Melissa Tacy
The well-known stationary phase formula gives us a way to precisely compute oscillating integrals so long as the symbol is regular enough (in comparison to the large parameter controlling the oscillation). However in a number of applications we find ourselves with symbols that are not suitably regular. In this paper we obtain decay bounds for such oscillatory integrals.
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We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $ \Bbb R^2$. We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations.
Let $P_h$ be a self-adjoint semiclassical pseudodifferential operator on a manifold $M$ such that the bicharacteristic flow of the principal symbol on $T^*M$ is completely integrable and the subprincipal symbol of $P_h$ vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes $u_h$ of $P_h.$ We show that on a nondegenerate rational invariant torus, Lagrangian regularity of $u_h$ (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional ``diffractive'' manner. In particular, in addition to …
We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are Hölder continuous with order $\frac1{2}^-$ away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.
We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two.
In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$.
In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-Δu=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1.
We investigate regularity estimates for the stationary Navier-Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved $C^{1,μ}$ estimate and identify the building blocks of the regularity theory, dubbed `Navier polynomials'. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regulari…
$H^2$-regularity for stationary and non-stationary Bingham problems with perfect slip boundary condition
本文证明了在完美滑移边界条件和边界上屈服应力为零的条件下，Bingham流体的定常与非定常问题具有$H^2$空间正则性，并通过先验估计和时间离散化建立了非定常问题的强可解性。
We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.
We use the scale $B^s_τ(L_τ(Ω))$, $1/τ=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.
In this paper, we prove that any $C^{1}$-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.
In this paper, we establish $\varepsilon$-regularity criteria at one scale for suitable weak solutions to the five dimensional stationary incompressible Navier-Stokes equations in both the unit ball $B_1$ and the unit half ball $B_1^+$, respectively, which improve previous results in [J. Math. Fluid Mech., 6(2004), pp 78--101] and [Comm. Pure Appl. Math., 41(1988), pp 437--458]. The proofs of the main results are based on an iteration argument, Campanato's method, and interpolation techniques.
We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of variations for the area functional.
In this article we establish $C^{3,α}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $Ω\subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding Euler-Lagrange equation classically on the reduced boundary. Moreover, we show that the singular set has zero $(n-1)$-dimensional Hausdorff measure. This complements the results in Choksi & Sternberg, in which the Euler-Lagrange equation was derived under the assumption of $C^2$-regularity of the topological boundary and the results in Sternberg & Topaloglu in which the …
A Regularity Theory for Stationary Two-valued Harmonic Functions
本文研究平稳二值调和函数的正则性，利用 Almgren 单调性公式和爆破方法证明其 Lipschitz 连续性，并刻画节点集的解析结构，可用于分离模型极限问题。
A regularity theorem for stationary measures
研究光滑流形上拉普拉斯-贝尔特拉米算子特征值关于Radon测度的变分问题，证明平稳测度关于经典体积测度绝对连续且密度由调和映射诱导。
Non-uniqueness of weak solutions to the 3-D stationary MHD equations in Besov space with negative regularity index
通过凸积分方法证明三维环面上分数阶稳态MHD方程存在无穷多个非平凡奇异解，从而在次临界Besov空间中建立弱解的非唯一性。
We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $Ω\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth boundary data, we show that if, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $Σ\subset\barΩ$, with $H^{n-4}(Σ)=0$, such that $u\in C^\infty(\barΩ\setminusΣ, N)$.
Small-energy regularity for stationary weak solutions of the Liouville equation
该研究在任意维度上证明了Liouville方程平稳弱解的小能量正则性，解决了Da Lio等人提出的开放问题，证明基于对几乎平稳场集中现象的分析。
On the Existence and Regularity for Stationary Boltzmann Equation in a Small Domain
研究小区域中硬势稳态Boltzmann方程在入射边界条件下的解的存在性与正则性，通过区域小性和法曲率条件构造了$W^{1,p}$（$1\leq p<3$）的真子集作为合适的解空间。
Using the harmonic map heat flow, we construct an energy class for wave maps $ϕ$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)
We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of $\mathbb{C}^{n}$. Namely, if $L$ $\subset$ $\mathbb{C}^{n}$ is a $C^{1}$ Lagrangian submanifold with weakly harmonic Lagrangian phase $θ,$ then $L$ must be smooth. In the process we also discuss a local version of the equation, which is a nonlinear fourth order double divergence equation of the potential function whose gradient graph defines the Hamiltonian stationary submanifolds locally, and we establish full regularity and removability of singular sets of capacity zero for weak solutions with $C^{1,1}$ norm below a di…
Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness
本文研究定常微极流体方程，推导速度场与微旋转场的耦合方程，探讨弱解存在性、正则性及唯一性，证明仅速度场满足无穷远衰减条件时平凡解唯一。
We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.
Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents
本文针对具一般系数的定常Fokker-Planck方程解（L²密度测度），利用Dirichlet形式和预解式，结合椭圆正则性结果，证明其密度具H¹·²正则性与Hölder连续性，突显相关理论在解正则性分析中的核心作用。
Regularity for the Geodesic X-Ray Transform in Nonsmooth Geometry
本文研究非光滑简单流形上测地线X射线变换的正则性，通过符号平滑和低正则伪微分算子映射性质，改进了$L^p$函数上变换的单射性结果。
It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(α)$ has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of $A$ on the scattering cotangent bundle of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of $λ(A…
Interior regularity for two-dimensional stationary $Q$-valued maps
本文证明二维平稳 $Q$-值映射具有 Hölder 连续性且奇异集维数至多为一，并建立了适用于所有维度的强集中紧性定理。
Regularity for Hamiltonian stationary equations in $\mathbb{R}_{n\leq 4}^{n}$
本文研究了复欧氏空间中 Hamiltonian 平稳方程解的正则性，证明了在 $n\leq 4$ 维时所有 Lagrangian 相位下的 $C^{1,1}$ 解均光滑，并给出了相应的 $C^{k,α}$ 估计。
In this paper, we prove the existence of classical solutions for second order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density…
In this paper we study the action of pseudo-differential operators acting on Gevrey spaces. We introduce classes of classical symbols with spatial Gevrey regularity. As the spatial Gevrey regularity of a symbol $p(\cdot,ξ)$ may depend on the frequency $ξ$, the action of the associated pseudo-differential operator $op(p)$ may induce a loss of regularity. The proof is based on a para-product decomposition.
Higher regularity estimate has been a challenging question for the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in [15] even for symmetric convex domains such as a disk or sphere. In this paper we answer this question in the affirmative by constructing the $C^{1,β}$ solutions away from the grazing boundary, for any $β<1$, to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwise…
Coulomb Gauges and Regularity for Stationary Weak Yang$-$Mills Connections in Supercritical Dimension
该研究证明5维平稳弱Yang-Mills场在1维Hausdorff测度为零的闭奇异集外光滑，通过构造受控Coulomb规范，将ε-正则性定理推广到弱联络变分类，拓展了相关正则性理论的适用范围。
On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach
本文基于恒等幂紧算子的广义 Fredholm 理论，研究有界三维凸区域中稳态玻尔兹曼方程的存在性与正则性，通过解算子正则化效应和双线性估计，建立了线性与非线性情形下的存在性理论。
We consider the piecewise-deterministic Markov process obtained by randomly switching between the flows generated by a finite set of smooth vector fields on a compact set. We obtain Hörmander-type conditions on the vector fields guaranteeing that the stationary density is: $C^k$ whenever the jump rates are sufficiently fast, for any $k<\infty$; unbounded whenever the jump rates are sufficiently slow and lower semi-continuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.
