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Register a new userOn the existence of helical invariant solutions to steady Navier-Stokes equations
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In this paper, we investigate the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a helically symmetric spatial domain. When data is assumed to be helical invariant and satisfies the compatibility condition, we prove this problem has at least one helical invariant solution.
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We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_{th}$ and $ a {bf u}$, especially we show that $|u_{th}(r,z)|leq cleft(f{log r}{r}right)^{f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $om_{th}$, integral representations of ${bf u}$ in terms of $bm{om}=textit{curl }{bf u}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $( a u_r, a u_z)$ and $(om_r, om_{z})$, where $bm{om}= om_r {bf e}_r + om_{th} {bf e}_{th}+ om_z {bf e}_z$. Another is the first decay rate estimates in the $Oz$-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.
In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in $mathbb{R}^3$. We obtain the optimal decay rate $|{bf u}(x)|leq frac{C}{|x|+1}$ for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component $u_{theta}$, that is, either $O(frac{1}{r+1})leq |u_{theta}(r,z)|leq frac{Clog(r+1)}{(r+1)^{1/2}}$ or $|u_{theta}(r,z)|leq frac{C r}{(rho+1)^3}$, where $rho=sqrt{r^2+z^2}$. In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: $|u_r(r,z)|+ |u_{z}(r,z)|leq frac{C}{rho+1}$. We do not require any small assumptions on the forcing term.
An old problem asks whether bounded mild ancient solutions of the 3 dimensional Navier-Stokes equations are constants. While the full 3 dimensional problem seems out of reach, in the works cite{KNSS, SS09}, the authors expressed their belief that the following conjecture should be true. For incompressible axially-symmetric Navier-Stokes equations (ASNS) in three dimensions: textit{bounded mild ancient solutions are constant}. Understanding of such solutions could play useful roles in the study of global regularity of solutions to the ASNS. In this article, we essentially prove this conjecture in the special case that $u$ is periodic in $z$. To the best of our knowledge, this seems to be the first result on this conjecture without unverified decay condition. It also shows that periodic solutions are not models of possible singularity or high velocity region. Some partial result in the non-periodic case is also given.
We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $a in L^2(partialOmega)$ satisfies ${1over 2pi}|int_{partialOmega}acdot |<1$. If $Omega$ is of class $C^{1,1}$, we can assume $ain W^{-1/4,4}(partialOmega)$. Moreover, we show that for every $D$--solution $(u,p)$ of the Navier--Stokes equations it holds $ abla p = o(r^{-1}), abla_k p = O(r^{epsilon-3/2}), abla_ku = O(r^{epsilon-3/4})$, for all $kin{Bbb N}setminus{1}$ and for all positive $epsilon$, and if the flux of $u$ through a circumference surrounding $complementOmega$ is zero, then there is a constant vector $u_0$ such that $u=u_0+o(1)$.
The asymptotic behavior of weak time-periodic solutions to the Navier-Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.
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