Lower bounds for the spectrum of the Laplace and Stokes operators

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.

Triebel-Lizorkin-Lorentz spaces and the Navier-Stokes equations

We derive basic properties of Triebel-Lizorkin-Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel-Lizorkin-Lorentz spaces to be of class $\mathcal{HT}$, to have property $(α)$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^\infty$-calculus. This is finally applied to derive local strong well-posedness for the Navier-Stokes equations on corresponding Triebel-Lizorkin-Lorentz ground spaces.

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

关于各类拉普拉斯算子在子流形上的限制

On the restriction of various Laplace operators on submanifolds

该研究围绕黎曼流形上Navier-Stokes方程研究中流形嵌入外围欧氏空间的场景，探究外围与流形上扩散算子的关联，给出了三维空间一般曲面对应关系的精确刻画。

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。

On the spectrum of the Stokes operator

We prove Li--Yau-type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier--Stokes equations.

Spectrum of generalized Hodge-Laplace operators on flat tori and round spheres

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

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,…

非局部时间问题：Rayleigh-Stokes型分数方程

Non-local time problem for the Rayleigh--Stokes type fractional equations

本文研究了Rayleigh-Stokes方程中Caputo导数的非局部时间问题，通过拉普拉斯变换和傅里叶方法分析了初始边界值问题的适定性。

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…

Vanishing properties of sign changing solutions to p-Laplace type equations in the plane

We study the nonlinear eigenvalue problem for the p-Laplacian, and more general problem constituting the Fucik spectrum. We are interested in some vanishing properties of sign changing solutions to these problems. Our method is applicable in the plane.

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…

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

We analyze the interior controllability problem for a nonlocal Schrödinger equation involving the fractional Laplace operator $(-Δ)^s$, $s\in(0,1)$, on a bounded $C^{1,1}$ domain $Ω\subset\mathbb{R}^n$. The controllability from a neighborhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. The results follow applying the multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian, and from an explicit computation of the spectrum of the operator in the one-dimensional c…

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.

The spectral estimates for the Neumann-Laplace operator in space domains

In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Poincaré-Sobolev inequalities that are obtained with the help of the composition operators theory for uniform Sobolev spaces. These composition operators are induced by a generalizations of conformal mappings that are mappings of bounded $2$-dilatation ($2$-quasiconformal mappi…

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。

Keller-Segel型系统的定量分析及其应用

Quantitative analysis and its applications for Keller-Segel type systems

该研究运用De Giorgi迭代获得Keller-Segel型系统解的精细上界估计，并将其用于弱解大时间行为、模型正则性及多类趋化流体系统的渐近稳定性与Hölder连续性分析。

Laplace-Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to o…

水弹性Stokes波的Benjamin-Feir谱

Benjamin-Feir spectrum of hydroelastic Stokes waves

针对二维有限深无旋Euler方程的小振幅水弹性Stokes波，在长波Floquet区域解析了从原点分岔的所有四个Bloch特征值，给出了Benjamin-Feir不稳定性的显式判据，并揭示了Wilton型共振消失的条件。

关于带混合插值Hardy项的p-Laplace局部与非局部算子的特征值与Fučík谱

On the eigenvalues and Fuč\'ık spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

本文研究带插值Hardy项的混合局部与非局部p-Laplacian算子特征值问题，建立了Hardy不等式，证明了特征值存在性与性质，并深入分析了Fučík谱、区域形状优化、特征函数正则性及Faber-Krahn不等式。

Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $δ>0$ and for an arbitrarily large $L>0$ …

On the Spectrum of geometric operators on Kähler manifolds

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin…

The Dirichlet isospectral problem for trapezoids

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result, which was only concerned with the Neumann Laplace spectrum.

Spectral invariants of the Stokes problem

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $Ω$ and the surface area of the boundary $\partial Ω$ in terms of the spectrum of the Stokes problem. As an application, we show that an $n$-dimensional ball is uniquely defined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

Spectral rigidity for spherically symmetric manifolds with boundary

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided th…

基于 $\mathrm{L}^p$ 的闭 manifold 上的 Sobolev 理论：向量值问题

$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Vector-valued problems

本文研究了闭 manifold 上向量值 PDEs 的适定性及 $\mathrm{L}^p$ 基础正则性，通过变分方法和经典理论，证明了 $\mathrm{W}^{m,p}$ 正则性，并应用于 Navier-Stokes 方程解的存在性。

输运-扩散方程与Navier-Stokes方程中宽频谱强迫力的重构

Reconstruction of wide spectrum forcing in transport-diffusion and Navier-Stokes equations

本文研究在系统状态不完整时重构未知驱动强迫力的问题，提出迭代和连续时间两种重构算法，证明了在观测分辨率足够高且参数适当调整时可保证收敛，并将可重构强迫力类别扩展至宽频谱范围。

Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations

Recently, the Navier-Stokes-Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we prove that the global attractor of the $D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one…

Maxwell和Stokes算子与椭圆微分复形相关

Maxwell's and Stokes' operators associated with elliptic differential complexes

提出一种基于同调代数微分算子复形生成偏微分方程系统的新技术，涵盖椭圆、抛物和双曲算子，可生成Maxwell、Navier-Stokes等现代数学物理方程。