Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $Ω$ subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary $\partialΩ$, friction coefficient $α(x)$ and the outflow $u_\infty$. We then study the regularity of the resolvent near the origin in the complex plane to develop $L^q$-$L^r$ decay estimates of the Oseen semigroup provided that $α(x)+u_\infty\cdotν(x)/2\geq 0$ for every $x\in\partialΩ$, where $ν(x)$ stands for the outward unit normal to the boundary $\partialΩ$.
We prove the existence of solution in a class H^2(Ω) x H^1(Ω) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H^{5/2}. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to a problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under additional assumption on the geometry of the boundary.
We prove new existence and uniqueness results in full Sobolev spaces for the steady-state Oseen problem in a smooth exterior domain of $\mathbb{R}^n$, $n\ge 2$. These results are then employed, on the one hand, in the study of analogous properties for the corresponding (linear) time-periodic case and, on the other hand and more significantly, to prove analogous properties for their nonlinear counterpart, at least for small data.
We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $C_0$-semigroup in $L^p$ space and then develop a series of $L^p$-$L^q$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $L^3$ and the steady flow are sufficiently small.
We consider the spatial-temporal behavior of the Navier-Stokes flow past a rigid body in $\mathbb{R}^3$. The present paper develops analysis in Lebesgue spaces with anisotropic weights $(1+|x|)^α(1+|x|-x_1)^β$, which naturally arise in the asymptotic structure of fluid when the translational velocity of the body is parallel to the $x_1$-direction. We derive anisotropically weighted $L^q$-$L^r$ estimates for the Oseen semigroup in exterior domains. As applications of those estimates, we study the stability/attainability of the Navier-Stokes flow in anisotropically weighted $L^q$ spaces to get t…
Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -- sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $Ω\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_Ω(t,s)\}_{t\geq s\geq0}$ on $L^p_σ(Ω)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $…
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…
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…
Consider the motion of a viscous incompressible fluid in a 3D exterior domain when a rigid body moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near the initial time as well as generation of the evolution operator was shown by Hansel and Rhandi (J. Reine Angew. Math. 2014) under reasonable conditions. In this paper we develop the $L^q$-$L^r$ decay estimates of the evolution operator and then apply them to the Navier-Stokes initial value problem.
In this paper, we consider the stationary Stokes equations in an exterior domain three-dimensional under a slip boundary condition without friction. We set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the solutions. In this work, we try to investigate the existence and uniqueness of the weak solutions related to this problem in $L^P-$theory when $p>3$. Our proof are based on obtaining inf-sup conditions that play a fundamental role.
An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with N…
We consider the two-phase flow model with slip boundary condition in a 3D exterior domains whose boundary is smooth. We establish the global existence of classical solutions of this system provided that the initial energy is suitably small. Moreover, the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.
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 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.
The author treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by $p-$potential function subject to slip boundary conditions in $\mathbb{R}^3_+$. Making use of the Oseen-type approximation to this model and the $L^\infty$-truncation method, one can establish the existence theorem of weak solutions for $p-$potential flow with $p\in(\frac{8}{5},2]$ provided that large initial data.
In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations contain vacuum states. Moreover, the large-time…
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 are concerned with the global existence of classical solutions to the barotropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. We demonstrate that the classical solutions exists globally in time provided that the initial total energy is suitably small. It is worth noting that the initial density is allowed to have large oscillations and contain vacuum states. For our purpose, some new techniques and methods are adopted to obtain necessary a priori estimates, especially the estimates on the boundary. Moveover, we also give the …
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).
In this article we derive Biot-Savar law for exterior 2D domain. No-slip condition in terms of vorticity is expressed in integral form. For non-stationary fluid dynamics equations no-slip condition generates affine invariant manifold and no-slip integral relations on vorticity can be transferred to Robin-type boundary condition.
We prove unique existence of mild solutions on $L^{\infty}_σ$ for the Navier-Stokes equations in an exterior domain in $ \mathbb{R}^{n}$, $n\geq 2$, subject to the non-slip boundary condition.
In this article we derive the explicit solution of 2-D Stokes system in exterior of the disc with no-slip condition on inner boundary and given velocity $\mathbf{v}_\infty$ at infinity. It turned out it is the first application of the associated Weber-Orr transform to mathematical physics in comparison to classical Weber-Orr transform which is used in many researches. From no-slip condition for velocity field we will obtain Robin-type boundary condition for vorticity. Then the initial-boundary value problem for vorticity will be solved with help of the associated Weber-Orr transform. Also the …
We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers.
This paper treats the stationary Stokes problem in exterior domain of $\mathbb{R}^3$ with Navier slip boundary condition. The behavior at infinity of the data and the solution are determined by setting the problem in $L^p$-spaces, for $p> 2$, with weights. The main results are the existence and uniqueness of strong solutions of the corresponding system.
