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 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 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.
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.
We prove that the Stokes semigroup is a bounded analytic semigroup on $L^{\infty}_σ$ of angle $π/2$ for two-dimensional exterior domains. This result is an end point case of the $L^{p}$-boundedness of the semigroup for $p\in (1,\infty)$, established by Borchers and Varnhorn (1993) and an extension of finite time $L^{\infty}$-estimates studied by the author and Giga (2014). The proof is based on the non-existence result of bounded steady flows (the Stokes paradox) and some asymptotic formula for the net force of the Stokes resolvent.
In this study, we consider a linearized compressible flow structure interaction PDE model for which the interaction interface is under the effect of material derivative term. While the linearization takes place around a constant pressure and density components in structure equation, the flow linearization is taken with respect to a non-zero, fixed, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms which causes the coupled system to be nondissipative. We analyze the long time dynamics in the sense of asymptotic (strong) stability …
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…
The asymptotic stability of two-dimensional stationary flows in a non-symmetric exterior domain is considered. Under the smallness condition on initial perturbations, we show the stability of the small stationary flow whose leading profile at spatial infinity is given by the rotating flow decaying in the scale-critical order $O(|x|^{-1})$. Especially, we prove the $L^p$-$L^q$ estimates to the semigroup associated with the linearized equations.
We consider the linear thermoelastic plate equations with free boundary conditions in uniform $C^4$-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $L^p$-spaces for all $p\in(1,\infty)$ and has maximal $L^q$-$L^p$-regularity on finite time intervals. On bounded $C^4$-domains, we obtain exponential stability.
In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.
The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. Another tool is a Drazin-like inverse operator $B$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This oper…
We show that the Stokes operator defined on $\mathrm{L}^p_σ (Ω)$ for an exterior Lipschitz domain $Ω\subset \mathbb{R}^n$ $(n \geq 3)$ admits maximal regularity provided that $p$ satisfies $| 1/p - 1/2| < 1/(2n) + \varepsilon$ for some $\varepsilon > 0$. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on $\mathrm{L}^p_σ(Ω)$ for such $p$. In addition, $\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions t…
Let $Ω\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-Δ)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if $T$ is a self-adjoint semigroup on $L^2(Ω)$ satisfying a fractional Gaussian estimate in the sense that $|T(t)f|\le e^{-t(-Δ)_{\RR^N}^s}|f|$, $0\le t \le 1$, $f\in L^2(Ω)$, then $T$ defines a bounded holomorphic semigroup of angle $\fracπ{2}$ that interpolates on $L^p(Ω)$, $1\le p<\infty$. Using a duality argument we prove that the same result also holds on the space…
In this paper, we study regularizing effects of the composition operator $S(t)\mathbb{P}\partial$ for the Stokes semigroup $S(t)$ and the Helmholtz projection $\mathbb{P}$ in a space of bounded functions. We establish new a priori $L^{\infty}$-estimates of the operator $S(t)\mathbb{P}\partial$ for a certain class of domains including bounded and exterior domains. They imply unique existence of mild solutions of the Navier-Stokes equations in a space of bounded functions.
The Stokes equation on a domain $Ω\subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\infty$. The situation is very different for the case $p=\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. Nevertheless it was recently proved by the first and the second author of this paper by a contradiction argument that the Stokes operator generates an analytic semigroup on spaces of bounded functions for a large class of domains. This paper presents a ne…
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…
Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $Ω\subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- γ(x) (ν\wedge B_{tan}) = 0, γ(x) > 0, \forall x \in Γ= \partial Ω.$ We study the case when $Ω= \{x \in {\mathbb R^3}:\: |x| > 1\}$ and $γ\neq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $γ(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The eigenvalues $λ_k$ of $G$ with ${\rm Re}\: λ_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{λ_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $\min_{x\in Γ} γ(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $…
