In this paper, the global strong axisymmetric solutions for the inhomogeneous incompressible Navier-Stokes system are established in the exterior of a cylinder subject to the Dirichlet boundary conditions. Moreover, the vacuum is allowed in these solutions. One of the key ingredients of the analysis is to obtain the ${L^{2}(s,T;L^{\infty}(Ω))}$ bound for the velocity field, where the axisymmetry of the solutions plays an important role.
We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $Π=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(Π)$ satisfying the decay condition of the swirl component $ru^θ_{0}\in L^{\infty}(Π)$.
In this article we consider the $α$--Euler equations in the exterior of a small fixed disk of radius $ε$. We assume that the initial potential vorticity is compactly supported and independent of $ε$, and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on $ε$. We prove that the solution of this problem converges, as $ε\to 0$, to the solution of a modified $α$--Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity.
We consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain B with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain B_R of the exterior domain B under certain artificial boundary conditions on the truncating boundary and then compare this solution with the solution in the exterior domain B to get the truncation error estimate.
We consider a free boundary problem of the Navier--Stokes equations in the three-dimensional Euclidean space with moving contact line, where the 90$^\circ$-contact angle condition is posed. We show that for given $T > 0$ the problem is local well-posed on $(0, T)$ provided that the initial data are small. In contrast to the strategy in Wilke (2013), we study the transformed problem in an $L^p$-in-time and $L^q$-in-space setting, which yields the optimal regularity of the initial data.
We study the incompressible limit of solutions to the compressible barotropic Navier-Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by the means of the RAGE theorem. The solution to the incompressible Navier-Stokes equations is identified as a limit.
In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like $\mathcal{O}(t^{(1/2)+\vare})$, for any $\vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $\mathcal{O}(t^{1/3})$. The purp…
In this paper we investigate the uniqueness of solutions of the steady planar Navier-Stokes equations with different boundary conditions in the exterior domain. For a class of incompressible flow with constant vorticity, we prove the uniqueness of the solution under the enhanced Navier boundary conditions. At the same time, some counterexamples are given to show that the uniqueness of the solution fails under the Navier boundary conditions. For the general incompressible flow with Dirichlet boundary condition, we prove various sufficient conditions for the uniqueness of the solution.
Global axisymmetric solutions and incompressible limit for the 3D isentropic compressible Navier-Stokes equations in annular cylinders with swirl and large initial data
We construct a maximal function associated with a family of skewed cylinders. These cylinders, which are defined as tubular neighborhoods of trajectories of a mollified flow, appear in the study of fluid equations such as the Navier-Stokes equations and the Euler equations. We define a maximal function subordinate to these cylinders, and show it is of weak type $(1, 1)$ and strong type $(p, p)$ by a covering lemma. As an application, we give an alternative proof for the higher derivatives estimate of smooth solutions to the three-dimensional Navier-Stokes equations.
We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R 3 and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary, Strichartz estimates, all matching the R 3 case.
We study a nonlinear, unsteady, moving boundary, fluid-structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by 2D incompressible Navier-Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (2-way coupling) via the kinematic and dynamic lateral boundary condition…
We consider the stationary incompressible Navier Stokes equation in the exterior of a disk B with non-zero Dirichlet boundary conditions on the disk and zero boundary conditions at infinity. We prove the existence of solutions for an open set of boundary conditions without symmetry.
We study the nonhomogeneous boundary value problem for the Navier--Stokes equations of steady motion of a viscous incompressible fluid in a three--dimensional exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric domains and data (without any smallness restrictions on the fluxes).
In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior domain.We describe, in particular, an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. The convergence of these type of approximation show in general a lack of strong convergence due to the presence of acoustic waves. In this paper we face this difficulty by taking care of the dispersive nature of these waves by means of the Strichartz estimates or waves equations satisfied by the pressure. We actually decompose the pressure in di…
In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis: first, that the initial exterior domain velocity converges strongly in $L^2$ to the full-space initial velocity and second, that the diameter of the obstacle is smaller tha…
We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in $\R^n$ with $n\ge 2$. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate $\exp(-Ct^2\log t)$ at infinity. Our proof is based on appropriate Carleman estimates.
In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity $ω_0$ and the circulation $γ$ of the initial flow around the obstacle. We prove that, if $γ$ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity $ω_0 + γδ$, where $δ$ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coeffi…
In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface. In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations.
We consider the 2D incompressible Navier-Stokes equations with Dirichlet boundary condition in the exterior of one obstacle. Assuming that the circulation at infinity of the velocity is sufficiently small, we prove that the large time behavior of the corresponding solution to the initial-boundary value problem is described by the Lamb-Oseen vortex. The later is the well-known explicit self-similar solution to the Navier-Stokes system in the whole space $\R^2$.
We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$.
We consider the incompressible Navier-Stokes equations with the Dirichlet boundary condition in an exterior domain of $\mathbb{R}^n$ with $n\geq2$. We compare the long-time behaviour of solutions to this initial-boundary value problem with the long-time behaviour of solutions of the analogous Cauchy problem in the whole space $\mathbb{R}^n$. We find that the long-time asymptotics of solutions to both problems coincide either in the case of small initial data in the weak $L^{n}$-space or for a certain class of large initial data.
