A note on functional inequalities and entropies estimates for some higher-order nonlinear PDEs

In this short note, we prove by simple arguments some functional inequalities in arbitrary space dimensions. We investigate the applications of our results in establishing some appropriate a priori estimates (entropy estimates) on the approximate solution of some models related to fluid dynamics system. In particular, we derive some entropy inequalities of the solution to the Navier–Stokes–Korteweg system, and to the fourth-order degenerate diffusion equation in higher dimensional spaces. As a by product, we show that the result obtained recently by D. Bresch, A. Vasseur and C. Yu [“Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities”, arXiv:1905.02701] for viscosity coefficients such that $\mu(\rho) = \rho^m$ and $\lambda(\rho) = 2(m-1)\rho^m$ for $\dfrac{2}{3} \lt m \lt 4$ could be generalized to the case $\dfrac{2}{3} \lt m \lt 6$ in the $3$-dimensional setting.

Keywords

a priori estimates, functional inequalities, Navier–Stokes–Korteweg equations, lubrification equations

2010 Mathematics Subject Classification

35B45, 35Q30

Full Text (PDF format)

Received 24 August 2020

Accepted 18 June 2021

Published 1 March 2023