Spring 2019
Time & Location: Typically talks will be on Fridays in Gibson Hall 126 at 3:30 PM.
Organizers: Glatt-Holtz, Nathan and Zhao, Kun
January 25
Clifford
February 1
Existence, continuation, and lower mass bounds for the Landau equation
Andrei TarfuleaUniversity of Chicago
Abstract:
Kinetic equations model gas and particle dynamics, specifically focusing on the interactions between the micro-, meso-, and macroscopic scales. Mathematically, they demonstrate a rich variety of nonlinear phenomena, such as hypoellipticity through velocity-averaging and Landau amping. The question of well-posedness remains an active area of research.
In this talk, we look at the Landau equation, a mathematical model for plasma physics arising from the Boltzmann equation as so-called grazing collisions dominate. Previous results are in the perturbative regime, or in the homogeneous setting, or rely on strong a priori control of the solution (the most crucial assumption being a lower bound on the density, as this prevents the elliptic terms from becoming degenerate).
We prove that the Landau equation has local-in-time solutions with no additional a priori assumptions; the initial data is even allowed to contain regions of vacuum. We then prove a "mass spreading" result via a probabilistic approach. This is the first proof that a density lower bound is generated dynamically from collisions. From the lower bound, it follows that the local solution is smooth, and we establish the mildest (to date) continuation criteria for the solution to exist for all time.
February 8
Results on some local energy solutions to the Navier-Stokes equations
Zach BradshawTulane University
Abstract:
Local energy solutions to the Navier-Stokes equations, that is, weak solutions which are uniformly locally square integrable, but not necessarily globally square integrable, have proven a useful class for studying regularity and uniqueness. In this talk we survey several recent results concerning the existence and properties of local energy solutions, including applications to self-similar solutions.
February 15
Applied and Computational
Dynamically constrained interpolation of large data sets in oceanography: computational aspects
Max YaremchukNavy Research Laboratory
Abstract:
Improving the quality of global ocean weather forecasts is a primary task of oceanographic research. The problem is currently treated as a statistically and dynamically consistent synthesis of the data streams arriving from satellites and autonomous observational platforms. Due to the immense size of the ocean state vector (10^8-10^9), operational algorithms combine variational optimization techniques with limited-size (10^2-10^3) ensembles simulating statistical properties of the error fields. A brief overview of current situation in operational state estimation/forcasting is presented with a special focus on selected problems requiring applied math research. These include efficient linearization and transposition of the operators describing evolution of the ocean state, sparse approximation of the inverse correlation matrices and consistent treatment of the state vector components with non-gaussian error statistics.
Location: Gibson Hall 325
Time: 10:00
February 22
Topic
Hayriye GulbudakUniversity of Louisiana at Lafayette
Abstract: TBA
March 29
Time Space Mimetic Differences Methods
Jose CastilloComputational Science Research Center - San Diego State University
Abstract:
Mimetic discretization methods provide a discrete analog of vector calculus and have been used in many applications very effectively. The mimetic operators take care of the spatial domain. For time dependent problems, different time discretizations have been used; however, it is not clear if this time discretization schemes are "mimetic" in time. Symplectic integrators have been developed for Hamiltonian systems which are represented by ordinary differential equations. With symplectic integrators there is energy conservation because of the existence of a conserved quantity close to the original Hamiltonian. We will present the coupling of mimetic difference operators with symplectic integrators for wave motion.
