1. Introduction

In the paper [²] the authors applied the viscosity-flux approximation method coupled with the maximum principle to obtain the a-priori time-independent L^∞ estimates for the approximation solutions of the following compressible polytropic gas dynamics system with an outer force:

(1)

with bounded measurable initial data

(2)

where the pressure function takes P = ρ^γ, γ > 1 denotes the polytropic exponent, ρ is the density of gas and u is the velocity. The source α(x, t) is a function of the space variable x and the time variable t. Based on the compactness framework from the compensated compactness theory [¹¹], [⁹], [⁴], [⁵], the pointwise convergence of the viscosity-flux approximation solutions and the global existence of the bounded entropy solutions for the Cauchy problem (1)-(2) were proved under the condition

(3)

where T(t) ∈ L¹ (0, +∞), X(x) ∈ L¹ (−∞, +∞).

In this paper, we study the isothermal gas, which is corresponding to the case of γ = 1. System (1) first appeared in [¹²]. When α(x, t) = 0, γ = 1, the study of the homogeneous system

(4)

has a long history. Under the Lagrangian coordinates, system (4) is equivalent to the system

(5)

where v = is the specific volume.

The first large data existence theorem for the Cauchy problem (5) with locally finite total variation initial data away from the vacuum (v = ∞) was obtained in [¹⁰] by using the Glimm’s scheme method (cf. [¹]). The ideas of compensated compactness developed in [⁹], [¹¹] were used in [³] to establish a global existence theorem for the Cauchy problem (4) with arbitrary large initial data including the vacuum (ρ = 0), with the use of the viscosity method.

To study the Cauchy problem (1)-(2), following [⁶], we consider the parabolic system

6)

with initial data

(7)

where δ > 0 denotes a regular perturbation constant, and ϵ > 0 is the viscosity coefficient. We mainly have the following theorem in this paper,

Theorem 1.1. I. Suppose α(x, t) is a suitable smooth function and it satisfies (3), where |X(x)|_L1(R) + |T(t)|_L1(R+) ≤ . Let the initial data (2) satisfy

(8)

where

(9)

are the Riemann invariants of (4)and M > 1 is a constant.Then, for fixed ϵ, δ, the Cauchy problem (6) and (7)has a global solution (ρ^ϵ,δ, u^ϵ,δ), ρ^ϵ,δ ≥ 2δ > 0, satisfying

(10)

II. There exists a subsequence of (ρ^ϵ,δ(x, t), u^ϵ,δ(x, t)), which converges pointwisely to a pair of bounded functions (ρ(x, t), u(x, t)) as δ, ϵ tend to zero, and the limit is a weak entropy solution of the Cauchy problem (1)-(2).

Remark. To construct the approximate solutions of the Cauchy problem (1)-(2), besides adding the classical viscosity parameter ϵ to the right-hand side of (1), we approximate the flux by adding a term depending on another parameter δ > 0 which vanishes when δ = 0. An obvious advantage of this kind of approximation on the flux functions is to obtain the positive lower bound ρ ≥ 2δ > 0, directly from the first equation in (6), which grantees that the term ρu² = is well defined. Moreover, for any weak entropy-entropy flux pair (η(ρ, m), q(ρ, m)) of system (4), we can easily prove that

(11)

with respect to the viscosity solutions (ρ^ϵ,δ, m^ϵ,δ) (cf. [⁷]).

2. Proof of Theorem 1.1

In this section, we shall prove Theorem 1.1. Multiplying (6) by and , respectively, we have

(12)

and

(13)

where

(14)

are two eigenvalues of the approximation system (4), and m = ρu denotes the momentum and (w, z) is given by (9).

Letting

(15)

where T(t), X(x) are given by (3), and z = B(x, t) + v, we have from (13) that

(16)

or

(17)

where ϵ₁ > 0 is a suitable small constant, a(x, t) = u − ρ_x and b(x, t) = X(x). Similarly, letting

(18)

and w = C(x, t) + s, we have from (2.1) that

(19)

where c(x, t) = u + ρ_x and d(x, t) = X(x).

First, we can choose ϵ = o(ϵ₁) (if necessary, we may smooth X(x) by a mollifier as the authors did in [²]) such that the following three terms on the left-hand side of (17) and (19)

(20)

and

(21)

Second, by using the maximum principle to the first equation in system (6), we have the a-priori estimate ρ^ϵ,δ ≥ 2δ.

Finally, by the calculations, we may obtain the some inequalities about the variables v and s.

Lemma 2.1.

(22)

where

(23)

(24)

(25)

and

(26)

Proof of Lemma 2.1. First, at the points (x, t) where ρ ≤ 1, the following terms on the left-hand side of (17)

(27)

Second, at the points (x, t) where ρ > 1, we have ≥ 1 − ϵ₂ > 0 for a small ϵ₂ > 0, and X(x) ln ρ = X(x)( (v + s) + M). Then,

(28)

Therefore, the first inequality in (22) is proved. Similarly, we may prove the second inequality in (22). Lemma 2.1 is proved.

Since b₂(x, t) ≤ 0, d₂(x, t) ≤ 0, under the conditions given in (8), it is clear that v(x, 0) ≤ 0, s(x, 0) ≤ 0, so, we may apply the maximum principle (cf. Lemma 2.4 given in [⁸]) to (22) to obtain the estimates v(x, t) ≤ 0, s(x, t) ≤ 0, and so the estimates in (10).

From the estimates in (10), we can use the Riemann invariants (9) to obtain the following uniformly bounded estimates on (ρ^ϵ,δ(x, t), m^ϵ,δ(x, t)) directly:

(29)

for a suitable positive constant M₁, which is independent of ϵ, δ and the time t. The existence result of the Cauchy problem (6)-(7) follows by applying the standard theory on the parabolic systems. Part I of Theorem 1.1 is proved. Furthermore, we may select a subsequence of (ρ^ϵ,δ(x, t), m^ϵ,δ,µ(x, t)), which converges pointwisely to a pair of bounded functions (ρ(x, t), m(x, t)) as δ, ϵ tend to zero by using the compactness framework given in [³], and prove that the limit (ρ(x, t), m(x, t)) is an entropy solution of the Cauchy problem (1)-(2), so Theorem 1.1 is proved.