Some exact blowup or global solutions for the non-isentropic Navier–Stokes equations with density-dependent viscosity

摘要

In this paper, we construct some exact solutions for the non-isentropic Navier–Stokes equations with density-dependent viscosity in R N . A class of exact solutions is obtained for γ = θ + 1 N ? 1 : (1) ρ ( t , x → ) = f x 1 + d 1 a ( t ) , x 2 + d 2 a ( t ) , … . , x N + d N a ( t ) a ( t ) N u i ( t , x → ) = a ? ( t ) a ( t ) ( x i + d i ) for i = 1 , 2 , … . , N S ( t , x → ) = - 1 N ln f x 1 + d 1 a ( t ) , x 2 + d 2 a ( t ) , … , x N + d N a ( t ) a ( t ) = t κ N + a 0 with an arbitrary scalar C 1 function f 0; constants a 0 N 0 and d i . In particular, for a 0 0 and the even dimensions, the solutions blow up in the finite time T = ? κNa 0 . For a 0 0, the constructed solutions are global. Here the main contribution is that we free the density function to be an arbitrary positive C 1 functions for the non-isentropic fluids. We note that our new method can work only for the non-isentropic fluids. In addition, the constructed exact solutions are useful for testing numerical methods for the system.