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Register a new userEnergy conservation via a combination of velocity and its gradient in the Navier-Stokes system
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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]}.
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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 $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<infty$ and $lim_{tto T}sqrt{T-t}||u(t)||_{L^infty}=infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $uin L^{2,infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,infty}([0,T];BMO(Omega))}<infty$ for arbitrary $Omegasubseteqmathbb{R}^3$ then the local energy equality is valid on $[0,T]timesOmega$. As a corollary, we also prove $||u||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}<infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)||_{BMO}$ must blow up at a rate faster than $frac{c}{sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}=M<infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}leq c_M$ then $u$ is regular on $(0,T]timesmathbb{R}^3$.
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 solution of the system, then $(Du, p)$ is bounded and piecewise continuous. The corresponding results for stationary Navier-Stokes systems are also established, from which the Lipschitz regularity of the stationary $H^1$-weak solution in dimensions $d=2,3,4$ is obtained.
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 Bernstein inequalities in Lorentz spaces, the embedding relation among various Lorentz type spaces, and Littlewood-Paley decomposition techniques. In addition, we establish several novel criteria in terms of the velocity or the gradient of the velocity in Lorentz spaces for energy conservation of the 3D Navier-Stokes equations. Particularly, we improve the classical Shinbrots condition for energy balance to allow both the space-time directions of the velocity to be in Lorentz spaces.
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}(0,T; L ^{q,infty}(mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=2$ $({3}/{2}<q<infty)$ or $| ablaPi|_{L^{p,infty}(0,T; L ^{q,infty}(mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=3$ $(1<q<infty)$ is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of $ abla u$ obtained here generalize known ones to allow the time direction to belong to Lorentz spaces.
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 to the Ginzburg-Landau free energy with physically relevant logarithmic potential. This model is studied in bounded smooth domain in R^d, d=2 and d=3, and is supplemented with a no-slip condition for the velocity, homogeneous Neumann boundary conditions for the order parameter and the chemical potential, and suitable initial conditions. We study uniqueness and regularity of weak and strong solutions. In a two-dimensional domain, we show the uniqueness of weak solutions and the existence and uniqueness of global strong solutions originating from an initial velocity u_0 in V, namely u_0 in H_0^1 such that div u_0=0. In addition, we prove further regularity properties and the validity of the instantaneous separation property. In a three-dimensional domain, we show the existence and uniqueness of local strong solutions with initial velocity u_0 in V.
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