On the spectrum of the Stokes operator

无界域上具有Hodge、Navier和Robin边界条件的Stokes算子的$H^\infty$-演算

$H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

本文研究了一致$C^{2,1}$无界域上Stokes算子，通过Hodge边界条件建立Hörmander泛函演算，并利用Robin型扰动为Navier型条件建立$H^\infty$-演算有界性，从而证明了极大$L^p$-正则性。

关于广义稳定斯托克斯算子的同伦公式，与de Rham复形相关

On a homotopy formula for generalized steady Stokes' operators, associated with the de Rham complex

本文研究了与de Rham复形相关的广义稳定斯托克斯算子的左、右及双侧基本解，并给出了该算子的同伦公式。

$H^\infty$-calculus for the surface Stokes operator and applications

We consider a smooth, compact and embedded hypersurface $Σ$ without boundary and show that the corresponding (shifted) surface Stokes operator $ω+A_{S,Σ}$ admits a bounded $H^\infty$-calculus with angle smaller than $π/2$, provided $ω>0$. As an application, we consider critical spaces for the Navier-Stokes equations on the surface $Σ$. In case $Σ$ is two-dimensional, we show that any solution with a divergence-free initial value in $L_2(Σ,\mathsf{T}Σ)$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.

Semigroup theory for the Stokes operator with Navier boundary condition on $L^p$ spaces

We consider the incompressible Navier-Stokes equations in a bounded domain with $\mathcal{C}^{1,1}$ boundary, completed with slip boundary condition. Apart from studying the general semigroup theory related to the Stokes operator with Navier boundary condition where the slip coefficient $α$ is a non-smooth scalar function, our main goal is to obtain estimate on the solutions, independent of $α$. We show that for $α$ large, the weak and strong solutions of both the linear and non-linear system are bounded uniformly with respect to $α$. This justifies mathematically that the solution of the Navi…

Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori in $\mathbb{R}^3$

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.

Muckenhoupt类之外的幂权重的Stokes算子

The Stokes Operator with Power Weights Outside the Muckenhoupt Class

本文在半空间无滑移边界条件下，研究Stokes预解问题在加权L^p空间中的估计，权重包括Muckenhoupt范围内外的幂权重，证明Stokes算子生成有界解析C_0半群并具有角度0的有界H^∞演算。

有界及外部 $C^1$ 区域中 Stokes 算子的预解估计

Resolvent Estimates for the Stokes Operator in Bounded and Exterior $C^1$ Domains

本文在有界及外部 $C^1$ 区域中建立 Stokes 算子的 $L^q$ 预解估计，并证明其在任意 $1<q<\infty$、$d\ge 2$ 时生成有界解析半群。

Two-Scale Convergence of Unsteady Stokes Type Equations

In this paper we study the homogenization of unsteady Stokes type equations in the periodic setting. The usual Laplace operator involved in the classical Stokes equations is here replaced by a linear elliptic differential operator of divergence form with periodically oscillating coefficients. Our mean tool is the well known two-scale convergence method.

Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle

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.

Behaviour of the Stokes operators under domain perturbation

Depending of the geometry of the domain, one can define --at least-- three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.

On the Blow-up criterion of Navier-Stokes equation associated with the Weinstein operator

In this paper we give Navier-Stokes system associated with the Weinstein operator $(NSW)$ (see Eq.\eqref{11}), We study the existence and uniqueness of solutions to equations (NSW) in $L_α^{p}\left(\mathbb{R}_{+}^{d+1}\right), 2 α+d+2<p \leq \infty$, and we proved some properties of the maximal solution of equation. If the maximum time $T^*$is finite, we establish that the growth of $\left\| u ( {t}) \right\|_{L ^ {p}_α} $ is at least of the order of ${\left(T^{*}-t\right)^{{-\frac{2 p}{p-2 α-d-2}}}},$ fo rall $t$ in $\left[0, T^{*}\right]$, also we give some blow-up results.

Resolvent Estimates and Maximal Regularity in Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders

We study resolvent estimate and maximal regularity of the Stokes operator in $L^q$-spaces with exponential weights in the axial directions of unbounded cylinders of ${\mathbb R}^n,n\geq 3$. For straights cylinders we obtain these results in Lebesgue spaces with exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for general cylinders with several exits to infinity we prove that the Stokes operator in $L^q$-spaces with exponential weight along the axial directions generates an exponentially decaying analytic semigroup and has maximal regularity. The pr…

Generic properties of the spectrum of the Stokes system with Dirichlet boundary condition in R^3

Let (SD_Ω) be the Stokes operator defined in a bounded domain Ωof R^3 with Dirichlet boundary conditions. We prove that, generically with respect to the domain Ωwith C^5 boundary, the spectrum of (SD_Ω) satisfies a non resonant property introduced by C. Foias and J. C. Saut to linearize the Navier-Stokes system in a bounded domain Ωof R^3 with Dirichlet boundary conditions. For that purpose, we first prove that, generically with respect to the domain Ωwith C^5 boundary, all the eigenvalues of (SD_Ω) are simple. That answers positively a question raised by J. H. Ortega and E. Zuazua. The proofs…

The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system

We study the inverse of the divergence operator on a domain $Ω\subset R^3$ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space $L^p(Ω)$ for any $1< p < 3$, with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.

