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 $\Vert {\rho }_0{\Vert }_{\infty }(\Vert {\rho }_0{\Vert }_3+\Vert {\rho }_0{\Vert }_{\infty }^2\Vert \sqrt{{\rho }_0}{u}_0{\Vert }_2^2)(\Vert \nabla u_0{\Vert }_2^2+\Vert {\rho }_0{\Vert }_{\infty }\Vert \sqrt{{\rho }_0}{E}_0{\Vert }_2^2)$ is sufficiently small, with the smallness depending only on the parameters $R,\gamma ,\mu ,\lambda ,$ and $\kappa$ in the system. The total mass can be either finite or infinite.

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Cited by in corpus (7)

• Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum
• Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier-Stokes equations with far field vacuum
• Local existence and uniqueness of heat conductive compressible Navier-Stokes equations in the presence of vacuum and without initial compatibility conditions
• Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum
• Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes equations
• Optimal decay of compressible Navier-Stokes equations with or without potential force
• Propagation of uniform boundedness of entropy and inhomogeneous regularities for viscous and heat conductive gases with far field vacuum in three dimensions