The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential

1 Introduction

In mathematical physics, the Navier-Stokes equation is known as one of the most fundamental equations in fluid mechanics. The compressible isentropic Navier-Stokes equations, which are the basic models describing the evolution of a viscous compressible fluid in a domain x ∈ Ω, read as follows:

where the unknowns ρ, u represent the density and the velocity of the fluid, respectively. Here D(u) = (∇u + (∇u)^T)/2 is the strain tensor and P(ρ) = aρ^γ (a > 0, γ > 1) is the pressure, μ and λ are the viscosity constants which satisfy the physical requirements μ ≥ 0 and 2μ + 3λ ≥ 0. The Navier-Stokes equations are the equations governing the motion of usual fluids like water, air, oil etc., under quite general conditions, and they appear in the study of many important phenomena, either alone or coupled with other equations. For instance, they are used in theoretical studies in aeronautical sciences, in meteorology, in thermo-hydraulics, in the petroleum industry, in plasma physics, etc. From the point of view of continuum mechanics the Navier-Stokes equations are essentially the simplest equations describing the motion of a fluid, and they are derived under a quite simple physical assumption, namely, the existence of a linear local relation between stresses and strain rates. The compressible isentropic Navier-Stokes equations (1.1) are derived from the conservation laws of mass and the balance of momentum, (for details see [1, 2]). While the physical model leading to the Navier-Stokes equations is simple, the situation is quite different from the mathematical point of view. In particular, because of the nonlinearity, the mathematical study of these equations is difficult and requires the full power of modern functional analysis. A major question is whether the solution remains smooth all the time. These and other related questions are interesting not only for mathematical understanding of the equations but also for understanding the phenomenon of turbulence.

There are huge literatures on the study of the existence and behavior of solutions to Navier-Stokes equations. Some of the previous relevant works in this direction can be summarized as follows. For instance, the 1D version of (1) were addressed by Kazhikhov et al. in [3] for sufficiently smooth data, and by Hoff [4] for discontinuous initial data, where the data is uniformly away from the vacuum; the existence of global weak solutions for isentropic flow were investigated by Lions in [5] by using the weak convergence method. In [6], the authors employed a new method to prove the existence and uniqueness of local strong solutions in the case where the initial data satisfies some compatibility conditions. The dynamics of weak solutions and vacuum states were investigated in [7] for the 1D compressible Navier-Stokes equations with density-dependent viscosity in bounded spatial domains or periodic domains. For other results we refer the reader to [8-13] and the references cited therein.

The above references mainly concerned the fluid which the relation between the stress and rate of strain is linear, namely, the Newtonian fluid. The study of non-Newtonian fluids (the relation between the stess and rate of strain is not linear) mechanics is of great significance because such fluids describe more realistic phenomenon. These flows are frequently encountered in many physical and industrial processes [14], such as porous flows of oils and gases [15], biological fluid flows of blood [16], saliva and mucus, penetration grouting of cement mortar and mixing of massive particles and fluids in drug production [17]. Many studies are based on the field of non-Newtonian flows, both theoretically and experimentally. In [18], Ladyzhenskaya first proposed a special form for the incompressible model, namely that the viscous stress tensor Γ_ij = (μ₀ + μ₁|E(∇u)^p−2|)E_ij(∇u). For μ₀ = 0, if p < 2, it is a pseudo-plastic fluid, and when p > 2, it is a dilatant fluid (see also [19]). From a Physics point of view, the model captures the shear thinning fluid for the case of 1 < p < 2, and captures the shear thickening fluid for the case of p > 2. In [20], the trajectory attractor and global attractor for an autonomous non-Newtonian fluid in dimension two was studied. The existence and uniqueness of solutions for non-Newtonian fluids were established in [21] by applying the Ladyzhenskaya’s viscous stress tensor model. Then the global existence and exponential stability of solutions to the one-dimensional full non-Newtonian fluids were investigated in [22]. Recently, in [23], the authors present a decoupling multiple-relaxation-time lattice Boltzmann flux solver for simulating non-Newtonian power-law fluid flows.

On the other hand, another basiclly important case is when the motion of compressible viscous isentropic flow is driven by a self-gravitational force. However, at the present, little is known yet on the strong solutions to system (1) with non-Newtonian self-gravitational potential on bounded domain, even in one dimensional case. In this paper we focus on the following 1D system of compressible non-Newtonian equations

in Ω_T = Ω × (0, T) with the initial and boundary conditions

Here, ρ, u, Φ denote the density, velocity and the non-Newtonian gravitational potential, respectively. P = aρ^γ (a > 0, γ > 1) is the pressure, the initial density ρ0≥0,43<p<2,q>2 are given constants. Our purpose is to some further light on problem (2)-(4). When 1 < p < 2, the second equation of (2) always has singularity. Secondly, we emphasize that the vacuum of initial density may exist. In the presence of vacuum, the parabolicity is lost. Moreover, the equations are strongly coupled with each other. We will investigate the existence and uniqueness of local strong solutions of (2)-(4) by overcoming the above difficulties, and the proof is inspired by the previous work in [6] and [21].

