اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSeveral almost critical regularity conditions based on one component of the solutions for 3D N-S Equations
60
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this article, we establish several almost critical regularity conditions such that the weak solutions of the 3D Navier-Stokes equations become regular, based on one component of the solutions, say $u_3$ and $partial_3u_3$.
قيم البحث
اقرأ أيضاً
In this paper, we consider the Cauchy problem to the 3D MHD equations. We show that the Serrin--type conditions imposed on one component of the velocity $u_{3}$ and one component of magnetic fields $b_{3}$ with $$ u_{3} in L^{p_{0},1}(-1,0;L^{q_{0}}(
B(2))), b_{3} in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))), $$ $frac{2}{p_{0}}+frac{3}{q_{0}}=frac{2}{p_{1}}+frac{3}{q_{1}}=1$ and $3<q_{0},q_{1}<+infty$ imply that the suitable weak solution is regular at $(0,0)$. The proof is based on the new local energy estimates introduced by Chae-Wolf (Arch. Ration. Mech. Anal. 2021) and Wang-Wu-Zhang (arXiv:2005.11906).
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc.
Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poissons equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poissons equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Greens function for Poissons equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
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)).
We prove global existence, uniqueness and stability of entropy solutions with $L^2cap L^infty$ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a unique continu
ous manner to all $L^2$ initial data. These weak solutions are found to satisfy one sided Holder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.
Several types of new regularity criteria for Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. Some of them are based on the third component $u_3$ of velocity under Prodi-Serrin index condition, another type is in terms of
$omega_3$ and $partial_3u_3$ with Prodi-Serrin index condition. And a very recent work of the authors, based on only one of the nine entries of the gradient tensor, is renovated.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
