Global Solutions to Isothermal System with Source

1. Introduction

In this paper, we studied the global entropy solutions for the Cauchy problem of isentropic gas dynamics system with source

with bounded measurable initial data

where ρ is the density of gas, u the velocity, P = P(ρ) the pressure. The function b(x, t) corresponds physically to the slope of the topography, α(x, t)ρ|u| to a friction term, where α(x, t) denotes a coefficient function and a(x) is a slowly variable cross section area at x in the nozzle.

The pressure-density relation is P(ρ) =, where γ > 1 is the adiabatic exponent and for the isothermal gas, γ = 1.

System (1) is of interest because it has different physical backgrounds. For the case of nozzle flow without the friction, namely b(x, t) = 0 and α(x, t) = 0, the global solution of the Cauchy problem was well studied (cf. [1, 2, 7, 9] and the references cited therein); When a(x) = 0 and α(x, t) = 0, the source term b(x, t) in System (1) is corresponding to an outer force [3, 8], and when b(x, t) = 0, a(x) = 0, α(x, t)u|u| in (1) corresponds physically to a friction term [5].

In this paper we study the isothermal case P(ρ) = ρ and prove the global existence of weak solutions for the Cauchy problem (1)-(2) for general bounded initial data. The main result is given in the following:

Theorem 1.1. Let P(ρ) = ρ, 0 < a_L ≤ a(x) ≤ M for x in any compact set x ∈ (−L, L), A(x) = − ∈ C¹ (R), 0 ≤ α(x, t) ∈ C¹ (R × R⁺) and |A(x)| + α(x, t) ≤ M, where M, a_L are positive constants, but aL could depend on L. Then the Cauchy problem (1)-(2) has a bounded weak solution (ρ, u) which satisfies system (1) in the sense of distributions

for all test function ϕ ∈ C¹ ₀ (R × R+), and

where (η, q) is a pair of entropy-entropy flux of system (1), η is convex, and ϕ ∈ (R× R⁺ − {t = 0}) is a positive function.

2. Proof of Theorem 1

In this section, we shall prove Theorem 1. Let v = ρa(x), then we may rewrite (1) as

By simple calculations, the two eigenvalues of (5) are

with corresponding Riemann invariants

To prove Theorem 1, we consider the Cauchy problem for the following parabolic system

with initial data

where δ > 0, ε > 0 denote a regular perturbation constant, the viscosity coefficient,

and G^δ is a mollifier.

Then

and

We multiply (8) by (w_v, w_m ) and (z_v, z_m ), respectively, to obtain

and

Letting z = +M t, w = +M t, where M is the bound of |A(x)|+α(x, t), we have from (13)-(14) that

and

Since α(x, t) ≥ 0, using the maximum principle to (15)-(16) (See Theorem 8.5.1 in [6] for the details), we have the estimates on the solutions (v ^δ,ϵ, m ^δ,ϵ ) of the Cauchy problem (8)-(9)

or

where M ₁ is a positive constant depending only on the bounds of the initial date. Therefore we have the following estimates from (17)

which deduce the following positive, lower bound of v ^δ,ϵ , by using the results in Theorem 1.0.2 in [6],

where c(t, δ, ϵ) could tend to zero as the time t tends to infinity or δ, ϵ tend to zero, and M₁(t) is a suitable positive function of t, independent of ε, δ.

With the uniform estimates given in (18) and (19), we may apply the compactness framework in [4] to obtain the pointwise convergence of the viscosity solutions

We rewrite (8) as

Multiplying a suitable test function ϕ to system (22) and letting ϵ go to zero, we can prove that the limit (ρ(x, t), u(x, t)) in (21) satisfies system (1) in the sense of distributions and the Lax entropy condition (3) and (4). So, we complete the proof of Theorem 1.

References

Author notes

xtwang2020@126.com

Additional information

To cite this article: X.T Wang, Global Solutions to Isothermal System with Source, Rev. Integr. temas mat. 39 (2021), No. 1, 51-55. doi: 10.18273/revint.v39n1-2021004

Revista Integración
ISSN: 0120-419X
Vol. 39
Num. 1
Año. 2021

Global Solutions to Isothermal System with Source

Xian-Ting Wang
Wuxi Institute of Technology,China