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Global existence of a classical solution for the isentropic nozzle flow

Shih-Wei Chou, Bo-Chih Huang, Yun-guang Lu, Naoki Tsuge

Abstract

Our goal in this paper is to prove the global existence of a classical solution for the isentropic nozzle flow. Regarding this problem, there exist some global existence theorems of weak solutions. However, that of classical solutions does not have much attention until now. When we consider the present problem, the main difficulty is to obtain the uniform bound of solutions and their derivatives. To solve this, we introduce an invariant region depending on the space variable and a functional satisfying the Riccati equation along the characteristic lines.

Global existence of a classical solution for the isentropic nozzle flow

Abstract

Our goal in this paper is to prove the global existence of a classical solution for the isentropic nozzle flow. Regarding this problem, there exist some global existence theorems of weak solutions. However, that of classical solutions does not have much attention until now. When we consider the present problem, the main difficulty is to obtain the uniform bound of solutions and their derivatives. To solve this, we introduce an invariant region depending on the space variable and a functional satisfying the Riccati equation along the characteristic lines.
Paper Structure (3 sections, 7 theorems, 48 equations, 3 figures)

This paper contains 3 sections, 7 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Invariant region
  3. Uniform bound of $z_x$ and $w_x$

Key Result

Theorem 1.1

We assume that Then, if we choose $M$ in eqn:H1 large enough, the initial boundary problem (P1) has a time global classical solution.

Figures (3)

  • Figure 1: $\Delta_{m,x}$ in $(z,w)$-plane
  • Figure 2: $\Delta_{r,x}$ in $(z,w)$-plane
  • Figure 3: $\Delta_{l,x}$ in $(z,w)$-plane

Theorems & Definitions (14)

  • Definition 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • ...and 4 more