T. Takahashi
We consider the large time behavior of the three-dimensional Navier-Stokes flow around a rotating rigid body. Assume that the angular velocity of the body gradually increases until it reaches a small terminal one at a certain finite time and it is fixed afterwards. We then show that the fluid motion converges to a steady solution as time $t\rightarrow\infty$.
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Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the question whether his steady slutions are attainable as limits for $t\to\infty$ of unsteady solutions starting from motionless state when $h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We study some generalized situation in which unsteady solutions start from large m…
A body $\mathscr B$ is started from rest by translational motion in an otherwise quiescent Navier-Stokes liquid filling the whole space. We show, for small data, that if after some time $\mathscr B$ reaches a spinless oscillatory motion of period $\cal T$, the liquid will eventually execute also a time periodic motion with same period $\cal T$. This problem is a suitable generalization of the famous Finn's starting problem for steady-states, to the case of time-periodic motions
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…
On the Kneser property of the three-dimensional Navier-Stokes equations with damping
研究带阻尼三维纳维 - 斯托克斯方程弱解可达集的连通性与紧致性，依控制阻尼项的参数值，在相空间弱或强拓扑下建立结果，还证全局吸引子连通及相关正则性等。
The aim of this work is to show the local null controllability of a fluid-solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier-Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid-rigid body system with Navier boundary conditions. This work i…
In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole `viscous incompressible fluid + rigid body' system is assumed to occupy the full plane $\R^{2}$. We prove the existence of global-in-time weak solutions with constant non-zero circulation at infinity.
In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.
The existence of weak solutions to the "viscous incompressible fluid + rigid body" system with Navier slip-with-friction conditions in a 3D bounded domain has been recently proved by Gérard-Varet and Hillairet in \cite{exi:GeH}. In 2D for a fluid alone (without any rigid body) it is well-known since Leray that weak solutions are unique, continuous in time with $ L^{2} $ regularity in space and satisfy the energy equality.In this paper we prove that these properties also hold for the 2D "viscous incompressible fluid + rigid body" system.
The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero bou…
Stress concentration between two adjacent rigid particles in Navier-Stokes flow
本文研究Navier-Stokes流中两个相邻凸刚性粒子间的应力集中问题，在二维和三维情形下建立了速度梯度与二阶导数的精确上下界，证明爆破速率最优，并分析了Cauchy应力的爆破行为。
We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the co…
Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions
研究了粘性流体中刚体在纳维-斯托克斯滑动边界条件下的稳态自推进运动，通过有限维推力空间分析了边界效应驱动的刚体运动机制。
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for …
Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equations
该研究针对二维各向同性Navier-Stokes方程，证明单方向耗散可保证自然能量水平下弱解的能量等式，克服方向正则性缺失的困难，进而推导弱-强唯一性原理。
Full-Tail Dynamical Rigidity Forced by Atomic Navier-Stokes Energy Concentration
针对趋近有限终端时间的三维纳维-斯托克斯流，证明点原子强制全尾动力学刚性，得到受约束奥森传播子饱和等结果，揭示延迟二阶算子预算在端点附近失效。
We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver-Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $L^p-L^q$ setting, globally in time, for small data in …
We consider the large time behavior of the Navier-Stokes flow past a rigid body in $\mathbb{R}^n$ with $n\geq 3$. We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow\infty$ in $L^q$ with $q\in[n,\infty]$. This is c…
We consider the motion of a rigid body, governed by the Navier-Stokes equations in a bounded domain. Navier's condition is prescribed on the boundary of the body. We give the global in a time solvability result of weak solution. The result permits a possibility of collisions of the body with the boundary of the domain.
We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navierslip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We provethe small-time exact controllability of the position and velocity of the solid when the control takes the form of adistributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body.The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls forwhich the ``Navier-Stokes+rigid body'' sys…
We study existence, uniqueness and asymptotic spatial behavior of time-periodic strong solutions to the Navier-Stokes equations in the exterior of a rigid body, $\mathscr B$, moving by time-periodic motion of given period $T$, when the data are sufficiently regular and small. Our contribution improves all previous ones in several directions. For example, we allow both translational, $\bfxi$, and angular, $\bfomega$, velocities of $\mathscr B$ to depend on time, and do not impose any restriction on the period $T$ nor on the averaged velocity, $\bar{\bfxi}$, of $\mathscr B$. If $\bfxi\not\equiv\…
The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Tosio Kato says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear i…
This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole $\mathbb{R}^3$. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.