On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet

In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.

Stokes系统响应算子图上单点的包络法

The enclosure method using a single point on the graph of the response operator for the Stokes system

本文针对时域Stokes系统的反障碍物问题，利用最新时域包络法，仅基于边界上有限时间内的速度与Cauchy应力数据，提出了两类提取未知障碍物几何信息的公式。

非光滑域中 Stokes 算子在 $L^\infty$ 中的预解估计

Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains

本文在非光滑域上建立了 Stokes 算子在有界螺线管函数空间中的预解估计，证明了其生成一致有界解析半群，并给出了尖锐的光滑性条件，关键步骤是尺度相关的压力-速度 $L^q$ 估计。

Asymptotic limit for the Stokes and Navier-Stokes problems in a planar domain with a vanishing hole

We show that the eigenvalues of the Stokes operator in a domain with a small hole converge to the eigenvalues of the Stokes operator in the whole domain, when the diameter of the hole tends to 0. The convergence of the eigenspaces and the convergence of the Stokes semigroup are also established. Concerning the Navier--Stokes equations, we prove that the vorticity of the solution in the perforated domain converges as the hole shrinks to a point $r$ to the vorticity of the solution in the punctured domain (i.e. the whole domain with the point $r$ removed). The main ingredients of the analysis ar…

Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation

We introduce a global scheme on the n-torus of a controlled incompressible Navier-Stokes equation in terms of a coupled controlled infinite ODE-system of Fourier-modes with smooth data. We construct a scheme of global approximations related to linear partial integrodifferential equations in dual space which are uniformly bounded in dual Sobolev spaces with polynomially decaying modes. The scheme is based on some infinite matrix algebra related to weakly singular integrals, and a Trotter-product formula for dissipative operators which leads to rigorous existence results and a uniform bound for …

Existence of weak solutions for inhomogeneous generalized Navier-Stokes equations

We prove existence of weak solutions for the fully inhomogeneous, stationary generalized Navier-Stokes equations for shear-thinning fluids. Our proof is based on the theory of pseudomonotone operators and the Lipschitz truncation method, whose application is presented as a general result. Our approach requires a smallness and a regularity assumption on the data; we show that this is inevitable in the framework of pseudomonotone operators.

On estimates for the Stokes flow in a space of bounded functions

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.

On critical spaces for the Navier-Stokes equations

The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the $L_p$-$L_q$ setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $\mathcal{H}^\infty$-calculus with $\mathcal{H}^\infty$-angle 0, and the real and complex interpolation spaces of these operators are identified.

The Stokes operator in two-dimensional bounded Lipschitz domains

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutio…

A Friedlander type estimate for Stokes operators

Let $Ω\subset \mathbb{R}^d$ be a bounded open connected set with Lipschitz boundary. Let $A^N$ and $A^D$ be the Neumann Stokes operator and Dirichlet Stokes operator on $Ω$, respectively. Further let $λ_1^N \leq λ_2^N \leq \ldots$ and $λ_1^D \leq λ_2^D \leq \ldots$ be the eigenvalues of $A^N$ and $A^D$ repeated with multiplicity, respectively. Then \[ λ_{n+1}^N < λ_n^D \] for all $n \in \mathbb{N}$.

外区域中广义 Oseen 演化算子的正则性质，及其在 Navier-Stokes 初值问题中的应用

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。

三维Lipschitz区域中Stokes算子的预解估计

Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain

通过改进已有方法，建立三维Lipschitz区域Stokes算子的L^p预解估计，证明其生成一致有界解析半群，补充了插值相关定理，修正了大p下估计可能失效的传统认知。

Analysis of Boundary-Domain Integral Equations to the Mixed BVP for a Compressible Stokes System with Variable Viscosity

The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.

A non-local approach to the generalized Stokes operator with bounded measurable coefficients

We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_σ (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.

