Global solutions to isothermal system in a divergent nozzle with friction

Applied Mathematics Letters 84 (2018) 176–180

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Global solutions to isothermal system in a divergent nozzle with friction Yun-guang Lu* K.K.Chen Institute for Advanced Studies, Hangzhou Normal University, PR China Universidad Industrial de Santander, Colombia

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Article history: Received 27 March 2018 Received in revised form 3 May 2018 Accepted 3 May 2018 Available online 29 May 2018

In this paper, we remove the restriction A′ (x) ≥ 0 in the paper ‘Lu (2011)’, the restriction z0 (x) ≤ 0 or w0 (x) ≤ 0 in the paper ‘Klingenberg and Lu (1997)’, and obtain the global existence of entropy solutions to the isothermal gas dynamics system in a divergent nozzle with friction. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Global L∞ solution Isothermal system Friction terms Flux approximation Compensated compactness

1. Introduction In this paper we studied the global entropy solutions for the Cauchy problem of isentropic gas dynamics system in a divergent nozzle with a friction, whose physical phenomena called “choking or choked flow”, occurs in the nozzle ⎧ a′ (x) ⎪ ⎪ ⎨ρt + (ρu)x = − a(x) ρu (1.1) ⎪ a′ (x) 2 ⎪ ⎩(ρu)t + (ρu2 + P (ρ))x = − ρu − αρu|u|, a(x) with bounded initial date (ρ(x, 0), u(x, 0)) = (ρ0 (x), u0 (x)),

ρ0 (x) ≥ 0,

(1.2)

where ρ is the density of gas, u the velocity, P = P (ρ) the pressure, a(x) is a slowly variable cross section area at x in the nozzle and α denotes a friction constant. For the polytropic gas, P takes the special form P (ρ) = γ1 ργ , where γ > 1 is the adiabatic exponent and for the isothermal gas, γ = 1. System (1.1) is of

*

Correspondence to: K.K.Chen Institute for Advanced Studies, Hangzhou Normal University, PR China. E-mail address: [email protected].

https://doi.org/10.1016/j.aml.2018.05.006 0893-9659/© 2018 Elsevier Ltd. All rights reserved.

Y.-g. Lu / Applied Mathematics Letters 84 (2018) 176–180

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interest because resonance occurs. This means there is a coincidence of wave speeds from different families of waves (See [1–4] and the references cited therein for the details). By simple calculations, two eigenvalues of system (1.1) are m √ ′ m √ ′ (1.3) λ1 = − P (ρ), λ2 = + P (ρ) ρ ρ with corresponding Riemann invariants ∫ ρ√ ′ P (s) m z(u, v) = ds − , s ρ 0

∫ w(u, v) = 0

ρ

√

P ′ (s) m ds + , s ρ

(1.4)

where m = ρu. When a′ (x) = 0, (1.1) is the river flow equations, a shallow-water model describing the vertical depth ρ and mean velocity u, where αρu|u| corresponds physically to a friction term, and its global, bounded solutions were obtained in [5]. When α = 0, i.e., the nozzle flow without friction, system (1.1) was well studied in [6–8] for the polytropic gas (γ > 1), and in [9] for the isothermal gas (γ = 1). In [9], a strong technique condition A′ (x) ≥ 0 was imposed when the author used the maximum principle to study the positive lower bound of the density ρ (the bound depends on the viscosity coefficient ε). Since the super-linear source terms in (1.1), when we prove the global existence, the main difficulty is to obtain L∞ estimates, of viscosity solutions, independent of the viscosity perturbation constant ε. With the help of the condition z0 (x) ≤ 0 or w0 (x) ≤ 0, the L∞ bound of (ρε , mε ) was obtained in [10,11] for the polytropic gas γ > 1. Since the case of γ = 1 is different from that of γ > 1, in this paper, we remove the conditions z0 (x) ≤ 0 or w0 (x) ≤ 0 in [10,11], and A′ (x) ≥ 0 for the isothermal case P (ρ) = ρ in [9], and prove the global existence of weak solutions for the Cauchy problem (1.1)–(1.2) for general bounded initial date. The main result is given in the following Theorem 1.1. Let P (ρ) = ρ, 0 < aL ≤ a(x) ≤ M for x in any compact set x ∈ (−L, L), A(x) = ′ (x) ∈ C 1 (R) and |A(x)| ≤ M , where M, aL are positive constants, but aL could depend on L. Then the − aa(x) Cauchy problem (1.1)–(1.2) has a bounded weak solution (ρ, u) which satisfies system (1.1) in the sense of distributions and ∫ ∞∫ ∞ η(ρ, m)ϕt + q(ρ, m)ϕx + A(x)(η(ρ, m)ρ ρu + η(ρ, m)m ρu2 )ϕdxdt ≥ 0, (1.5) 0

−∞

where (η, q) is a pair of entropy–entropy flux of system (1.1), η is convex, and ϕ ∈ C0∞ (R × R+ − {t = 0}) is a positive function. Remark 1.2. The global existence of symmetrical weak solutions of the isothermal gas dynamics system (1.1) without a friction in the Lagrangian coordinates was well studied in [12–14] by using the Glimm scheme method [15,16]. Remark 1.3. The homogeneous case of isothermal system (1.1) (a′ (x) = 0, α = 0) in the Euler coordinates was studied in [17] by using the compensated compactness theory [18]. 2. Proof of Theorem 1.1 Let v = ρa(x), then we may rewrite (1.1) as { vt + (vu)x = 0 (vu)t + (vu2 + v)x + A(x)v + αvu|u| = 0.