In [Lacave, IHP, ana, to appear (2008)] the author considered the two dimensional Euler equations in the exterior of a thin obstacle shrinking to a curve and determined the limit velocity. In the present work, we consider the same problem in the viscous case, proving convergence to a solution of the Navier-Stokes equations in the exterior of a curve. The uniqueness of the limit solution is also shown.
We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. We assume that the initial velocity is a finite-energy and L^q-summable perturbation of the Oseen vortex with circulation α, where 1 < q < 2. If α is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation α. This is a global stability result, which holds for arbitrarily large perturbations of the Oseen vortex, and our smallness assumption on the circulation is independent of the domain under cons…
Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities…
We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall.
We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as $t \to \infty$. This result was obtained in collaboration with Yasunori Maekawa (Kobe University).
We consider a free boundary problem of compressible-incompressible two-phase flows with phase transitions in general domains of $N$-dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The compressible fluid and the incompressible fluid are separated by either compact or non-compact sharp moving interface, and the surface tension is taken into account. In our model, the compressible fluid and incompressible fluid are occupied by the Navier-Stokes-Korteweg equations and the Navier-Stokes equations, respectively. This paper shows that for given $T > 0$ …
We consider weak (''Leray'') solutions to the stationary Navier-Stokes system with Oseen and rotational terms, in an exterior domain. It is shown the velocity may be split into a constant times the first column of the fundamental solution of the Oseen system, plus a remainder term decaying pointwise near infinity at a rate which is higher than the decay rate of the Oseen tensor. This result improves the theory by M. Kyed, Asymptotic profile of a linearized flow past a rotating body, Q. Appl. Math. 71 (2013), 489-500.
In some problems of fluid mechanics, it is possible to be confronted with data that are not regular, that is why we are interested here in the search for the so-called very weak solutions for the stationary Stokes problem with Navier-type boundary conditions in a three-dimensional exterior domain. The problem describes the flow of a viscous and incompressible fluid past an obstacle where we assume that the fluid may slip on the boundary of the obstacle. Because the flow domain is unbounded, we set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the soluti…
We study the incompressible Navier-Stokes equations in the two-dimensional strip $\mathbb{R} \times [0,L]$, with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, we prove that the solution remains uniformly bounded for all times, and that the vorticity distribution converges to zero as $t \to \infty$. We deduce that, after a transient period, a laminar regime emerges in which the solution rapidly converges to a shear flow governed by the one-dimensional heat equation. Our approach is constructive and gives explicit estimates on the size of the solution …
Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
We provide general sufficient conditions for branching out of a time-periodic family of solutions from steady-state solutions to the two-dimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem …
We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for $t \in R$ or $t \in (0,\infty)$. In the latter case it is coupled with small initial data in weak $L^{3}$. As a corollary, the unique existence of time-periodic solutions is shown for the small periodic boundary data. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically discrete…
We consider an elliptic equation with unbounded drift in an exterior domain, and obtain quantitative uniqueness estimates at infinity, i.e. the non-trivial solution of $-\triangle u+W\cdot\nabla u=0$ decays in the form of $\exp(-C|x|\log^2|x|)$ at infinity provided $\|W\|_{L^\infty(\mathbb{R}^2\setminus B_1)}\lesssim 1$, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by Kenig-Wang \cite{KW2015} in the whole space. As an application, the asymptotic behavior of an incompressible fluid around a bounded obstacle is also considered. Specially f…
We consider the incompressible Navier-Stokes equations in the cylinder $\R \times \T$, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not know if such solutions stay uniformly bounded for all times, we prove that they converge in an appropriate sense to the family of spatially homogeneous equilibria as $t \to \infty$. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the $L^\infty$ norm of th…
We investigate the equations of anisotropic axisymmetric incompressible viscous fluids in the exterior of a cylinder of $\R^3$, rotating around an inhomogeneous vector $B(t, r)$. We prove uniform local existence with respect to the Rossby number in suitable anisotropic Sobolev spaces. We also obtain the propagation of the isotropic Sobolev regularity.
Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in 3D, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier-Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator $T(t,s)$ in the space $L^q$ generated by the linearized non-autonomous system was proved by Hansel and Rhandi [26] and the large time behavior of $T(t,s)f$ in $L^r$ for $(t-s)\to\infty$ was then developed by the pre…
We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder. Assuming that the derivatives of density, velocity, external force with respect to the third co-ordinate are sufficiently small in some norms we prove large time regular solutions without any restriction on the existence time. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point th…
In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are de…
We establish several boundary $\varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the Dirichlet boundary condition on the flat boundary. Our proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur [18], Choi-Vasseur [2], and Phuc-Guevara [6].
We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier-Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.
In their classical work [20], Caflisch and Sammartino established the inviscid limit and boundary layer expansions of vanishing viscosity solutions to the incompressible Navier-Stokes equations for analytic data on a half-space. It was then subsequently announced in their Comptes rendus article [4] that the results can be extended to include analytic data on an exterior circular domain, however the proof appears missing in the literature. The extension to an exterior domain faces a fundamental difficulty that the corresponding linear semigroup may not be contractive in analytic spaces as was t…
