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Register a new userTraveling wave solutions to the free boundary incompressible Navier-Stokes equations
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In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e. time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an over-determined Stokes problem and its under-determined adjoint, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions.
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We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $epsilon^{3/8 - delta}$ near the boundary and $epsilon^{3/4 - delta}$ in the interior, where $delta, delta$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.
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.
In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $dot{H}^{-alpha}(mathbb{R}^{3})$ or $dot{H}^{-alpha}(mathbb{T}^{3})$ with $0<alphaleq 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $tgeq0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlovi{c} and Staffilani on (SIMA, [1])in which $alpha$ is restricted to $0<alpha<frac{1}{4}$.
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 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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