arXiv subjects

Genqian Liu

Publications and source records attributed to Genqian Liu.

At least 19 recordsLinked to original sources

arXiv2025

Existence of smooth solutions of the Navier-Stokes equations in three-dimensional Euclidean space

Based on the essential connection of the parabolic inertia Lam\'{e} equations and Navier-Stokes equations, we prove the existence of smooth solutions of the incompressible Navier-Stokes equations in three-dimensional Euclidean space $\mathbb{R}^3$ by showing the existence and uniqueness of smooth solutions of the parabolic inertia Lam\'{e} equations and by letting a Lam\'{e} constant $\lambda$ tends to infinity (the other Lam\'{e} constant $\mu>0$ is fixed).

math.AP↗

arXiv2024

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

This paper is devoted to establish the most essential connections of the eigenvalue problems for the Laplace operator, Lamé operator, Stokes operator, buckling operator and clamped plate operator. We show that the $k$-th Stokes (respectively, Laplace) eigenvalue is the limit of the $k$-th Lamé eigenvalue for the Dirichlet or traction boundary condition as the Lamé coefficient $λ$ tends to $+\infty$ (respectively, to $-μ$). Furthermore, we establish the eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Laplace operator, the Lamé operator, the Stokes operator and buckling operator with the Dirichlet and traction boundary conditions.

math.DG↗

arXiv2024

Spectral asymptotics for linear elasticity with mixed boundary conditions

In this note, we shall show that the two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions which were given by Capoferri and Mann in \cite{CaMa-24} essentially are old well-known results due to T. Branson, P. Gilkey, B. Ørsted and A. Pierzchalski in \cite{BGOP}. In addition, we further point out that the so-called ``general formulae'' in \cite{SaVa-97, CaFrLeVa-23} and the calculations for several examples in \cite{CaMa-24, CaFrLeVa-23} are all wrong.

math.AP↗

arXiv2024

On an algorithm for two-term spectral asymptotic formulas

In the book [Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Amer. Math. Soc., Providence, RI, 1997], a key and central ``algorithm'' was established, by which the coefficients of two-term asymptotic expansions of the eigenvalue counting functions can be explicitly calculated for many partial differential operators under an additional geometric assumption. In this paper, we give a counter-example to this ``algorithm'' by discussing the case of elastic eigenvalues. This implies that the most conclusions in the above book written by Yu. Safarov and D. Vassiliev are fundamentally wrong because they are based on the erroneous ``algorithm''.

math.DG↗

arXiv2024

Two-term spectral asymptotics in linear elasticity on a Riemannian manifold

In this note, by explaining two key methods that were employed in \cite{Liu-21} and by giving some remarks, we show that the proof of Theorem 1.1 in \cite{Liu-21} is a rigorous proof based on theory of strongly continuous semigroups and pseudodifferential operators. All remarks and comments to paper \cite{Liu-21}, which were given by Matteo Capoferri, Leonid Friedlander, Michael Levitin and Dmitri Vassiliev in \cite{CaFrLeVa-22}, are incorrect. The so-called "numerical counter-examples" in \cite{CaFrLeVa-22} are useless examples for the two-term asymptotics of the counting functions of the elastic eigenvalues. Clearly, the conclusion and the proof of \cite{Liu-21} are completely correct.

math.SP↗

arXiv2023

Remarks on paper "Two-term spectral asymptotics in linear elasticity''

In this note, we shall point out that all ``numerically calculations'' and figures in \cite{CaFrLeVa-23} are wrong because these calculations are based on some incorrect formulas. Furthermore, by pointing out several serious errors in \cite{CaFrLeVa-23} and especially by Section 7, Proposition 7.1, Remarks 7.2--7.3, and Section 8 (a result of A. Pierzchalski and B. Ørsted) we show that the conclusions published by Matteo Capoferri, Leonid Friedlander, Michael Levitin and Dmitri Vassiliev (J Geom Anal (2023)33:242) as well as the main ``algorithm'' theory of the book \cite{SaVa-97} are completely wrong. Finally, we explain the correctness of proof of Theorem 1.1 in our paper \cite{Liu-21} by giving some remarks and putting the whole proof in Appendix (see also \cite{Liu-22b} and \cite{Liu-22c}).

math.SP↗

arXiv2022

Geometric invariants of spectrum of the Navier-Lamé operator

For a compact connected Riemannian $n$-manifold $(Ω,g)$ with smooth boundary, we explicitly calculate the first two coefficients $a_0$ and $a_1$ of the asymptotic expansion of $\sum_{k=1}^\infty e^{-t τ_k^\mp}= a_0t^{-n/2} \mp a_1 t^{-(n-1)/2}+a_2^\mp t^{-(n-2)/2} +\cdots+ a_m^\mp t^{-(n-m)/2} +O(t^{-(n-m-1)/2})$ as $t\to 0^+$, where $τ^-_k$ (respectively, $τ^+_k$) is the $k$-th Navier-Lamé eigenvalue on $Ω$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body $Ω$ and the surface area of the boundary $\partial Ω$ in terms of the spectrum of the Navier-Lamé operator. This gives an answer to an interesting and open problem mentioned by Avramidi in \cite{Avr10}. More importantly, our method is valid to explicitly calculate all the coefficients $a_l^\mp$, $2\le l\le m$, in the above asymptotic expansion. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lamé spectrum among all bounded elastic bodies with smooth boundary.

math.DG↗

arXiv2022

Determining Lamé coefficients by elastic Dirichlet-to-Neumann map on a Riemannian manifold

