Dimension reduction for compressible Navier–Stokes equations with density-dependent viscosity

In this paper, we investigate the Navier–Stokes equations describing the motion of a compressible viscous fluid confined to a thin domain Ωε=Iε×(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega _{\varepsilon }=I_{\varepsilon }\times (0, 1)$\end{document}, Iε=(0,ε)⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{ \varepsilon }=(0, \varepsilon )\subset \mathbb{R}$\end{document}. We show that the strong solutions in the 2D domain converge to the classical solutions of the limit 1D Navier–Stokes system as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \to 0$\end{document}.

Although all fluid flows are in general two-dimensional, in many cases the specific shape of the physical domain enforce major changes in the density and velocity only in one direction. A typical example is the fluid flow confined to a thin domain that can be effectively described by using only spatial variable. We consider a family of shrinking domains: where I ⊂ R is a unit interval. Under suitable conditions on the initial data it is natural to expect that the strong solution (ρ ε , u ε ) of (1.1)-(1.2) on Ω ε tends, as ε → 0, to a classical solution (ρ, u) of the 1D system on (0, 1): (1.4) The boundary conditions (1.2) naturally lead to the no-slip boundary conditions for the velocity, i.e., Since we are interested in smooth solutions of the 1D equations, we complement the system of equations (1.4) and (1.5) with the initial conditions Hereinafter we use the notation x = (x 1 , y) ∈ R 2 , y ∈ R and denote the derivative in x 2 by ∂ y .
In this paper we give a rigorous justification of the convergence (ρ ε , u ε ) → (ρ, u) as ε → 0. As far as we know, the limit passage for fluid flows has not yet been rigorously investigated and there is only a handful of results on related problems. Since incompressibility in one dimension does not allow for any movement, such a limit makes little sense for 1D incompressibility flows. However, dimension reduction to 2D-planar flows was examined in [11,[17][18][19]; see also the references given therein.
The case of a compressible barotropic fluid was studied by many authors. Vodák in [22] studied the steady and nonsteady Navier-Stokes system for barotropic compressible flow. For three-dimensional system, Bella, Feireisl and Novotný in [1] considered the motion of a compressible viscous fluid confined to a cavity shaped as a thin rod Ω ε = εQ × (0, 1), Q ⊂ R 2 , they showed that the weak solutions in the 3D domain converge to (strong) solutions of the limit 1D Navier-Stokes system as ε → 0. Březina-Kreml-Mácha in [2] studied the dimension reduction for the full Navier-Stokes-Fourier system in a thin long pipe Ω ε = εQ × (0, 1) ∈ R 3 , where Q is an open rectangular domain in R 2 , they showed that the weak solutions of 3D system on Ω ε tend, as ε → 0, to a classical solution of 1D system on (0, 1). Ducomet-Caggio-Nečasová-Pokorný in [5] investigated the rotating Navier-Stokes-Fourier-Poisson system confined to a straight layer Ω ε = ω × (0, ε), where ω is a 2-D domain, they showed that the weak solutions in the 3d domain converge to the strong solution of the 2-D system on ω → 0 as ε → 0 on the time interval, where the strong solution exists.
In elasticity theory, the analysis of similar dimension reduction problems leans on variants of the Korn inequality which controls the gradient of velocity v by its symmetric part, specifically, Two problems have arisen in (1.7). Firstly, the kernel of the linear operator v → ∇ x v + ∇ t x v, v · n| ∂Ω ε has to be empty, in particular, the "bottom" set I must not be rotationally symmetric. Secondly, for any fixed ε > 0, even if (1.7) holds, the constant c(ε) blows up for ε → 0 unless some necessary restrictions are imposed on the field v, and this is true even if the set I is not rotationally symmetric, cf. the interesting paper by Lewicka and Müller in [14].
Bella, Feireisl and Novotný in [1] obtained their result for a regular planar domain since they avoid the use of Kron's inequality by exploring the structural stability of the family of solutions of the barotropic Navier-Stokes system. It is not difficult to see that the problems arising in the context of compressible fluids would need a stronger analogue of (1.7), namely In view of the above-mentioned difficulties related to the validity of (1.7) or (1.8), our approach relies on the structural stability of the family of solutions of the barotropic Navier-Stokes system encoded in the relative entropy inequality introduced in [6,8]. This method is basically independent of the specific form of the viscous stress and of possible "dissipative" bounds for the Navier-Stokes system.
The paper is organized as follows. In Sect. 2 we introduce the relative entropy inequality and formulate our main result. In Sect. 3, we establish convergence towards the target system (1.4).

