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Register a new userNavier-Stokes equations, symmetric and uniform analytic solutions in phase space
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Qixiang Yang
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For incompressible Navier-Stokes equations, Necas-Ruzicka-Sverak proved that self-similar solution has to be zero in 1996. Further, Yang-Yang-Wu find symmetry property plays an important role in the study of ill-posedness. In this paper, we consider two types of symmetry property. We search special symmetric and uniform analytic functions to approach the solution and establish global uniform analytic and symmetric solution with initial value in general symmetric Fourier-Herz space. For two kinds of symmetry of initial data, we prove that the solution has also the same symmetric structure. Further, we prove that the uniform analyticity is equivalent to the convolution inequality on Herz spaces. By these ways, we can use symmetric and uniform analytic functions to approximate the solution.
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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 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.
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.
The paper deals with the Navier-Stokes equations in a strip in the class of spatially non-decaing (infinite-energy) solutions belonging to the properly chosen uniformly local Sobolev spaces. The global well-posedness and dissipativity of the Navier-Stokes equations in a strip in such spaces has been first established in [S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip. Glasg. Math. J., 49 (2007), no. 3, 525--588]. However, the proof given there contains rather essential error and the aim of the present paper is to correct this error and to show that the main results of that paper remain true.
In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Greens function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.
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