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Communications in Mathematical Sciences
Volume 21 (2023)
Number 7
Dissipative solutions to the compressible isentropic Navier–Stokes equations
Pages: 1961 – 1987
DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n7.a10
Authors
Abstract
The dissipative solutions to the compressible isentropic Navier–Stokes equations are introduced in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions ($\href{https://global.oup.com/academic/product/mathematical-topics-in-fluid-mechanics-9780199679218}{[\textrm{P.-L. Lions, Oxford Science Publication, Oxford, 1996}]}$, Section 4.4). We establish the existence of the dissipative solutions for the compressible Navier–Stokes equations, which is carried out by an approximate scheme for a modified Brenner model with artificial diffusion and artificial pressure at the same level. Moreover, we prove that the weak solution of the compressible isentropic Navier–Stokes equations is a dissipative solution.
Keywords
compressible isentropic Navier–Stokes equations, dissipative solutions, weak-strong uniqueness
2010 Mathematics Subject Classification
35Q30, 76N10
Received 28 June 2021
Received revised 3 January 2023
Accepted 27 January 2023
Published 9 October 2023