This paper studies the stability of a stationary solution of the Navier-Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier-Stokes system governing the stationary solution which belongs to the weak $L^3$-space $L^{3,\infty}$. Under the condition that the initial datum belongs to a solenoidal $L^{3 , \infty}$-space, we prove that if both the $L^{3,\infty}$-norm of the initial datum and the $L^{3,\infty}$-norm of the stationary solution are sufficiently small then the system admits a unique global-in-time stro…
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 paper, we consider the initial-boundary value problem of three-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms in an exterior domain with Navier-slip boundary condition and with far-field vacuum. This problem is related to the motion of the compressible viscous flow past a rotating obstacle. We establish the global existence and uniqueness of classical solutions, provided that the initial mass is small. The initial data and the angular velocity of the obstacle are allowed to have large oscillations.
The nonhomogeneous Navier-Stokes equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with nonslip or slip boundary conditions. We prove that the strong solutions exists globally in time provided that the gradient of the initial velocity is suitably small. Here the initial density is allowed to contain vacuum states. Moreover, after developing some new techniques and methods, the large-time behavior of the strong solutions with exponential decay-in-time rates is also obtained.
Motivated by Gilbarg-Weinberger's early work on asymptotic properties of steady plane solutions of the Navier-Stokes equations on a neighborhood of infinity \cite{GW1978} , we investigate asymptotic properties of steady plane solutions of this system on a half-neighborhood of infinity with finite Dirichlet integral and Navier-slip boundary condition, and obtain that the velocity of the solution grows more slowly than $\sqrt{\log r}$, while the pressure converges to $0$ along each ray passing through the origin.
The full compressible magnetohydrodynamic system in three-dimensional exterior domains is investigated. For the initial-boundary-value problem of this system with slip boundary condition for the velocity, adiabatic one for the temperature, and perfect one for the magnetic field, the global existence and uniqueness of classical solutions is established, under the condition that the initial data are of small energy but possibly large oscillations. In particular,the initial density and temperature are both allowed to vanish. Moreover, the large-time behavior of the classical solutions is also obt…
The full compressible Navier-Stokes system (FNS) describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid in a three-dimensional (3D) exterior domain is studied. For the initial-boundary-value problem with the slip boundary conditions on the velocity and the Neumann one on the temperature, it is shown that there exists a unique global classical solutions with the initial data which are of small energy but possibly large oscillations. In particular, both the density and temperature are allowed to vanish initially. This is the first result about classical…
Time-periodic solutions to the Navier-Stokes equations that govern the flow of a viscous liquid past a three-dimensional body moving with a time-periodic velocity are investigated. The net motion of the body over a full time-period is assumed to be non-zero. In this case the appropriate linearization of the system is constituted by the time-periodic Oseen equations in a three-dimensional exterior domain. A priori Lp estimates are established for this linearization. Based on these estimates, existence of a solution to the fully non-linear Navier-Stokes problem is obtained by the contraction map…
Local well-posedness of strong solutions to the compressible Navier-Stokes equations with degenerate viscosities and far field vacuum in 3D exterior domains
Strong solutions to the initial-boundary-value problem of compressible MHD equations with degenerate viscosities and far field vacuum in 3D exterior domains
We consider a fluid-structure interaction system composed by a rigid ball immersed into a viscous incompressible fluid. The motion of the structure satisfies the Newton laws and the fluid equations are the standard Navier-Stokes system. At the boundary of the fluid domain, we use the Tresca boundary conditions, that permit the fluid to slip tangentially on the boundary under some conditions on the stress tensor. More precisely, there is a threshold determining if the fluid can slip or not and there is a friction force acting on the part where the fluid can slip. Our main result is the existenc…
The asymptotic behavior of the vorticity for the steady incompressible Navier-Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero. It is well known that the asymptotic behavior of the velocity field is given by the fundamental solution of the Oseen system which is the linearization of the Navier-Stokes equation around $\boldsymbol{u}_{\infty}$. The vorticity has the property of decaying algebraically inside a parabolic region called the wake and exponentially outside. The previously proven asymptotic…
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…
Stokes flow equations, used to model creeping flow, are a commonly used simplification of the Navier--Stokes equations. The simplification is valid for flows where the inertial forces are negligible compared to the viscous forces. In infinite domains, this simplification leads to a fundamental paradox. In this work we review the Stokes paradox and present new insights related to recent research. We approach the paradox from three different points of view: modern functional analysis, numerical simulations, and classical analytic techniques. The first approach yields a novel, rigorous derivation…
On the axisymmetric Navier-Stokes flow passing a cone with the total-slip boundary condition
School of Mathematics and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）；School of Mathematics, Southeast University（东南大学数学系）；Department of mathematics, University of California, Riverside（加州大学河滨分校数学系）
Let $D$ be the exterior of a cone inside a ball, with its altitude angle at most $π/6$ in $\mathbb{R}^3$, which touches the $x_3$ axis at the origin. For any initial value $v_0 = v_{0,r}e_{r} + v_{0,θ} e_θ + v_{0,3} e_{3}$ in a $C^2(\overline{D})$ class, which has the usual even-odd-odd symmetry in the $x_3$ variable and has the partial smallness only in the swirl direction: $ | r v_{0, θ} | \leq \frac{1}{100}$, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition has a finite-energy solution that stays bounded for all time. In particular, no fin…
