Some Exact Blowup Solutions to the Pressureless Euler Equations in Rⁿ

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The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in Rⁿ:% [c]c% ρ(t,x)=f(1a(t)ˢi=1 N∑xᵢˢ)a(t)ⁿ,u(t,x%)=·a(t)a(t)x, a(t)=a₁+a₂t. eq234% where the arbitrary function f≥0 and f∈ C¹; s≥1, a₁>0 and a₂ are constants. In particular, for a₂<0, the solutions blow up on the finite time T=-a₁/a₂. Moreover, the functions (eq234) are also the solutions to the pressureless Navier-Stokes equations.