The unstable spectrum of the Navier-Stokes operator in the limit of vanishing viscosity

A general class of linear advective PDEs, whose leading order term is of viscous dissipative type, is considered. It is proved that beyond the limit of the essential spectrum of the underlying inviscid operator, the eigenvalues of the viscous operator, in the limit of vanishing viscosity, converge precisely to those of the inviscid operator. The general class of PDEs includes the equations of incompressible fluid dynamics. Hence eigenvalues of the Navier-Stokes operator converge in the inviscid limit to the eigenvalues of the Euler operator beyond the essential spectrum.

Stokes Resolvent Estimates in Spaces 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…

Existence theorem of a weak solution for Navier-Stokes type equations associated with de Rham complex

Let $ \{d_q, Λ^{q} \} $ be de Rham complex on a smooth compact closed manifold $X$ over $ \mathbb{R}^3 $ with Laplacians $Δ_{q} $. We consider operator equations, associated with the parabolic differential operators $\partial_t + Δ_2 + N^{2} $ on the second step of complex with nonlinear bi-differential operator of zero order $ N^{2} $. Using by projection on the next step of complex we show that the equation has unique solution in special Bochner-Sobolev type functional spaces for some (small enough) time $ T^* $.

Explicit characterisation of the fractional power spaces of the Dirichlet Laplacian and Stokes operators

We identify explicitly the fractional power spaces for the $L^2$ Dirichlet Laplacian and Dirichlet Stokes operators using the theory of real interpolation. The results are not new, but we hope that our arguments are relatively accessible.

Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface

This paper is devoted to the well-posedness of the stationary $3$d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of Gérard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some stro…

On the stability phenomenon of the Navier-Stokes type Equations for Elliptic Complexes

Let ${\mathcal X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\geq 2$, $E^i$ be smooth vector bundles over $\mathcal X$ and $\{A^i,E^i\}$ be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with $\{A^i,E^i\}$ on the scale of anisotropic Hölder spaces over the layer ${\mathcal X} \times [0,T]$ with finite time $T > 0$. Using the properties of the differentials $A^i$ and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm op…

An Open Mapping Theorem for the Navier-Stokes Equations

We consider the Navier-Stokes equations in the layer ${\mathbb R}^n \times [0,T]$ over $\mathbb{R}^n$ with finite $T > 0$. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all $t \in [0,…

Maximal $L_p$-$L_q$ regularity for the Stokes equations with various boundary conditions in the half space

We prove resolvent $L_p$ estimates and maximal $L_p$-$L_q$ regularity estimates for the Stokes equations with Dirichlet, Neumann and Robin boundary conditions in the half space. Each solution is constructed by a Fourier multiplier of $x'$-direction and an integral of $x_N$-direction. We decompose the solution such that the symbols of the Fourier multipliers are bounded and holomorphic. We see that the operator norms are dominated by a homogeneous function of order $-1$ for $x_N$-direction. The basis are Weis's operator-valued Fourier multiplier theorem and a boundedness of a kernel operator. W…

Symmetry and monotonicity results for solutions of vectorial $p$-Stokes systems

In this paper we shall study qualitative properties of a $p$-Stokes type system, namely $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$},$$ where ${\boldsymbol Δ}_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $Ω$ and the function $\boldsymbol{ f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to procee…

Periodic Lp estimates by R-boundedness: Applications to the Navier-Stokes equations

General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic $L^p$ estimates of maximal regularity type are established from $\mathscr{R}$-bounds of the family of solution operators ($\mathscr{R}$-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier-Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.

R Boundedness, Maximal Regularity and Free Boundary Problems for the Navier-Stokes Equations

In this lecture note, we study free boundary problems for the Navier-Stokes equations with and without surface tension. The local well-posedness, the global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the Lp in time and Lq in space framework. The tool in proving the local well-posedness is the maximal Lp-Lq regularity for the Stokes equations with non-homogeneous free boundary conditions. The approach here of proving the maximal Lp-Lq regularity is based on the R bounded solution operators of the generalized resolvent problem for the Stokes equations wi…

Distribution Dependent SDEs for Navier-Stokes Type Equations

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

Decay of solutions to the linearized free surface Navier-Stokes equations with fractional boundary operators

In this paper we consider a slab of viscous incompressible fluid bounded above by a free boundary, bounded below by a flat rigid interface, and acted on by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is well-known that equilibria are asymptotically stable but that the rate of decay to equilibrium depends heavily on whether or not surface tension forces are accounted for at the free interface. The aim of the paper is to better understand the decay rate by studying a generalization of the linearized dynamics in which the surface tension operator is replaced by a more ge…

Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC 2010: 35Q30, 58J65

Existence and smoothness of the Navier-Stokes equations and semigroups of linear operators

Based on Leray's formulation of the Navier-Stokes equations and the conditions of the exact linear representation of the nonlinear problem found in this paper, a compact explicit expression for the exact operator solution of the Navier-Stokes equations is given. It is shown that the introduced linear operator for Leray's equations is the generator of one-parameter contraction semigroup. This semigroup yields the existence of a unique and smooth classical solution of the associated Cauchy problem of Navier-Stokes equations in space $\mathbb{R}^3$ under smooth initial conditions.

