Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLerays backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces
102
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, it is shown that there does not exist a non-trivial Lerays backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces $dot{mathcal{M}}^{q,1}(mathbb{R}^{3})$ provided $3/2<q<6$, or in $dot{mathcal{M}}^{q,l}(mathbb{R}^{3})$ provided $6leq q<infty$ and $2<lleq q$. This generalizes the corresponding results obtained by Nev{c}as-Rr{a}uv{z}iv{c}ka-v{S}ver{a}k [19, Acta.Math. 176 (1996)] in $L^{3}(mathbb{R}^{3})$, Tsai [25, Arch. Ration. Mech. Anal. 143 (1998)] in $L^{p}(mathbb{R}^{3})$ with $pgeq3$,, Chae-Wolf [3, Arch. Ration. Mech. Anal. 225 (2017)] in Lorentz spaces $L^{p,infty}(mathbb{R}^{3})$ with $p>3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in $dot{mathcal{M}}^{q,frac{12-2q}{3}}(mathbb{R}^{3})$ with $12/5leq q<3$ and in $L^{q, infty}(mathbb{R}^3)$ with $12/5leq q<6$.
rate research
Read More
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Lerays blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,infty}(-1,0;L^{q,l}(mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<infty$ and $qleq l <infty$ is finite.
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $ abla_{h}{u}$ (or $ abla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; dot{B}_{p,r}^{s}(mathbb{R}^{3}))$, where $ abla_{h}=(partial_{x_{1}},partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $partial_3u_3$.
We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
In this article, we establish sufficient conditions for the regularity of solutions of Navier-Stokes equations based on one of the nine entries of the gradient tensor. We improve the recently results of C.S. Cao, E.S. Titi (Arch. Rational Mech.Anal. 202 (2011) 919-932) and Y. Zhou, M. Pokorn$acute{y}$ (Nonlinearity 23, 1097-1107 (2010)).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
