Organizers: Glatt-Holtz, Nathan and Zhao, Kun

January 25

February 1

**Andrei Tarfulea**University 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

**Zach Bradshaw**Tulane 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

**Max Yaremchuk**Navy 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 15

**Speaker**Institution

**Abstract**: TBA

February 22

**Hayriye Gulbudak**University of Louisiana at Lafayette

**Abstract**: TBA

March 8

**Speaker**Institution

**Abstract**: TBA

March 22

**Speaker**institution

**Abstract**: TBA

March 29

**Jose Castillo**Computational Science Research Center - San Diego State University

**Abstract**:

April 5

**Speaker**Institution

**Abstract**: TBA

April 12

**Speaker**institution

**Abstract**: TBA

April 19

April 26

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu