Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

Pointwise wave behavior of the non-isentropic Navier–Stokes equations in half space

Pages: 795 – 827

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n3.a8

Authors

Hai-Liang Li (School of Mathematical Sciences and Academy for Multidisciplinary Studies, Capital Normal University, Beijing, China)

Hou-Zhi Tang (School of Mathematical Sciences and Academy for Multidisciplinary Studies, Capital Normal University, Beijing, China)

Hai-Tao Wang (School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSE and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China)

Abstract

In this paper, we aim to study the global well-posedness and pointwise behavior of the classical solution to one-dimensional non-isentropic compressible Navier–Stokes equations in half space. Based on $H^s$ energy method, we first establish the global existence and uniqueness. To derive the accurate pointwise estimate of the solution, Green’s function for the initial boundary value problem is investigated. It is shown that Green’s function can be expressed in terms of a fundamental solution to the Cauchy problem. Then applying Duhamel’s principle and nonlinear analysis yields the space-time estimate of the solution under some suitable assumptions on the initial data, which exhibits the rich wave structure. As a corollary, we prove that the solution converges to the equilibrium state at an algebraic time decay rate $(1+t)^{-1/2}$ in $L^\infty$ norm with respect to the spatial variable.

Keywords

Navier–Stokes equations, pointwise estimate, Green’s function, half space, nonisentropic

2010 Mathematics Subject Classification

35B40, 35M13, 76N10

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This work is partially supported by National Key R&D Program of China (No. 2022YFA1007300).

The work of the first author and the second author are supported by the National Natural Science Foundation of China (No. 11931010 and No. 11871047), by the key research project of Academy for Multidisciplinary Studies, Capital Normal University, and by the Capacity Building for Sci-Tech Innovation- Fundamental Scientific Research Funds (No. 007/20530290068).

The third author is supported by National Nature Science Foundation of China under Grant Nos. 12031013 and 12161141004, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDA25010403.

Received 30 January 2022

Received revised 25 June 2022

Accepted 14 August 2022

Published 27 February 2023