Global Entropy Solutions to the Gas Flow in General Nozzle

Abstract

We are concerned with the global existence of entropy solutions for the compressible Euler equations describing the gas flow in a nozzle with general crosssectional area, for both isentropic and isothermal fluids. New viscosities are delicately designed to obtain the uniform bound of approximate solutions. The vanishing viscosity method and compensated compactness framework are used to prove the convergence of approximate solutions. Moreover, the entropy solutions for both cases are uniformly bounded independent of time. No smallness condition is assumed on initial data. The techniques developed here can be applied to compressible Euler equations with general source terms. 2010 AMS Classification: 35L45, 35L60, 35Q35.

The global existence of L∞$L^{\infty }$ solutions to isentropic Euler equations in general nozzle

“…When $$ \gamma \ge 3$$, the technique introduced in [1] to obtain the a priori $$L^{\infty }$$ estimates of viscosity solutions does not work because the necessary conditions $$a_{12} \le 0$$ and $$a_{21} \le 0$$, to guarantee the maximum principle (cf. Lemma 3.1 in [1]), are not true.…”

“…In this paper, we apply our method introduced in [18] to give a simple proof of the global existence of the entropy solutions for general nozzle and to extend the results of [1] for any adiabatic exponent $$ \gamma >1$$.…”

“…In [1], the authors introduced the following approximate system, which is different from the viscosity method introduced in [18], $$$$\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{l}\rho _{t}+(\rho u)_{x} = - \frac{a^{\prime }(x)}{a(x)} \rho u + \varepsilon \rho _{xx} \\[9pt] (\rho u)_t+ (\rho u^2+ P(\rho ))_x = - \frac{a^{\prime }(x)}{a(x)} \rho u^{2} + \varepsilon (\rho u)_{xx} - 2 \varepsilon b(x) \rho _{x}, \end{array} \right.} \end{equation}$$$$to study the general nozzle for more general gases $$ 1 \le \gamma \le 3$$.…”

Soluciones globales para sistema isotérmico con fuente

“…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].…”

“…where M, a L are positive constants, but a L 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…”