Department of Mathematics

Analysis and PDE

  •  Affine motion of 3d compressible ideal fluids surrounded by vacuum
  •  11/04/2016
  •  4:10 PM - 5:00 PM
  •  1502 Engineering Building
  •  Tom Sideris, University of California at Santa Barbara; Thomas C. Sideris, University of California, Santa Barbara

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial free boundary value problem for compressible fluids surrounded by vacuum in 3d. The general problem is known to be locally well-posed. For affine motion, the equations of motion reduce to a globally solvable Hamiltonian system of ordinary differential equations on GL_+(3,R). The domain occupied by the fluid at each time is an ellipsoid whose diameter grows linearly in time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid domain is determined by a semi-definite quadratic form of rank r = 1,2, or 3, corresponding to a collapse of the ellipsoid along 2,1, or 0 of its principal axes. When the adiabatic index \gamma determining the pressure law lies in the interval ( 4/3 , 2 ), the rank of this form is 3 and there is a scattering operator, i.e. a bijection between states at minus and plus infinity. However, larger values of \gamma can lead to collapsed asymptotic states of rank 1 or 2.



Department of Mathematics
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