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This is a dissertation on the motion of incompressible charged and non charged particles in a fluid. Specifically, we are concerned with the affine motion of such two dimensional fluids. The physical quanitities of the fluid are derived in terms of the deformation gradient which reduces the Incompressible Euler Equations (EE) and the Incompressible Ideal Magnetohydrodynamical (MHD) equations to ordinary differential equations on SL$(2,\mathbb{R})$. The EE and MHD become the equations of a free particle and harmonic oscillator, respectively, constrained to SL$(2,\mathbb{R})$ with the magnetic field strength acting as a bifurcation parameter between the two types of dynamics. We analyze the geometry of SL$(2,\mathbb{R})$ and completely characterize the behavior of all affine solutions. Inspired by the decay of the pressure for affine solutions to EE we analyze a related system of quadratically coupled transport equations. By smoothing the equation we show local well posedness in a generalized Sobolev space along with coupled energy estimates for a low and high energy. These estimates are inherited by the non-smoothed solution which, along with weighted energy estimates, allow us to show global well posedness for small data.

We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases, however, the motivation of such a collision kernel is to treat the (extreme) case of Coulomb potentials, on which this particular form of collision operator is well defined. We show that the scaled angle-potential singular concentration in a grazing collisions limit makes the Boltzmann operator converge in the sense of distributions to the Landau operator acting on the Boltzmann solutions, and also that solutions of this type of Boltzmann equation converge to solutions of the Landau equation that conserve mass, momentum and energy.