Huyuan Chen, Hichem Hajaiej
In this paper, we develop the Littman-Stampacchia-Weinberger duality approach to obtain global W^1,p estimates for a class of elliptic problems involving Leray-Hardy operators and measure sources in a distributional framework associated with a dual formulation with a specific weight function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we prove Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields with sharp constants. This complements the former work by Costin-Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields.
In this paper, we present some suitable distributional identities of the solutions for nonhomogeneous elliptic equations involving the Hardy-Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems by the distributional identities. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy-Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.
In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
The purpose of this paper is to study the eigenvalues $\{λ_{μ,i} \}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$ -Δu+μ|x|^{-2}u=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm on}\ \ \partialΩ,$$ where $-Δ+\fracμ{|x|^2}$ is the Hardy-Leray operator with $μ\geq -\frac{(N-2)^2}{4}$ and $Ω$ is a smooth bounded domain with $0\inΩ$. We provide lower bounds of $\{λ_{μ,i} \}_i$ together with the Li-Yau's one for $μ>-\frac{(N-2)^2}{4}$ and Karachalio's one for $μ\in [-\frac{(N-2)^2}{4},0)$. Secondly, we obtain Cheng-Yang's type upper bounds for $λ_{μ,k}$. Finally, we get the Weyl's limit of eigenval…
We show that the sharp constant in the classical $n$-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for $n=2$ without the axisymmetry assumption.
We study existence and stability of solutions of (E 1) --$Δ$u + $μ$ |x| 2 u + g(u) = $ν$ in $Ω$, u = 0 on $\partial$$Ω$, where $Ω$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $μ$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $ν$ is a Radon measure on $Ω$. We show that the situation differs according $ν$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.
In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate $\pp$-Laplace operator.
We prove the existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$ (-Δ)^s u=\vartheta\frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb{R}^N).$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using of the moving plane method, in a nonlocal setting, on the whole $\mathbb{R}^N$ and by some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem…
We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient cond…
We study the kernel function of the operator u $\rightarrow$ L $μ$ u = --$Δ$u + $μ$ |x| 2 u in a bounded smooth domain $Ω$ $\subset$ R N + such that 0 $\in$ $\partial$$Ω$, where $μ$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $μ$ u = 0 in $Ω$ with boundary data $ν$ + k$δ$ 0 , where $ν$ is a Radon measure on $\partial$$Ω$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L …
Sharp Hardy-Leray inequality for solenoidal fields
本文计算了 $\mathbb{R}^N$ 上无源向量场Hardy-Leray不等式的最佳常数，在无对称性假设下推导并简化了该常数，并证明其在积分有限的函数空间内无法达到。
Fractional heat equation involving Hardy-Leray Potential
本文研究了含Hardy-Leray势的非局部抛物方程，证明了临界存在指数与Fujita型指数，并据此划分了非负上解不存在、局部解存在、正解有限时间爆破及整体解存在的参数条件。
We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy's inequality in the case $p=n\geq2.$ In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p=2<n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186--233, and for $p>n\geq1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421--441.
Eventual regularity and asymptotic behavior of Leray-Hopf weak solutions for the Hall-MHD system
研究不可压缩、粘性、电阻性Hall-MHD系统，证明二维Leray-Hopf弱解在有限时间后变光滑，三维中构造出磁涡度场最终获得额外正则性的弱解，并在初始数据低频频谱假设下结合广义Fourier分裂方法导出速度场和磁场解耦的代数衰减率。
Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness
本文刻画了二维非齐次 Navier-Stokes 系统在正时间下变为强解的 Leray-Hopf 解，基于强能量不等式与压力正则性建立了弱-强唯一性结果，并为该领域近期进展提供了统一框架。
Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions
针对临界耗散广义Leray问题，利用极大平滑、椭圆正则性与分数阶热半群作用，证明弱解全局正则性及最优衰减，并据此构造三维Navier-Stokes自相似解。
In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4} if n = 3, and t* <= K_{4} nu^{-3} || u(.,0) ||_{L2}^{2} if n = 4 (in particular, by reducing the current best known values for the constants K_{3}, K_{4}). Some related results of clear interest are also included (derived) in our discussion.
It has recently become common to study many different approximating equations of the Navier-Stokes equation. One of these is the Leray-$α$ equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution $u$ in the equation with $(1-α^2\triangle)u$ the operator $(1-α^2\triangle)$. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form $|ξ|^γ$ ($γ=2$ is the standard Navier-Stokes equation), and recently in [14] Tao also considered multipliers of the form $|ξ|^γ/g(|ξ|)$, where $g$ is (e…
We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case L^2_w(0,T;L^\infty(R^3)). Particularly, an open problem proposed in [KK] was solved.
