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In the spirit of recent work cite{[NNT]},it is shown that $vin L^{frac{2p}{p-1}}(0,T; L^{frac{2q}{q-1}}(mathbb{T}^{3})) $ and $ abla vin L^{p}(0,T; L^{q}(mathbb{T}^{3})) $ imply the energy equality in homogeneous incompressible Navier-Stokes equations and together with bounded density with positive lower bound yields the energy conservation in the general compressible Navier-Stokes equations. This unifies the known energy conservation criteria via the velocity and its gradient in incompressible Navier-Stokes equations. This also helps us to extend the conditions via the velocity or gradient of the velocity for energy equality from the incompressible fluid to compressible flow and improves the recent results due to Nguyen-Nguyen-Tang cite[Nonlinearity 32 (2019)]{[NNT]} and Liang cite[Proc. Roy. Soc. Edinburgh Sect. A (2020)]{[Liang]}.
In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<in
We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a weak soluti
In this paper, we derive some new Gagliardo-Nirenberg type inequalities in Lorentz type spaces without restrictions on the second index of Lorentz norms, which generalize almost all known corresponding results. Our proof mainly relies on the Bernstei
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $|Pi|_{L^{p,infty}
The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation associated