Presenter: Iryna Petrenko, Florida International University
Authors: Iryna Petrenko
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 18, 2022; 10:40 am

Asymptotic behavior of stochastic Navier-Stokes and Schrodinger equations

We consider the asymptotic limits of two dimensional incompressible stochastic Navier Stokes equation and one dimensional stochastic Schrodinger equation. These limits include large and moderate deviations, Central limit theorem, and the law of the iterated logarithm. For large and moderate deviations, we will discuss both the Azencott method and the weak [Read More...]

Presenter: Parisa Fatheddin, Ohio State University, Marion
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 18, 2022; 11:05 am

Intermittency in turbulence and the 3D Navier-Stokes 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 [Read More...]

Presenter: Aseel Farhat, Florida State University
Authors: Aseel Farhat, Zoran Grujic
Symposium Year: 2022
Session: Deterministic and Probabilistic Approaches for Nonlinear PDEs
Presentation Time: June 18, 2022; 11:30 am