We study existence and regularity of weak solutions for the following PDE $$ -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. $$ where $A(x,s) = A_+(x)χ_{\{s>0\}}+A_-(x)χ_{\{s\le 0\}}$ and $f(x,s) = f_+(x)χ_{\{s>0\}}+f_-(x)χ_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}μ\le A_{\pm} \le μ$, $A_{\pm}\in C(Ø)$ and $f_{\pm}\in L^N(Ø)$, we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C^{0,α}_{loc}$ for every $α\in (0,1)$ with precise estimates. Our methods relies on similar techniques as those deve…
Limit behaviors of blow up solutions for impressible Navier-Stokes equations are obtained.
Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation
本文研究具超耗散 α∈(1,5/4) 的不可压缩 Navier-Stokes 方程 Leray-Hopf 弱解的部分正则性，证明爆破点集的 Hausdorff 维数与盒计数维数具有由 α 决定的显式上界。
In this paper, we investigate the Leray problem for steady Navier-Stokes system under full slip boundary conditions in a two dimensional channel with straight outlets. The existence of solutions with arbitrary flux in a general channel with slip boundary conditions is established, which tend to the shear flows at far fields. Furthermore, if the flux is suitably small, the solutions are proved to be unique. One of the crucial ingredients is to construct an appropriate flux carrier and to show a Hardy type inequality for flows with full slip boundary conditions.
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -Δu+\fracμ{|x|^2} u=u^p\quad {\rm in}\quad Ω\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partialΩ. \end{equation} We classify the isolated singularities and obtain the existence, the stability of positive solutions of (\ref{0}). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.
In this paper, we establish some $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$ \limsup\limits_{\varrho\rightarrow0} \varrho^{1-\frac{2}{p}-\sum\limits^{3}_{j=1}\frac{1}{q_{j}}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \leq\varepsilon, ~~\frac{2}{p}+\sum\limits^{3}_{j=1}\frac{1}{q_{j}} \leq2~~~~~\text{with}~q_{j} > 1;\\$$$$ \sup_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}} <2\quad \text{with}\, 1<q_{j}<\infty;$$ $$\…
It is a simple consequence of the Cafarelli-Kohn-Nirenberg theory that every possible singularity in a thin Haussdorff-measurable set of a Leray- Hopf solution of the incompressible Navier Stokes equation is on the tip of a small open cone, where the solution is smooth. Using global regularity results for weak Hopf-Leray solutions this potential singularity can be analyzed by investigation of the asymptotic behavior at infinite time of a solution of a related initial-boundary value problem posed in transformed coordinates on a cylinder. Next to some new consequences such as global regularity o…
Global Regularity of weak solutions to the generalized Leray equations and its applications
本文研究三维广义Leray方程弱解的全局正则性，通过消失黏度、分数阶算子非局部效应、Besov空间差分刻画与bootstrap论证提升正则性，结合非局部Stokes线性理论得到逐点估计，还获得经典Navier-Stokes方程自相似解的最优衰减估计，解决了Tsai提出的相关问题。
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.
Regularity of a bulk-surface Cahn-Hilliard model driven by Leray velocity fields
本文研究对流Cahn-Hilliard系统，在更一般的速度场正则性下证明弱解适定性，并针对Leray型速度场证明强解适定性，方法基于新的体-表面椭圆系统理论。
In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschi…
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
If $u$ is a Leray-Hopf solution to the Navier-Stokes equations with the initial data in $ L^2(\mathbb{R}^3)$, then $u$ is regular.
A Hybrid High-Order method for variable-exponent Leray-Lions problems with convergence under minimal regularity
针对变指数Leray-Lions问题，提出基于多面体网格的混合高阶（HHO）离散方法，证明其在最小正则性下收敛，数值实验验证了方法对奇异解及指数1-6的有效性。
Global regularity of Leray-Hopf weak solutions to 3D Navier-Stokes equations
本文通过构造带稀疏逆对数权重的超临界空间并叠加高频能量估计，证明了初值属于H^{1/2}的三维Navier-Stokes方程Leray-Hopf弱解具有全局正则性。
Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian
本文研究带奇异权重与强吸收的 $\infty$-Laplacian Hardy-Hénon 型方程，证明粘性解的显式改进正则性指数与非退化性，并推广零障碍、死核结果及给出 Liouville 型应用。
Motivated by Kolmogorov's theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber $Λ(t)$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we also we prove that…
Energy Superposition and Regularity for 3D Navier-Stokes Equations in the Largest Critical Space
该研究提出能量叠加新方法，证明了最大临界空间中三维Navier-Stokes方程Leray-Hopf弱解的正则性，简化了已知正则准则的证明，方法可推广到其他超临界非线性PDE。
We correct an inaccuracy in a previous article [Auscher, Pascal; Bernicot, Frédéric; Zhao, Jiman. Maximal regularity and Hardy spaces. Collect. Math. 59 (2008), no. 1, 103-127.]