For the Lamé operator $\mathcal{L}_{λ,μ}$ with variable coefficients $λ$ and $μ$ on a smooth compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the elastic Dirichlet-to-Neumann map $Λ_{λ,μ}$. We show that $Λ_{λ,μ}$ uniquely determines partial derivatives of all orders of the Lamé coefficients $λ$ and $μ$ on $\partial M$. Moreover, for a nonempty open subset $Γ\subset\partial M$, suppose that the manifold and the Lamé coefficients are real analytic up to $Γ$, we prove that $Λ_{λ,μ}$ uniquely determines the Lamé coefficients on the whole manifold $\bar{M}$.

math.SP↗

arXiv2022

One can know the area and total curvatures of the boundary by hearing the resonances of a Stokes flow

By calculating full symbol for the Dirichlet-to-Neumann map $Λ$ of a Stokes flow, we establish the asymptotic expansion of the trace of the heat kernel for $Λ$. We also give a useful procedure, by which all coefficients of the asymptotic expansion can be explicitly calculated. These coefficients are the Steklov spectral invariants of $Λ$, which provide precise geometric information of the boundary for the Stokes flow. In particular, the first two coefficients show that the area and (total) mean curvature of the the boundary can be known by the Steklov eigenvalues of $Λ$.

math.DG↗

arXiv2022

Asymptotic Expansion of the Heat Trace of the Thermoelastic Dirichlet-to-Neumann map

This paper is devoted to study the asymptotic expansion of the heat trace of the Dirichlet-to-Neumann map for the thermoelastic equation on a Riemannian manifold with doundary. By providing a method we can obtain all the coefficients of the asymptotic expansion. In particular, we explicitly give the first two coefficients involving the volume and the total mean curvature of the boundary.

math.AP↗

arXiv2022

Determination of isometric real-analytic metric and spectral invariants for elastic Dirichlet-to-Neumann map on Riemannian manifolds

In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We prove that for a strong convex or extendable real-analytic manifold with boundary, the elastic Dirichlet-to-Neumann map $Ξ_g$ uniquely determines the metric $g$ of $Ω$ in the sense of isometry, thereby solving an open problem for the uniqueness of the metric under real-analytic setting. Furthermore, by calculating the symbol representation of the resolvent operator $(Ξ-τI)^{-1}$ we can explicitly obtain all coefficients $a_0, a_1 \cdots, a_{n-1}$ of the asymptotic expansion $\sum_{k=1}^\infty e^{-t τ_k}\sim \sum_{m=0}^{n-1} a_m t^{m+1-n} +o(1)$ as $t\to 0^+$, where $τ_k$ is the $k$-th eigenvalue of the elastic Dirichlet-to-Neumann map $Ξ_g$ (i.e., $k$-th elastic Steklov eigenvalue). These coefficients (spectral invariants) provide important geometric information for the manifold, which give an answer to another open problem for the elastic Steklov spectral asymptotics.

math.AP↗

arXiv2022

Asymptotic Expansions of The Traces of the Thermoelastic Operators

We obtain the asymptotic expansions of the traces of the thermoelastic operators with the Dirichlet and Neumann boundary conditions on a Riemannian manifold, and give an effective method to calculate all the coefficients of the asymptotic expansions. These coefficients provide precise geometric information. In particular, we explicitly calculate the first two coefficients concerning the volumes of the manifold and its boundary. As an application, by combining our results with the isoperimetric inequality we show that an $n$-dimensional geodesic ball is uniquely determined up to isometry by its thermoelastic spectrum among all bounded thermoelastic bodies with boundary.

math.SP↗

arXiv2021

The Payne conjecture for Dirichlet and Buckling eigenvalues

We prove the long-standing Payne conjecture that the $k^{\text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{\text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover, we show that the Payne conjecture is still true for $n$-dimensional case ($n\ge 2)$.

math.AP↗

arXiv2021

Spectral Invariants of the Magnetic Dirichlet-to-Neumann Map on Riemannian Manifolds

This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic Steklov eigenvalues.

math.AP↗

arXiv2021

Uniqueness of the scatterer for electromagnetic field with one incident plane wave

In this paper, we solve a longstanding open problem for determining the shape of an obstacle from the knowledge of the electric (or magnetic) far field pattern for the scattering of time-harmonic electromagnetic field. We show that the electric (or magnetic) far field patten ${\mathbf{E}}^\infty(\boldsymbolβ, {\boldsymbolα}_0, k_0)$ (or ${\mathbf{H}}^\infty (\boldsymbolβ, {\boldsymbolα}_0, k_0)$), known for all $\boldsymbolβ\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${\boldsymbolα}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely. The boundary condition on $\partial D$ is either the perfect conductor or the impedance one.

math.AP↗

arXiv2021

Uniqueness in inverse elastic scattering with one incident wave

In this paper, we give a positive answer to a longstanding open problem for determining the shape of an obstacle from the knowledge of the far field pattern for the scattering of time-harmonic elastic wave. We show that the elastic far field pattern by an incoming plane wave with a fixed frequency, a fixed incident direction and a fixed polarization determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely. The boundary condition on $\partial D$ is either the Dirichlet, or the Neumann, or the Robin one.

math.AP↗

arXiv2021

Uniqueness of scatterer in inverse acoustic obstacle scattering with a single incident plane wave

In this paper, we give a positive answer to a challenging open problem for recovering unknown obstacle (which is usually referred to as a scatterer) by acoustic wave probe associated to the Helmholtz equation. We show that the acoustic scattering amplitude $A(β, α_0, k_0)$, known for all $β\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${α}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely (The boundary condition on $\partial D$ is either the Dirichlet, or Neumann, or the impedance one).

math.AP↗

arXiv2020

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.

math.AP↗