Preliminaries, main result
In this section, we first introduce the relative entropy inequality, and then give the solutions of the target systems (1.1) and (1.4), finally, we state our main result.

Relative entropy inequality
The proof of our main theorem is based on the method of the relative entropy (see [8], Dafermos in [3], Germain in [9] and Mellet, Vasseur in [16]). The relative entropy where the potential H(ρ) is defined (modulo a linear function) through along with the relative entropy inequality and the remainder F ε reads Here, the functions r, U are arbitrary smooth, r strictly positive, and U satisfying the noslip boundary conditions (1.5). It is easy to check that (2.3) is satisfied as an equality as soon as solution ρ, u is smooth enough.

Solutions of target systems (1.1) and (1.4)
The existence of global-in-time strong solutions to the two-dimensional Navier-Stokes system (1.1) with complete slip boundary conditions was established by Vaigant and Kazhikhov in [21]. It reads as follows.

Proposition 2.1
Let Ω ⊂ R 2 be a rectangular domain. Assume that If the initial data (ρ 0 , u 0 ) are such that where m 0 and M 0 are some positive constants, and the agreement conditions Remark 2.1 From [1,21] and [6], we know that, for two-dimensional compressible barotropic Navier-Stokes system, the global-in-time solution (ρ ε , u ε ) of Eqs. (1.1) enjoys the regularity specified in Proposition 2.1, the relative entropy inequality in (2.3) is satisfied for ρ = ρ ε , u = u ε and any pair of the test functions and by means of density arguments, the class of test functions can be extended to less regular (r, U).
For one-dimensional compressible Navier-Stokes system (1.4)-(1.6) with no-slip boundary conditions, it has been discussed by many mathematicians. Kazhikhov and Shelukhi in [13] (for polytropic perfect gas with constant viscosity) and Kawohl in [12] (for real gas with μ = μ(ρ)) got global classical solutions for large initial data with inf ρ 0 > 0, respectively. Ding, Wen and Zhu in [4] obtained the global existence of classical solutions to the compressible Navier-Stokes equations in 1D when the initial data are large and the density dependent viscosity. For completeness, we state the proposition as follows.

Main result
We are ready to state our main result.

Proof of Theorem 2.1
Let ρ, ρ > 0 be two positive constants such that Then each measurable function h can be written as We have the following lemmas.

Lemma 3.1 Let ρ, ρ > 0 be two positive constants and let
Then there exists some constant c > 0 independent of ρ ε such that The proof of the lemma can be found in [2,7] and is therefore omitted here for simplicity.
Proof By the definition of H(ρ ε ), we have Then that is, According to Lemma 5.1 in [7] there exists a constant c > 0 such that On the other hand, we know from Lemma 3.1 that which together with (3.2) gives Similarly to the above inequality, one has In conclusion, we obtain which together with (3.1) and (3.2) completes the proof.

Lemma 3.3
Let Ω ε = (0, ε) × (0, 1) and u ε ∈ W 1,2 (Ω ε ) be such that u ε · n = 0 on ∂Ω ε . Then there exists some positive constant c independent of ε such that Proof By straightforward calculation, we can get Thus, using (3.4) and integrating by parts on Ω ε , we have By the definition of S(∇ x u ε ), we get from (3.5) where the (scaled) relative entropy functional and the remainder F ε reads In order to handle the integrals on the right-hand side of (3.9), we proceed in several steps as follows.