Franz Gmeineder, Daniel Spector
We establish a family of inequalities that allow one to estimate the $\mathrm{L}^{q}$-norm of a matrix-valued field by the $\mathrm{L}^{q}$-norm of an elliptic part and the $\mathrm{L}^{p}$-norm of the matrix-valued curl. This particularly extends previous work by Neff et al. and, as a main novelty, is applicable in the regime $p=1$.
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For a bounded three-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditi…
Korn's inequality in anisotropic Sobolev spaces
本文证明经典Korn不等式及其涉及非线性连续映射的推广形式在各向异性Sobolev空间中成立，并指出Poincare不等式在该空间中同样有效。
Korn-Maxwell-Sobolev inequalities for general incompatibilities
本文在超线性增长条件下为广义不相容场建立尖锐的强制性 Korn–Maxwell–Sobolev 不等式，统一推广已有结果，并首次覆盖高阶 inc 张量情形。
Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative
Fractional Korn's inequalities without boundary conditions
本文在分数阶Sobolev空间中建立了无需边界条件的分数阶Korn第一和第二不等式，将适用范围从零Dirichlet边界条件推广到具有$C^1$边界或小Lipschitz常数边界的区域，并证明了平面凸区域情形，同时猜想可推广至任意Lipschitz区域。
This investigation develops basic methods for the multi-scale analysis for problems in thin porous layers. More precisely, we provide tools for the homogenization in case of "tangentially" periodic structures and dimensional reduction letting the layer thickness tend to zero proportional to the scale parameter $ε$. A crucial point is the identification of scale limits of functions $v_ε$ in subsets of function spaces characterized by uniform a priori estimates with respect to $ε$, arising for solutions of differential equations in heterogeneous media with thin layers, e.g., of a Navier-Stokes s…
Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities
该研究给出一般不相容性下极限Korn-Maxwell-Sobolev不等式成立的精确条件，将相关估计推广到p=1的临界情形，解决了此前的一个公开问题。
Necessary and sufficient conditions are exhibited for a Korn type inequality to hold between (possibly different) Orlicz norms of the gradient of vector-valued functions and of the deviatoric part of their symmetric gradients. As a byproduct of our approach, a positive answer is given to the question of the necessity of the same sufficient conditions in related Korn type inequalities for the full symmetric gradient, for negative Orlicz-Sobolev norms, and for the gradient of the Bogovskii operator.
This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains $Ω\subset{\mathbb R}^n$ presenting power type cusps of integer dimension $m\le n-2$, we prove the existence of solutions of the equation ${div}{\bf u}=f$ in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be impro…
Constant rank operators in Korn-Maxwell-Sobolev inequalities
本文研究降阶常秩算子的Korn-Maxwell-Sobolev不等式，通过在不等式左侧引入投影修正，将Gmeineder等人处理降阶椭圆算子的技术推广至常秩情形。
Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
本文刻画了适用于任意维度且不相容矩阵场的最优 Korn-Maxwell-Sobolev 不等式，推广了线性弹性中的 Korn 型不等式，并特别对二维情形进行了精细分析。
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both impro…
Korn and Poincaré-Korn inequalities for thin domains in $GSBD^p$
本文在 $GSBD^p$ 空间中证明了薄域的 Korn 第一不等式，常数 $C h^{-p}$ 在含平坦区域时最优，并处理了带边界条件的情形。
Stein-Weiss, and power weight Korn type Hardy-Sobolev Inequalities in $L^1$ norm
本文将 $L^1$ Stein-Weiss 不等式推广至正非整数幂权数据，以消失矩假设替代协同抵消条件，并在整数权情形建立更弱的 Korn 型 Hardy-Sobolev 不等式及对偶估计改进实例。
We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to proving a fractional version of the classi…
A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $θ_{h}$ of one base is considered; the diameter of $θ_{h}$ is of the same order as the plate relative thickness $h\ll1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support $θ_{h}$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $ch^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\…
A global solvability result of the Cauchy problem of the two-species Vlasov-Maxwell-Landau system near a given global Maxwellian is established by employing an approach different than that of [5]. Compared with that of [5], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov-Maxwell-Landau system for the case of $γ>-3$ and the one-species Vlasov-Maxwell-Boltzmann system for the case of …
Relaxation in Sobolev spaces and $L^1$ spectral gap of the 1D dissipative Boltzmann equation with Maxwell interactions
本文研究具有Maxwell相互作用的1D耗散Boltzmann方程在$H^s$和$L^1$空间中的松弛与收敛性，通过谱收缩分析和谱隙估计扩展了弱拓扑下的经典结果。
We show that in dimensions $n \geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Pal…
In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to the general case $ m^2 > 0 $ the results of Krieger-Sterbenz-Tataru ($d=4,5 $) and Rodnianski-Tao ($ d \geq 6 $), who considered the case $ m=0$. We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation str…
Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces
该研究针对可压缩Euler与Euler-Maxwell两类松弛系统，在Sobolev空间中结合新渐近展开方法与流函数技术，推导了不同初始数据情形下的全局收敛速率并证明强收敛性，给出时间一致误差估计。
Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension
本文在有界John域的加权Sobolev空间上证明了散度算子右逆、Poincaré、Korn及Fefferman-Stein等不等式，其距离权重指数仅需满足$βp>-(n-\dim_A(\partialΩ))$，条件较文献更宽松。
The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{L^1} ] is shown to hold if and only if (A(D)) is elliptic and canceling. Here (A(D)) is a homogeneous linear differential operator (A(D)) of order (k) on (\mathbb{R}^n) from a vector space (V) to a vector space (E). The operator (A(D)) is defined to be canceling if [\bigcap_{ξ\in \mathbb{R}^n \setminus {0}} A(ξ)[V]={0}.] This result implies in particular the classical Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and Hodge-Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. I…
Spectral Compactness and Critical Lorentz Defects for Maxwell--Ohm Evolution with Prescribed Velocity
本文为Maxwell--Ohm演化建立谱弱-强稳定性准则，证明在临界Sobolev指标处稳定性尖锐失效，并构造出Lorentz力收敛到非零测度的反例。
Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary
本文研究了具有吸收边界条件的非线性Maxwell系统，证明了小数据解的全局存在性和指数衰减性，并通过迹估计和正则性分析改进了线性自治情况下的结果。
Endpoint Sobolev inequalities for vector fields and cancelling operators
本文刻画了满足端点Sobolev不等式的注入椭圆向量微分算子，其充要条件为消去条件，并统一了多个经典不等式，证明基于共消去算子核。
We prove the propagation of regularity, uniformly in time, for the scaled solutions of one-dimensional dissipative Maxwell models. This result together with the weak convergence towards the stationary state proven by Pareschi and Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1 norm towards it depending on the regularity of the initial data. In the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. This generalizes a recent result of Carlen, Carrillo and Carvalho (arXiv:0805.1051v1), in which, for we…
The time local and global well-posedness for the Maxwell-Schr{ö}dinger equations is considered in Sobolev spaces in three spatial dimensions. The Strichartz estimates of Koch and Tzvetkov type are used for obtaining the solutions in the Sobolev spaces of low regularities. One of the main results is that the solutions exist time globally for large data.
Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields
本文通过消没条件刻画使端点Sobolev估计成立的微分算子，统一并推广了Gagliardo-Nirenberg、Korn-Sobolev及Hodge复形等经典不等式，并借助对偶估计与调和分析技术证明，同时引入弱条件刻画$L^\infty$估计。
We prove a family of Sobolev inequalities of the form $$ \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} \le \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} $$ where $A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-order homogeneous linear differential operator with constant coefficients, $u$ is a vector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpeni…
Optimizers in Sobolev-curl inequalities
研究涉及 p-curl 算子的 Sobolev 型不等式，证明临界情形下极小元的存在性，并引入新的变分方法处理拟线性不定问题。
The Maxwell-Klein-Gordon equation $ \partial^α F_{αβ} = -Im(Φ\overline{D_β Φ}) $ , $ D^μD_μ Φ= m^2 Φ$ , where $F_{αβ} = \partial_α A_β - \partial_β A_α$, $D_μ = \partial_μ - iA_μ $, in the (3+1)-dimensional case is known to be unconditionally well-posed in energy space, i.e. well-posed in the natural solution space. This was proven by Klainerman-Machedon and Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge. The main purpose of the present paper is to establish that for both gauges this also holds true for data $Φ(0)$ in Sobolev spaces $H^s$ with less regularity, i.e.…
Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations
本文研究广义 Navier-Stokes-Maxwell 方程的适定性、磁螺度守恒、无粘极限及渐近稳定性，在不同分数阶拉普拉斯参数下建立了全局或局部适定性，并揭示了 NSM 与 MHD 在磁场收敛性和衰减速率上的差异。
KMS Inequalities: From Elliptic Operators to Constant Rank
本文介绍了Korn-Maxwell-Sobolev (KMS)不等式，并将其从椭圆算子扩展至常秩算子，通过引入修正项以处理更复杂的向量场结构，同时展示了其在两种情况下的应用。
Charge-free well-posedness and charge-driven norm inflation for the Euler-Maxwell system
研究二维不可压缩 Euler-Maxwell 系统，发现投影无电荷系统在亚光速条件下具有局部适定性，而未投影系统中电荷密度导致涡量范数膨胀，揭示了高斯约束对正则性与膨胀的区分作用。
We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized spherical harmonics expansion suited for Maxwell equations. In particular, our solution theory will allow us to give meaning…
Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures
该研究针对周期奇异结构上的Maxwell方程组，开发新的Floquet分解分析工具，证明了阶数最优的范数预解式收敛估计，无需谱分析且不依赖现有均匀化方法。
In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $μ_0^{-1/2}\operatorname{curl} η(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} μ_0^{-1/2} - μ_0^{1/2}\nabla ν(\mathbf{x}/\varepsilon) \operatorname{div} μ_0^{1/2}$, where $μ_0$ is a constant positive matrix, a matrix-valued function $η(\mathbf{x})$ and a real-valued function $ν(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (τ{\mathcal L…
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