On global classical solutions to 1D compressible Navier‐Stokes equations with density‐dependent viscosity and vacuum

Abstract

For the initial boundary value problem of compressible barotropic Navier-Stokes equations in one-dimensional bounded domains with general density-dependent viscosity and large external force, we prove that there exists a unique global classical solution with large initial data containing vacuum. Furthermore, we show that the density is bounded from above independently of time, which yields the large time behavior of the solutions as time tends to infinity. More precisely, the density and the velocity converge to the steady states in L p and in W 1,p (1 ≤ p < +∞), respectively. Moreover, the decay rate in time of the solutions is shown to be exponential. Finally, we also prove that the spatial gradient of density will blow up as time tends to infinity when the vacuum states appear initially even at one point. KEYWORDS compressible Navier-Stokes equations, density-dependent viscosity, global classical solutions, large-time behavior, vacuum MSC CLASSIFICATION 35Q35; 35A09; 35B45; 35B65

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

“…whereC is independent of δ and η. Moreover, similar to [17], we can prove that there exists someC depending on δ and η such that…”

“…In the presence of vacuum, Ding-Wen-Zhu [4] considered the global existence of classical solutions to 1D compressible Navier-Stokes equations in bounded domains, provided that µ ∈ C 2 [0, ∞) satisfies 0 <μ ≤ µ(ρ) ≤ C(1 + P (ρ)). (1.4) Recently, for general µ, we [17] establish not only the global existence but also the large-time behavior for classical solutions containing vacuum to the initial boundary value problem for 1D compressible Navier-Stokes equations. For the Cauchy problem (1.1)-(1.3) without external force (f = 0), Ye [23] studies the global classical large solutions under the following restriction on µ(ρ): µ(ρ) = 1 + ρ β , 0 ≤ β < γ.…”

“…As mentioned in many papers (see [7,12]), the main difficulty comes from initial vacuum, density-dependent viscosity, large external force, and the unboundedness of the domain. Motivated by our previous result [17], we find that the key issue is to derive both the time-independent upper bound of the density and the time-depending derivative ones of the solutions. Motivated by [11], we localize the problem on bounded domains and use the method develop in our previous work [17] to get the upper bound of the density independently of x and thus the uniform bound of density on the whole R due to the arbitrariness of x (see Lemma 2.3).…”

“…Motivated by our previous result [17], we find that the key issue is to derive both the time-independent upper bound of the density and the time-depending derivative ones of the solutions. Motivated by [11], we localize the problem on bounded domains and use the method develop in our previous work [17] to get the upper bound of the density independently of x and thus the uniform bound of density on the whole R due to the arbitrariness of x (see Lemma 2.3). Furthermore, with the help of far field behavior of ρ, we can bound the L 2 -norm of u in terms of ρ 1/2 u L 2 and u x L 2 (see (2.23)).…”

Blow‐up criterion for the modified compressible Navier–Stokes equations with density‐dependent viscosity

“…When the viscosity is constant, the free boundary problems for one-dimensional CNS equations were investigated in [9][10][11][12] (also see [13][14][15] for the Cauchy problem) and among others, where the global existence of weak solutions was proved. The case where the viscous gas is in contact with the vacuum when the viscosity depends on the density was considered in [16,17]. Also, a local existence theorem was obtained by Liu et al and Makino (see [18,19]), where the initial density was assumed to be connected to vacuum with discontinuities and viscosity depending on density.…”