Some exact blowup solutions to the pressureless Euler equations in R~N

摘要

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 R~N: {ρ(t,x)=(f(1/(a(t)~5)∑_(i-1)~Nx_i~5))/(a(t)~N), u(t,x)-(a(t))/(a(t))x, a(t)=a_1+a_2t. where an arbitrary function f ≥ 0 and f ∈ C~1; s ≥ 1, a_1 > 0 and a_2 are constants. This new structure of the solutions fully covers the previous well-known one in radial symmetry: ρ(t,r)=(f(r/a(t)))/(a(t)~N),v(t,r)=(a(t))/(a(t))r. In particular, for a_2 < 0, the similar solutions blow up in the finite time T= a_1/a_2. Moreover, the functions (1) are also the solutions to the pressureless Navier-Stokes equations. Our exact solutions could provide the data for testing numerical methods. Alternatively, the exact solutions can be used as a primary estimation of the solutions for the Euler-Poisson equations if some initial conditions are satisfied.