# Applied Mathematics Seminar

In this talk, we are interested in the Landau equation which is a kinetic model in plasma physics that describes the evolution of the density function $$f = f(t, x, v)$$ representing at time $$t\in \mathbb{R}^{+}$$, the density of particles at position $$x\in T^{3}$$ (the 3-dimensional unit periodic box) and velocity $$v\in \mathbb{R}^{3}$$. We study the Landau equation, depending on the Knudsen Number and its limit to the incompressible Navier-Stokes-Fourier equation on the torus. We prove uniform estimate of some adapted Sobolev norm and get existence and uniqueness of solution for small data. These estimates are uniform in the Knudsen number and allow to derive the incompressible Navier-Stokes-Fourier equation when the Knudsen number tends to 0.