Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

Dissipative solutions to the compressible isentropic Navier–Stokes equations

Pages: 1961 – 1987



Liang Guo (School of Mathematics and Statistics, Henan University, Kaifeng, China)

Fucai Li (Department of Mathematics, Nanjing University, Nanjing, China)

Cheng Yu (Department of Mathematics, University of Florida, Gainesville, Fl., U.S.A.)


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 ($\href{}{[\textrm{P.-L. Lions, Oxford Science Publication, Oxford, 1996}]}$, 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.


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