We present pointwise space-time decay estimates for the velocity part of solutions to the time-dependent Oseen system in 3D, with Dirichlet boundary conditions and vanishing velocity at infinity. In addition, similar estimates are derived for solutions to the time-dependent incompressible Navier-Stokes system with Oseen term, and for solutions to the stability problem associated with the stationary incompressible Navier-Stokes system with Oseen term.
Traveling wave solutions to the free boundary incompressible Navier-Stokes equations with Navier boundary conditions
针对带广义 Navier 滑移边界的自由边界不可压缩 Navier-Stokes 系统，利用隐函数方法构造行波解，并证明滑移参数趋于零时其收敛于无滑移解。
Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications
本文证明高维（n≥4）Navier-Stokes方程满足|u(x)|≤C/|x|的稳态解必为零，通过加权能量估计给出初等简短证明，并应用于奇异性消除与无穷远渐近行为。
We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier-Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole plane. We also prove that the…
Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
该研究证明了五维定常Navier-Stokes方程的临界Morrey刚性定理，推导了五维可去奇点准则，还通过有限能量论证证明了四维对应的三次Morrey刚性定理。
We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness res…
On the "viscous incompressible fluid + rigid body" system with Navier conditions
本文研究了带有Navier滑移条件的“粘性不可压缩流体+刚体”系统，证明了全局弱解的存在性，展示了解的正则性、无限惯性极限，以及在粘度趋于零时弱解向无粘系统光滑解的最优收敛性。
Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
本文证明在五维及以上空间中，满足横向界$|u(x)|\leq C |x'|^{-1}$的稳态Navier-Stokes方程光滑解必为零，并利用局部正则性定理消除线奇异性，推广了外部区域的渐近展开结果。
We consider the motion of compressible Navier-Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are non zero. This kind of pressure model is largely employed in various physical and industrial applications. We prove the existence of weak solution to the system in the exterior domain.
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…
On Navier-Stokes equations arising from the rotation of an obstacle in a fluid
本文研究了流体中旋转障碍物引起的修正纳维-斯托克斯方程，通过加权Sobolev空间分析解在远场的行为，并证明了解的存在性和正则性。
For a natural number $m \ge 2$, we study $m$ layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the …
We study the spatial decay of time-periodic Navier-Stokes flow at the rate $|x|^{-1}$ with/without wake structure in 3D exterior domains when a rigid body moves periodically in time. In this regime the existence of time-periodic solutions was established first in the 2006 paper by Galdi and Silvestre, however, with little information about spatial behavior at infinity so that uniqueness of solutions was not available. This latter issue has been addressed by Galdi, who has recently succeeded in construction of a unique time-periodic solution with spatial behavior mentioned above if translationa…
We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the motion of the rigid body is described by the conservation law of linear and angular momentum. Under the assumption that the diameter of the rigid body tends to zero and that the density of the rigid body goes to infinity, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier-Stokes equations in the full space without rigid body.
Vanishing viscosity limit of the 2D stationary Navier-Stokes equations outside a rotating disc and its application
本文研究了二维旋转圆盘外静止纳维-斯托克斯方程的消失粘性极限问题，通过高阶渐近近似构造了解并验证了边界层展开的有效性，同时解决了尤多维奇提出的一个开放问题。
We consider a coupled PDE-ODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.
In this paper, we investigate the global solvability of the chemotaxis-Navier-Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with boundary conditions that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state $(0,\hat{c},\mathbf{0})$, where $\hat{c}$ is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work…
On the regularity of weak solutions to the fluid-rigid body interaction problem
本文研究三维流体-刚体相互作用问题，将 Navier-Stokes 方程的 Prodi-Serrin 正则性准则推广到流体-刚体系统，证明满足该条件的弱解具有 $W^{2,p}$ 和 $W^{1,p}$ 正则性，并在刚体加速度几乎处处有界时得到 $C^{\infty}$ 光滑性。
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.