We state the main results as follows:

The rest of the paper is organized as follows. After stating the notations, in Section 2, we first present some useful lemmas, then the analysis of a priori estimates for smooth solutions are derived. In Section 3, we give the proof of existence, and finally complete the proof of uniqueness of the main theorem in Section 4.

In what follows, we use the following abbreviations for simplicity of notation:

Throughout this paper, we will omit the variables t, x of functions if it does not cause any confusion. We use C to denote a generic constant that may vary in different estimates.

2 A priori estimates for smooth solutions

In this section, we provide some known facts that will be used in the proof of the main result.

To prove the existence of strong solutions, we require some more regular estimates. Next, we derive some a priori estimates for smooth solutions which are crucial to prove the local existence of strong solutions.

Let (ρ, u, Φ) be a smooth solution of (2)-(4) and ρ₀ ≥ δ, where 0 < δ ≪ 1 is a positive number, m₀ := ∫_Ωρ₀(x)dx be initial mass and m₀ > 0. Throughout the paper, we will denote

As stated above, we need to estimate the uniform bound of the approximate solutions. Now, we consider the following linearized problem

with the initial and boundary conditions

where

and u₀ ∈ H01 ∩ H² is the smooth solution of the boundary value problem

We will prove the existence of solutions for (5)-(9) by virtue of the uniform estimates which do not depend on ε, depending only on M₀, and prove the limit of the approximate solutions is the solution of problem (2)-(4) with vacuum.

It follows from (10) and Young’s inequality that there exists a constant C depending only on M₀, such that

We construct an auxiliary function

Our derivation will be based on the local boundedness of Ψ(t). Before we estimate each term of Ψ, we need to do the estimate of |u_xx|_L². Firstly, multiplying (7) by Φ and integrating over Ω and using Young’s inequality,

Then, it follows from equation (6) and (5) that

Taking it by L²-norm, using Young’s inequality, we obtain

which along with (11), implies that

3 Proof of the existence

Since a priori estimates for higher regularity have been derived, the existence of strong solutions can be established by a standard argument, in the case of bounded domains, we construct approximate solutions via a semi-discrete Galerkin scheme, derive uniform bounds and thus obtain solutions by passing to the limit. Our method that constructed approximate systems is similar to that in [6]. To implement a semi-discrete Galerkin scheme, we take our basic function space as X = H01(Ω) ∩ H²(Ω) and its finite-dimensional subspaces as X = span{^,1, ^,2, Π, ^,m} ⊂ X ∩ C²(Ω). Hence φ^m is the mth eigenfunction of the general elliptic operator defined on X.

Let ρ₀, u₀, Φ₀, and f be functions satisfying the hypothes of theorem, assume for the moment that ρ0δ ∈ C(Ω) and ρ₀δ≥δ in Ω for some constant δ > 0. We can construct an approximate solution for any v ∈ X^m, φ ∈ C²(Ω), such that

The initial and boundary conditions are

Under the hypotheses of the theorem, similarly, for any fixed δ > 0, we may get the similar estimate

Since C does not depend on ε, δ and m (for any m > M, M is dependent on the approximate velocity of the initial condition). We can deduce from the uniform estimate(24) that (ρ^m, u^m, Φ^m) converges, up to an extraction of subsequences. Let m → ∞. We obtain the following estimates in the obvious weak sense

For each small δ > 0, ρ0δ = J_δ * ρ₀ + δ is a mollifier on Ω, and u0δ∈H01(Ω)∩H2(Ω) is the unique solution of the boundary value problem

where gδ∈C0∞,and |gδ|L2(Ω)≤|g|L2(Ω),limδ→0+|gδ−g|L2=0.

We deduce that (ρ^δ, u^δ, Φ^δ) is a solution of the following initial and boundary value problem

where 43<p<2,q>2.

Together with Lemma 2.2, there is s subsequence {uεj,δ} of {uε,δ}, as εj→0,{u0εj,δ}→u0δ in H01(Ω) ∩ H²(Ω). Also there is a subsequence {u0δj} of {u0δ}, such that as δj→0+,u0δj→u0 in H01(Ω)∩H2(Ω). With ρ0δ = J_δ * ρ₀ + δ, we have, as δj→0+,Px(ρ0δj)−(ρ0δj)12gδj→Px(ρ0)−(ρ0)12g in L2(Ω). By (25), there exists a subsequence {u0δj} of {u0δ}, such that as δj→0+,(|u0xδj|p−2u0xδj)x→(|u0x|p−2u0x)x in L2(Ω). Hence, u₀ ∈ H01(Ω) ∩ H²(Ω) satisfies equation −(|u0x|p−2u0x)x+Px(ρ0)=(ρ0)12g for a.e. x ∈ Ω and is a unique solution. According to above uniform estimates, by the lower semi-continuity of various norms, we have

where C is a positive constant depending only on M₀.

4 Proof of the uniqueness

We now prove the uniqueness results. Let (ρ, u, Φ), (ρ̄, ū, Φ̄) be two solutions of the problem (2)-(4). Combining (2)₁ and (2)₂,

we show that