We show that a steady-state solution ${\bf U}$ to the system of equations of a Navier-Stokes flow past a rotating body is nonlinearly unstable if the associated linear operator $\cal L$ has a part of the spectrum in the half-plane $\{λ\in\mathbb C;\ {\rm Re}\, λ>0\}$. Our result does not follow from known methods, %on spectral instability, mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied. As an auxiliary result of independent interest, we also show that the uniform growth bound of the $C_0$--semigroup ${\rm e}^{{\cal L} t}$ is equal to the spectral bound of operator $\mathcal L$.
Cagnetti, Gomes, Mitake and Tran (2013) introduced a new idea to study the large time behavior for degenerate viscous Hamilton--Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton--Jacobi equations. We establish the convergence result under rather general assumptions.
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.
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…
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…
This paper provides asymptotic structure at spatial infinity of plane steady Stokes flow in exterior domains when the obstacle is rotating with constant angular velocity. The result shows that there is no longer Stokes paradox due to the rotating effect.
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 a boundary-value problem describing the steady motion of a two-component mixture of viscous compressible heat-conducting fluids in a bounded domain. We make no simplifying assumptions except for postulating the coincidence of phase temperatures (which is physically justified in certain situations), that is, we retain all summands in equations that are a natural generalization of the Navier-Stokes-Fourier model of the motion of a one-component medium. We prove the existence of weak generalized solutions of the problem.
We study the two-dimensional stationary Navier-Stokes equations with rotating effect in the whole space. The unique existence and the asymptotics of solutions are obtained without the smallness assumption on the rotation parameter.
The Navier-Stokes equations for compressible barotropic flow in the stationary three dimensional case are considered. It is assumed that a fluid occupies a bounded domain and satisfies the no-slip boundary condition. The existence of a weak solution under the assumption that the adiabatic exponent satisfies $γ>1$ is proved. These results cover the cases of monoatomic, diatomic, and polyatomic gases.
We investigate the instability and stability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $Ω$ of class $C^2$. When the steady density is heavier with increasing height (i.e., the Rayleigh-Taylor steady-state), we show that the steady-state is linear unstable (i.e., the linear solution grows in time in $H^2$) by constructing a (standard) energy functional and exploiting the modified variational method. Then, by introducing a new energy functional and using a careful bootstrap argument, we further show that the stea…
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…
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 a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a $C^2$ boundary. We use a stream-function formulation after the manner of N. N. Frolov [Math. Notes, \textbf{53}(5-6), 650--656, 1993] in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. With a fixed point argument we show the existence of solutions to the resul…
We prove the existence and uniqueness of strong solutions to the steady isentropic compressible Navier-Stokes equations with inflow boundary conditions for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary condition on inflow and outflow part of the boundary while the full Navier boundary conditions on the wall $Γ_{0}$ for the velocity filed are considered. For our result, there are no restrictions on the amplitude of friction coefficients $α$, and the appropriately large hypothesis for the viscous coefficients $μ$ is enough. One …
In [F. Jiang, S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014) 831--863], Jiang et.al. investigated the instability of Rayleigh--Taylor steady-state of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $Ω$ of class $C^2$. In particular, they proved the steady-state is nonlinearly unstable under a restrictive condition of that the derivative function of steady density possesses a positive lower bound. In this article, by exploiting a standard energy functional a…
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods…
From the steady Stokes and Navier-Stokes models, a penalization method has been considered by several authors for approximating those fluid equations around obstacles. In this work, we present a justification for using fictitious domains to study obstacles immersed in incompressible viscous fluids through a simplified version of Brinkman's law for porous media. If the scalar function $ψ$ is considered as the inverse of permeability, it is possible to study the singularities of $ψ$ as approximations of obstacles (when $ψ$ tends to $\infty$) or of the domain corresponding to the fluid (when $ψ= …
We study existence, uniqueness, regularity and asymptotic spatial behavior of a Navier-Stokes flow past a body moving by a time-periodic translational motion of period $T$, and with zero average. For example, $\mathscr B$ moves in an oscillating fashion. The flow is also time-periodic with same period $T$. However, sufficiently ``far" from the body, the oscillatory component decays faster than the averaged component, so that the flow shows there a distinctive steady-state character. This provides a rigorous proof of the ``steady streaming" phenomenon.
Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes
Faculty of Mathematics and Physics of the Charles University（查理大学数学物理学院）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）
In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is related to "cellular instabilities" occuring in detonation and MHD. Our results extend to multiple dimensions the results of Franz Achleitner and Peter Szmolyan (Saddle-node bifurcation of viscous profiles) on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results o…
In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the follo…
In this work we investigate the existence of weak solutions for steady flows of generalized incompressible and homogeneous viscous fluids. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the exponent $q$ that characterizes the flow depends on the space variable: $q=q(\mathbf{x})$. For the associated boundary-value problem we prove the existence of weak solutions for any variable exponent $q\geqα>\frac{2N}{N+2}$, where $α=\mathrm{ess}\inf q$. This work improves all the known existence results in the sense that the lowest possible bound of $q$ is attai…
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.
We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized…
This paper solves the global well-posedness and stability problem on a special $2\frac12$-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in $L^p$-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion, and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced …
We study the asymptotic spatial behavior of the vorticity field, $ω(x,t)$, associated to a time-periodic Navier-Stokes flow past a body, $\mathscr B$, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, $\mathcal R$, $ω$ decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by $\barω$ its time-average over a period and by $ω_P:=ω-\barω$ its purely periodic component, we prove that inside $\mathcal R$, $\barω$ has the same algebraic decay as that known for the associated steady-state problem, whereas $ω_P$ decays eve…
In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ with a moving flat boundary $\{Y=0\}$. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the s…
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).
The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded domains is proposed, based on choosing a family of approximate steady states, and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family of approximate steady states is analyzed by means of a system of an ODE for the parameter $ξ$ that describes the family, cou…
We analyze the steady motion of a viscous incompressible fluid in a three-dimensional channel containing an obstacle through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet, together with the standard no-slip assumption on the obstacle and on the remaining walls of the domain. Explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructi…
The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of the gas flow. While the steady Navier-Stokes equations with smooth divergence-free external force always have at least one…
Lie symmetry group method is applied to study the boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
In this paper we consider the 3-dimensional incompressible Oldroyd-B model. First, we establish two results of the global existence for different kinds of the coupling coefficient $k$. Then, we prove that the solutions $(u,τ)$ are globally steady when $k^m\rightarrow k>0$, though $(u,τ)$ corresponds to different decays for different kinds of $k>0~$. Finally, we show that the energy of $u(t,x)$ will have a jump when $k\rightarrow 0$ in large time, which implies a non-steady phenomenon. In a word, we find an interesting physical phenomenon of \eqref{1} such that smaller coupling coefficient $k$ …
Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of …
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
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.
Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) en…
We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence o…
In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ under the assumption of a moving boundary at $\{Y=0\}$. The validity of the boundary layer expansion and the convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the nonshear flows.
We study solutions to the obstacle problem for the stationary Navier--Stokes system in a~two dimensional exterior domain (flow past a prescribed body). We prove that the classical Leray solution to this problem is always nontrivial. No additional condition (on symmetry or smallness, etc.) is assumed. This is a complete extension of a~classical result of C.J. Amick (Acta Math. 1988) where nontriviality was proved under symmetry assumption.
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $Ω= Ω_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,ρ) \in W^2_p(Ω) \times W^1_p(Ω)$, where $v$ is the velocity of the fluid and $ρ$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar ρ\equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar ρ)$. The term $u \cdot \nabla w$ in the continuity eq…
We prove the existence of unique regular solutions of steady-state buoyancy-driven flows of viscous incompressible heat-conducting fluids in 3D open channels under mixed boundary conditions. The model takes into account the phenomena of the viscous energy dissipation.
The outflow problem for the viscous full two-phase flow model in a half line is investigated in the present paper. The existence, uniqueness and nonlinear stability of the steady-state are shown respectively corresponding to the supersonic, sonic or subsonic state at far field. This is different from the outflow problem for the isentropic Navier-Stokes equations, where there is no steady-state for the subsonic state. Furthermore, we obtain either exponential time decay rates for the supersonic state or algebraic time decay rates for supersonic and sonic states in weighted Sobolev spaces.
We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result fo…