We consider a single disk moving under the influence of a 2D viscous fluid and we study the asymptotic as the size of the solid tends to zero.If the density of the solid is independent of $\varepsilon$, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal $L^p-L^q$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier-Stokes equations in $\R^2$.
In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on…
In this paper, we study the problem concerning the approximation of a rigid obstacle for flows governed by the stationary Navier-Stokes equations in the two-dimensional case. The idea is to consider a highly viscous fluid in the place of the obstacle. Formally, as the fluid viscosity goes to infinity inside the region occupied by the obstacle, we obtain the original problem in the limit. The main goal is to establish a better regularity of approximate solutions. In particular, the pointwise estimate for the gradient of the velocity is proved. We give numerical evidence that the penalized solut…
Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body
本文研究匀速平移刚体外部时间周期不可压缩Navier-Stokes方程的解，利用时间周期Oseen基本解推导表示公式，并据此获得速度场和压力场的完整渐近结构信息。
We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entr…
The vanishing limit of a rigid body in three-dimensional viscous incompressible fluid
研究了三维不可压缩粘性流体中小刚体收缩为无质量点时的消失极限，基于半群估计和不动点方法证明流体-刚体系统解收敛于全空间 Navier-Stokes 方程解。
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 wav…
Rigid body in compressible flow with general inflow-outflow boundary data
该研究针对固定有界域内可压缩等熵粘性流体中的刚体运动问题，给定边界入出流条件与流固交界面无滑移条件，证明了刚体不接触域边界的时段内该耦合问题弱解的存在性。
Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
本文研究静止纳维-斯托克斯方程在三维空间中的李普希茨型结果，证明在仅对速度径向分量施加积分条件时，任何$\dot{H}^1(\mathbb{R}^3)$解都是平凡的，并在变指数设定中建立唯一性结果。
Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary
本文研究移动刚体外部三维Navier--Stokes流，证明线性化解由半群控制并满足L^q-L^r估计，进而在小速度和小初值条件下得到全局温和解的存在唯一性。
Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary
研究一般不可压缩纳维-斯托克斯-傅里叶系统自由边界问题的行波解，考虑了重力、外部力源和边界应力及热源作用，以及温度依赖粘度和毛细系数的影响，证明了小数据下问题的well-posed性。
We prove that traveling waves in viscous compressible liquids are a generic phenomenon. The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, viscous fluid, modeled by the free boundary compressible Navier-Stokes equations in dimension $n \ge 2$. The bottom boundary of the fluid is flat and rigid, while the top is a moving free boundary. A constant gravitational field acts normal to the flat bottom. We allow external forces to act in the fluid's bulk and external stresses to act on its free surface. These are posited to be in traveling wave form…
In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is …
Regularity properties of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes initial value problem
本文研究三维外区域中广义 Oseen 演化算子缺失的时间导数、Hölder 及加权估计，并将其用于证明 Navier-Stokes 初值问题局部唯一强 Lq 解的新存在性定理。
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…
We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous proof of Zhukovskiy's Theorem, which states that in the limit of time going to infinity, the relative fluid velocity tends to zero and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. The existence of global weak solutions for this system was established previously. In particular, we prove that every weak solution of this type is subject to Zhukovskiy's Theorem. Independently of …
This paper is concerned with the free boundary problem for the Navier Stokes equations without surface tension in the $L_p$ in time and $L_q$ in space setting with $2 < p < \infty$ and $N < q < \infty$. A local in time existence theorem is proved in a uniform $W^{2-1/q}_q$ domain in the $N$-dimensional Euclidean space ${\Bbb R}^N$ ($N \geq 2$) under the assumption that weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the assumption that $Ω$ is bounded additionally. This was already proved by Solonnikov \cit…
In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by $ω_0 = αe_z δ_{x = y = 0}$, where $δ_{x=y= 0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $α\in \mathbb R$ is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of m…
Large-Time Behavior of a Rigid Body of Arbitrary Shape in a Viscous Fluid Under the Action of Prescribed Forces and Torques
本文研究任意形状刚体在无界 Navier-Stokes 流体中受给定力和力矩作用的问题，证明了在小初值和外力矩随时间 $L^2$ 衰减条件下全局强解的存在性及其渐近收敛到零，将以往仅适用于球体的结果推广到任意形状。
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