On scaling invariance and type-Ⅰ singularities for the compressible Navier-Stokes equations
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摘要:
We find a new scaling invariance of the barotropic compressible Navier-Stokes equations.Then it is shown that type-Ⅰ singularities of solutions with lim sup t↗T|divu(t,x)|(T-t) ≤ κ,can never happen at time T for all adiabatic number γ ≥ 1.Here κ > 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under ρ(t,x) ≤M/(T-t)κ,M < ∞.This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.