On some properties of the curl operator and their consequences for the Navier-Stokes system

We investigate some geometric properties of the $\operatorname{curl}$ operator, based on its diagonalizationand its expression as a non-local symmetry of the pseudo-derivative $(-Δ)^{1/2}$ among divergence-free vector fieldswith finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectorsof $(-Δ)^{-1/2}\operatorname{curl}$.The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one. Having observed that the non-linearity of Navier-Stokes has the structure of a cross-productand its weak (dis…

齐次加权函数空间中外区域上的Stokes算子：从弱理论到$\u2708^\u221e$-演算再到分数次定义域

The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains

本文研究光滑外区域上带Muckenhoupt权的齐次加权Sobolev空间中的Stokes算子，证明其有界$\u2708^\u221e$-演算性质，刻画分数次幂定义域的复插值结构，还得到加权变分不等式、半群衰减估计及$L^p$-极大正则性等结果。

On the spectral properties for the linearized problem around space-time periodic states of the compressible Navier-Stokes equations

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of $\mathbb{R}^n$ ($n=2,3$), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as $t\rightarrow\infty$.

On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations

The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dotΩ=Ω_+\cupΩ_-$, where $Ω_\pm$ are uniform $W_r^{2-1/r}$ domains of $N$-dimensional Euclidean space $\mathbf{R}^N$ ($N\geq 2$, $N<r<\infty$). More precisely, given a uniform $W_r^{2-1/r}$ domain $Ω$ with two boundaries $Γ_\pm$ satisfying $Γ_+\capΓ_-=\emptyset$, we suppose that some hypersurface $Γ$ divides $Ω$ into two sub-domains, that is, there exist domains $Ω_\pm\subsetΩ$ such that $Ω_+\capΩ_-=\emptyset$ and $Ω\setminusΓ=Ω_+\cupΩ_-$, where $Γ\cap…

Faber-Krahn 不等式对 Stokes 算子是否成立？

Is the Faber-Krahn inequality true for the Stokes operator?

本文研究不可压缩 Dirichlet-Stokes 算子第一特征值的最小化问题，证明球体在二维满足一、二阶最优性条件，但在三维不满足，从而说明 Faber-Krahn 不等式在二维可能成立、在三维不成立。

特征值不等式及Lamé算子和Stokes算子热迹的三项渐近公式

Eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Lamé operator and Stokes operator

本文建立了Laplace、Lamé、Stokes、屈曲和夹紧板算子特征值问题的联系，证明Stokes和Laplace特征值是Lamé特征值的极限，并给出了特征值不等式及热迹的三项渐近公式。

粗糙区域与无界区域上 Stokes 算子的 $H^\infty$-演算

$H^\infty$-calculus for Stokes operators on rough and on unbounded domains

本文基于 Kunstmann–Weis 抽象比较原理，统一证明并概述粗糙及无界区域上 Stokes 算子的有界 $H^\infty$-演算，并新证有界凸区域中 Neumann 型边界条件情形。

Semi-classical propagation of singularity for Stokes system

We study the quasi-mode of Stokes system posed on a smooth bounded domain with Dirichlet boundary condition. We prove that the high energy L2 norm of solutions concentrate on the bi-characteristic of Laplace operator as matrix-valued Radon measure. Moreover, we prove that the support of such measure is invariant under Melrose-Sjöstrand flow.

无限深度流体弹性Stokes波的调制谱

Modulational spectrum of infinite-depth hydroelastic Stokes waves

该研究分析无限深度流体弹性Stokes波的调制谱，通过解耦特征值推导调制稳定性判据，识别弹性弯曲产生的稳定岛，揭示无限深度问题的奇异性。

Navier-Stokes-Poisson系统中小振幅冲击轮廓的谱稳定性

Spectral stability of shock profiles for the Navier-Stokes-Poisson system

研究Navier-Stokes-Poisson系统中小振幅冲击轮廓的谱稳定性，通过扩展的Evans函数框架证明零特征值的简并性。

On the stability of shear flows of Prandtl type for the steady Navier-Stokes equations

In this paper, we prove the stability of shear flows of Prandtl type as $ \big(U(y/\sqrtν),0\big)$ for the steady Navier-Stokes equations under a natural spectral assumption on the linearized NS operator. We develop a direct energy method combined with compact method to solve the Orr-Sommerfeld equation.