We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the $B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a lower-order term depending on the gradient.
For the incompressible Euler equations the pressure formally scales as a quadratic function of velocity. We provide several optimal regularity estimates on the pressure by using regularity of velocity in various Sobolev, Besov and Hardy spaces. Our proof exploits the incompressibility condition in an essential way and is deeply connected with the classic Div-Curl lemma which we also generalise as a fractional Leibniz rule in Hardy spaces. To showcase the sharpness of results, we construct a class of counterexamples at several end-points.
Constrained large solutions to Leray's problem in a distorted strip with the Navier-slip boundary condition
本文研究了二维无限扭曲带中带Navier-slip边界条件的定常Navier-Stokes系统Leray问题，证明了解的存在性、唯一性、正则性和渐近行为，并发现小摩擦系数下速度总通量可相对较大。
In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.
Hardy Space and Exponential Decay of Inhomogeneous Navier-Stokes System in the 2D Global domain
研究二维全局域中非齐次不可压缩Navier-Stokes系统，利用相消结构和Hardy空间估计，证明速度指数衰减及强解的全局适定性和流映射正则性。
On the Hessian Hardy-Sobolev Inequality and Related Variational Problems
本文通过Hessian泛函的下降梯度流证明了Hessian积分的Hardy-Sobolev不等式，将其应用于相关变分问题解的存在性与正则性研究，扩展了Hessian方程的变分理论。
In this paper, we show the equivalence between the boundedness of the Riesz transform $dΔ^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the scaled Poincaré inequalities hold. Here, $H^p$ is a Hardy space of exact $1-$forms, naturally associated with the Riesz transform.
This paper establishes global weighted Calderón-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stokes operator consist of symmetric and skew-symmetric parts, which are both discontinuous. Moreover, the skew-symmetric part is not required to be bounded and therefore it could be singular. Sufficient conditions on the coefficients are provided to ensure the global weighted $W^{1,p}$-regularity estimates for weak solutions of the systems. As a direct…
A priori error estimates for a Hybrid High-Order method on variable-exponent Leray-Lions problems
该研究推导变指数Leray-Lions问题的混合高阶（HHO）离散化的先验误差估计，建立局部与全局误差估计，通过数值实验验证结果，表明通量正则性假设不可省略。
An inhomogeneous p-laplacian equation with a Hardy potential
本文研究了具有Hardy势的非齐次p-laplace方程解的存在性与正则性，明确了参数取值范围及函数空间条件。
We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli's type estimates for the Stokes systems with variable coefficients. A new $ε$-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regular…
Forward self-similar solutions to the MHD equations: existence and pointwise estimates
本文针对三维全空间 MHD 方程，利用 Leray-Schauder 定理与爆破论证构造全局时间光滑前向自相似解，并借助加权 Sobolev 空间中 Leray 系统弱解的正则性获得其逐点估计。
We provide characterizations for boundedness of multilinear Fourier operators on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0<q\le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1<q\le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and T…
In this paper we consider the regularity problem of the Navier-Stokes equations in $ \R^{3} $. We show that the Serrin-type condition imposed on one component of the velocity $ u_3\in L^p(0,T; L^q(\R^{3} ))$ satisfying $ \frac{2}{p}+ \frac{3}{q} <1$, $ 3<q \le +\infty$ implies the regularity of the weak Leray solution $ u: \R^{3} \times (0,T) \rightarrow \R^{3} $ with the initial data belonging to $ L^2(\Bbb R^3) \cap L^3(\R^{3})$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.
We show that for an $L^2$ drift $b$ in two dimensions, if the Hardy norm of $\text{div }b$ is small, then the weak solutions to $Δu+b\cdot\nabla u=0$ have the same optimal Hölder regularity as in the case of divergence-free drift, that is, $u\in C^α_{\text{loc}}$ for all $α\in (0, 1)$.
We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of Hölder regularity, higher than what is given by the classical exponent $1/K$.
Quasi-linear elliptic equations with superlinear convection
本文研究了涉及Leray-Lions算子和超线性对流项的拟线性椭圆方程解的存在性与正则性，涵盖p-Laplacian方程，并推广了Boccardo等人（2024年）的部分结果。
Hierarchical exact controllability for a parabolic equation with Hardy potential
本文通过Stackelberg-Nash策略研究具有Hardy势的抛物方程的分层精确可控性，在线性和半线性情形下分别利用Carleman不等式和不动点定理证明了领导者的存在性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。