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Open Access Publications from the University of California

Scholarly Works (2 results)

  • Thesis
  • Peer Reviewed
UC Santa Barbara Electronic Theses and Dissertations (2019)

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.

    Cover page: Affine Solutions of Two Dimensional Magnetohydrodynamics and Related Quadratically Coupled Transport Equations
    • Article
    • Peer Reviewed
    UC Santa Barbara Previously Published Works (2020)
    The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in SL (2 , R). In the case of perfect fluids, the motion is given by geodesic flow in SL (2 , R) with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in SL (2 , R). A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as t→ ± ∞. For MHD, solutions are bounded and generically quasi-periodic and recurrent.
      Cover page: Affine Motion of 2d Incompressible Fluids Surrounded by Vacuum and Flows in SL(2,R)
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