Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Green's function and pointwise convergence for compressible Navier-Stokes equations


Authors: Shijin Deng and Shih-Hsien Yu
Journal: Quart. Appl. Math. 75 (2017), 433-503
MSC (2010): Primary 35Q30, 65M80
DOI: https://doi.org/10.1090/qam/1461
Published electronically: February 2, 2017
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Abstract: In this paper we introduce a program to construct the Green's function for the linearized compressible Navier-Stokes equations in several space dimensions. This program contains three components, a procedure to isolate global singularities in the Green's function for a multi-spatial-dimensional problem, a long wave-short wave decomposition for the Green's function and an energy method together with Sobolev inequalities. These three components together split the Green's function into singular and regular parts with the singular part given explicitly and the regular part bounded by exponentially sharp pointwise estimates. The exponentially sharp singular-regular description of the Green's function together with Duhamel's principle and results of Matsumura-Nishida on $ L^\infty $ decay yield through a bootstrap procedure an exponentially sharp space-time pointwise description of solutions of the full compressible Navier-Stokes equations in $ {\mathbb{R}}^n(n=2,3)$.


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Additional Information

Shijin Deng
Affiliation: Department of Mathematics, Shanghai Jiao Tong University
Email: matdengs@sjtu.edu.cn

Shih-Hsien Yu
Affiliation: Department of Mathematics, National University of Singapore
Email: matysh@nus.edu.sg

DOI: https://doi.org/10.1090/qam/1461
Received by editor(s): November 20, 2016
Published electronically: February 2, 2017
Article copyright: © Copyright 2017 Brown University

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