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 (2)/(3) < m < 4 could be generalized to the case (2)/(3) < m < 6 in the 3-dimensional setting.
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