(2.1)

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The two eigenvalues of (2.1) are λ1 = u − 1,

λ2 = u + 1

(2.2)

with corresponding Riemanna invariants m m , w(v, m) = ln v + , m = vu. v v Now we consider the Cauchy problem for the related parabolic system with a flux approximation { vt + ((v − 2δ)u)x = εvxx (vu)t + ((v − δ)u2 + v − 2δ ln v)x + A(x)v + αvu|u| = ε(vu)xx , z(v, m) = ln v −

(2.3)

(2.4)

with initial data (v(x, 0), v(x, 0)u(x, 0)) = (v0δ (x), v0δ (x)uδ0 (x)),

(2.5)

where δ > 0, ε > 0 denote a regular perturbation constant, the viscosity coefficient, (v0δ (x), uδ0 (x)) = (a(x)ρ0 (x) + 2δ, u0 (x)) ∗ Gδ

(2.6)

and Gδ is a mollifier. Then the hyperbolic part of (2.4) has the same Riemann invariants and the entropies like that of system (1.1) (See [19] for the details). Moreover, (v0δ (x), uδ0 (x)) ∈ C ∞ (R) × C ∞ (R),

(2.7)

v0δ (x) ≥ 2δ,

(2.8)

and v0δ (x) + |uδ0 (x)| ≤ M.

We multiply (2.4) by (wv , wm ) and (zv , zm ), respectively, to obtain zt + λδ1 zx − A(x) − αu|u| = εzxx − ε(zvv vx2 + 2zvm vx mx + zmm m2x )

(2.9)

εv 2 2ε 2ε = εzxx + vx zx − 2x ≤ εzxx + vx zx v v v and wt + λδ2 wx + A(x) + αu|u| = εwxx − ε(wvv vx2 + 2wvm vx mx + wmm m2x )

(2.10)

2ε εv 2 2ε = εwxx + vx wx − 2x ≤ εwxx + vx wx , v v v v−2δ δ where λδ1 = u − v−2δ v , λ2 = u + v . Letting z = z¯ + M t, w = w ¯ + M t, where M is the bound of |A(x)|, we have from (2.9)–(2.10) that

z¯t + λδ1 z¯x + α|u|(¯ z − w) ¯ ≤ ε¯ zxx +

2ε vx z¯x v

(2.11)

2ε vx w ¯x . v

(2.12)

and w ¯t + λδ1 w ¯x + α|u|(w ¯ − z¯) ≤ εw ¯xx +

Using the maximum principle to (2.11)–(2.12) (See the proof of Lemma 2.4 in [20] for the details), we have the estimates on the solutions (v δ,ε , mδ,ε ) of the Cauchy problem (2.4)–(2.5) z¯(v δ,ε , mδ,ε ) ≤ M1 ,

w(v ¯ δ,ε , mδ,ε ) ≤ M1

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or z(v δ,ε , mδ,ε ) ≤ M1 + M t = M (t),

w(v δ,ε , mδ,ε ) ≤ M1 + M t = M (t),

(2.13)

where M1 is a positive constant depending only on the bounds of the initial date. Since the initial date v0δ (x) ≥ 2δ, from the first equation in (2.4), we have the a priori estimate v δ,ε ≥ 2δ. The estimates in (2.13) give us the following L∞ bound 2δ ≤ v δ,ε ≤ M1 (t),

ln v δ,ε − M (t) ≤ uδ,ε ≤ M (t) − ln v δ,ε ,

|mδ,ε | ≤ M2 (t),

(2.14)

where Mi (t), i = 1, 2 are two suitable positive functions of t, independent of ε. By applying the general contracting mapping principle to an integral representation of (2.4), we can first obtain a local solution of the Cauchy problem (2.4)–(2.5), then using the lower, positive estimate and the L∞ estimates given in (2.14), we can obtain the existence and uniqueness of smooth solution of the Cauchy problem (2.4)–(2.5) (the details can be found in [21] ). Finally applying the convergence frame given in [17] we have the pointwise convergence (v δ,ε (x, t), mδ,ε (x, t)) → (v(x, t), m(x, t)) a.e., as ε, δ → 0

(2.15)

(ρε,δ (x, t), (ρε,δ uε,δ )(x, t)) → (ρ(x, t), (ρu)(x, t)) a.e., as ε, δ → 0.

(2.16)

or

We rewrite (2.4) as ⎧ a′ (x) a′ (x) a′′ (x) u ⎪ ⎪ ⎨ρt + (ρu)x = − a(x) ρu + ερxx + 2ε a(x) ρx + ε a(x) ρ + 2δ a(x) ⎪ a′ (x) 2 a′ (x) a′′ (x) ln(aρ) ⎪ ⎩(ρu)t + (ρu2 + ρ)x = − ρu + ε(ρu)xx + 2ε (ρu)x + ε (ρu) + 2δ , a(x) a(x) a(x) a(x) ′

(2.17)

′′

(x) (x) where, for simplicity, the indexes ε, δ are dropped. Since aa(x) and aa(x) are bounded for x in any compact support set (−L, L), then multiplying a suitable test function ϕ to system (2.17), we can prove that the limit (ρ(x, t), u(x, t)) in (2.16) satisfies system (1.1) in the sense of distributions and the Lax entropy condition (1.5). So, we complete the proof of Theorem 1.1.

Acknowledgments This paper is partially supported by a Qianjiang professorship of Zhejiang Province of China, a research project from UIS, Bucaramanga, Colombia and a Humboldt fellowship of Germany. References

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