Global Small Solutions of Heat Conductive Compressible Navier–Stokes Equations with Vacuum: Smallness on Scaling Invariant Quantity

Abstract

In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions, that the scaling invariant quantity ρ 0 ∞ ( ρ 0 3 + ρ 0

Local well-posedness to the 2D Cauchy problem of full compressible magnetohydrodynamic equations with vacuum at infinity

“…Yet the well-posedness theory of multi-dimensional problems becomes much more complicated and challenging in the presence of vacuum (that is, the flow density is zero) despite its fundamental importance both physically and theoretically in understanding the behaviors of the solutions of compressible viscous flows. Substantial difficulties arise due to the strong degeneracy of the hyperbolic-parabolic system in the case where a vacuum state appears (see [3,16,22,25,29,30,38,40]).…”

Global strong solution for 3D compressible heat-conducting magnetohydrodynamic equations revisited

“…Wen and Zhu [24] showed global existence of strong solutions with far-field vacuum under the condition that the initial mass is properly small in certain sense. Meanwhile, Li [12] obtained a new type of global strong solutions under some smallness condition on the scaling invariant quantity. Very recently, Liang [16] established global strong solutions when the initial energy is small.…”

Entropy bounded solutions to the one-dimensional compressible Navier-Stokes equations with zero heat conduction and far field vacuum

“…Local well-posedness of strong solutions was proved in [4][5][6]34]. Global existence of strong solutions, of small energy but allowing large oscillations and vacuum, was first proved by Huang-Li-Xin [16] for the isentropic case, and generalized later by the authors in [15,24,39] for the full case.…”