Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

Abstract

We consider the Cauchy problem for the full compressible Navier-Stokes equations with vanishing of density at infinity in R 3 . Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data is more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of R 3 and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first result so far for the global existence of solutions to the full compressible NavierStokes equations when density vanishes at infinity (in space). In addition, the exponential decay rate of the strong solution is of independent interest.

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

“…For the case that the initial density allows vacuum, global existence of weak solutions was first proved in [41,42], see [1,[15][16][17]26] for further developments, but the uniqueness is still an open problem. Local well-posedness of strong solutions was proved in [8][9][10], and the global well-posedness, with small initial data, was proved in [22], and see [21,37,58] for further developments.…”

“…respectively. However, since the explicit dependence of ε 0 on ρ 0 ∞ , θ 0 ∞ , √ ρ 0 θ 0 2 , and ∇u 0 2 are not derived in [21,58], the scaling invariant quantities, on which the smallness guarantees the global well-posedness, can not be identified there.…”

“…Note that the dissipation estimates of the form T 0 ∇u 2 2 ≤ C, which can be guaranteed by the basic energy estimates for the isentropic case, is crucial in the arguments of [22]. To overcome this difficulty, some kinds of dissipative estimates were recovered for the full compressible Navier-Stokes equations in [21] and [58], for the cases that with non-vacuum and vacuum far field, respectively, by using the entropy inequality and the conservation of mass. Notcing that the entropy inequality, one of the keys in [21], holds only for the case that with non-vacuum far field, and the finiteness of the mass is required in [58], and recalling that we consider the case that with vacuum far field and allowing possible infinite mass, the arguments in [21,58] do not work for our case.…”

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.…”

Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D

“…Global existence of strong solutions to the multi-dimensional compressible Navier-Stokes equations, with small initial data, in the presence of initial vacuum, was first proved by Huang-Li-Xin [17] for the isentropic case (see also for further developments), and later by Huang-Li [16] and Wen-Zhu [48] for the nonisentropic case; in a recent work, the author [28] proved the global well-posedness result under the assumption that some scaling invariant quantity is small. Due to the finite blow-up results in [49,50], the global solutions obtained in [16,28,48] must have unbounded entropy if the initial density is compactly supported; however, if the initial density has vacuum at the far field only, one can expect the global entropy-bounded solutions, see the recent work by the author and Xin [30,31].In all the global well-posedness results [1,21,23,24,29,52,53], for the heat conductive compressible Navier-Stokes equations in 1D, the density was assumed uniformly away from vacuum. For the vacuum case, global well-posedness of heat GLOBAL STRONG SOLUTIONS 1D COMPRESSIBLE NAVIER-STOKES 3 conductive compressible Navier-Stokes equations in 1D was proved by with the heat conductive coefficient κ ≈ 1 + θ q , for positive q suitably large, and by the author [27] with κ ≡ Const.The aim of this paper is to study the global well-posedness of strong solutions to the one-dimensional non-heat conductive compressible Navier-Stokes equations, i.e., system (1.1)-(1.3), with constant viscosity, in the presence of vacuum; this is the counterpart of the paper [27] where the heat conductive case was considered.…”