Some exact blowup solutions to the pressureless Euler equations in RN

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: equation(1)ρ(t,x→)=f1a(t)s∑i=1Nxisa(t)N,u→(t,x→)=a˙(t)a(t)x→,a(t)=a1+a2t,where an arbitrary function f ⩾ 0 and f ∈ C1; s ⩾ 1, a1 > 0 and a2 are constants.This new structure of the solutions fully covers the previous well-known one in radial symmetry: equation(2)ρ(t,r)=f(r/a(t))a(t)N,V(t,r)=a˙(t)a(t)r.In particular, for a2 < 0, the similar solutions blow up in the finite time T = −a1/a2.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.

Research highlights
► Some exact solutions are given in non-radial symmetry to the pressureless Euler equations in RN.
► This new structure of the solutions fully covers the previous well-known one in radial symmetry.
► The exact solutions could provide the data for testing numerical methods.