Quasi-stationary distribution for kinetic SDEs with low regularity coefficients
本文研究低正则性系数下动力学随机微分方程，通过证明Harnack不等式并结合Lyapunov条件，建立了柱形区域中拟平稳分布的存在唯一性，并给出两类满足条件的典型情形。
Optimal Regularity for Hölder continuous Hamiltonian Stationary Lagrangian graphs
本文证明Hölder指数大于1/3且相位超临界时哈密顿驻点拉格朗日图光滑，并构造指数为1/3的奇异解，证明正则性阈值最优，与特殊拉格朗日理论形成对比。
Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries
本文研究了高度振荡粗糙John域上平稳Navier-Stokes方程的大尺度正则性，通过构造一阶和二阶边界层证明了包括大尺度Calderón-Zygmund、Lipschitz及高阶$C^{1,\gamma}$和$C^{2,\gamma}$估计在内的正则性结果。
Some remarks on the regularity of weak solutions for the stationary Ericksen-Leslie and MHD systems
本文研究了三维空间中受外力作用的定常 Ericksen-Leslie (SEL) 和 MHD 两个椭圆耦合系统，为 SEL 系统提出了基于齐次 Morrey 空间的弱解正则性新准则，并利用 Gevrey 类证明了 MHD 系统有限能量弱解的解析性。
$C^{1,α}$ Regularity For Stationary Mean-Field Games With Logarithmic Coupling
本文利用 Hopf--Cole 变换和 Morrey 空间方法，证明了环面上具有 Lipschitz 非齐次扩散与对数型耦合的稳态平均场博弈存在 $C^{1,α}$ 解，填补了低正则性与光滑结果之间的研究空白。
In a $d-$dimensional strip with $d\geq 2$, we study the non-stationary Stokes equation with no-slip boundary condition in the lower and upper plates and periodic boundary condition in the horizontal directions. In this paper we establish a new maximal regularity estimate in the real interpolation norm \begin{equation*} ||f||_{(0,1)}=\inf_{f=f_0+f_1}\left\{\left\langle\sup_{0<z<1} |f_0|\right\rangle+ \left\langle\int_0^{1} |f_1| \frac{dz}{(1-z)z}\right\rangle\right\}\,, \end{equation*} where the brackets $\langle\cdot\rangle$ denotes the horizontal-space and time average. The norms involved in …
Resolvent Approach to Atangana--Baleanu Evolution Equations: Laplace Symbols, Mild Solutions, and Regularity
本文提出基于Atangana-Baleanu导数的解析方法，通过引入分数解析和Mittag-Leffler核，建立了AB型演化方程的弱解表示，并推导了稳定性与正则性估计。
We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier …
Here we develop a regularity theory for a polyconvex functional in $2\times2-$dimensional compressible finite elasticity. In particular, we consider energy minimizers/stationary points of the functional $I(u)=\int\limits_Ω{\frac{1}{2}|\nabla u|^2+ρ(\det\nabla u)\;dx},$ where $Ω\subset\mathbb{R}^2$ is open and bounded, $u\in W^{1,2}(Ω,\mathbb{R}^2)$ and $ρ:\mathbb{R}\rightarrow\mathbb{R}_0^+$ smooth and convex with $ρ(s)=0$ for all $s\le0$ and $ρ$ becomes affine when $s$ exceeds some value $s_0>0.$ Additionally, we may impose boundary conditions. The first result we show is that every stationar…
In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing $\mathbf{Q}_{Q}(\mathbb{R}^{n})$-valued functions in the Sobolev space $\mathcal{Y}_{2}(Ω, \mathbf{Q}_{Q} (\mathbb{R}^{n}))$, where the domain $Ω$ is open in $\mathbb{R}^{m}$. In this article, we introduce the class of weakly stationary-harmonic $\mathbf{Q}_{Q} (\mathbb{R}^n)$-valued functions. These functions are the critical points of Dirichlet integral under smooth domain-variations and range-variations. W…
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…
The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.
We study the regularity of Radon measures $μ$ which satisfy that there exists a function $h_μ$ in $H^1(Ω)$, stationary harmonic such that $Δh_μ=μ$ in $Ω$ (here $Ω$ is an open set of $\mathbb{R}^2$). Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family $(u_\varepsilon)_\varepsilon$ of solutions of the Ginzburg-Landau system without magnetic field. Under these conditions we prove that locally there exists a harmonic function $H$ such that the support of the measure is contained in the set of zeros of $H$. Using the local structure o…
For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $ΣM$, and any compact Riemannian manifold $N$, we show an $ε$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $ψ\in W^{1,p}$ for some $p>2n/3$, th…
We consider the nonlinear Schrödinger equation $-Δu + V(x) u = Γ(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-Δ+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,Γ$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable Hölder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional solution of the more general equat…
In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical.
Partial regularity for steady double phase fluids
该研究针对散度型非线性椭圆方程组，探究其在双相增长条件下的部分Hölder正则性，为定常双相非牛顿流体的建模提供数学理论支撑。
In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers $(ρ_\eps)_\eps$, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter $\eps$ with the non-regularity of the symbol.
$H^s_x$ regularity of solutions to the stationary Boltzmann equation with the incoming boundary condition
本文研究了带有入射边界条件的 stationary Boltzmann 方程解的正则性，针对不同势场情况，证明了解在不同 Sobolev 空间中的正则性，利用速度平均引理和双线性估计扩展了线性问题的结果。
Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation
本文研究了稳态线性Boltzmann方程在不同几何条件下的$W^{1,p}$正则性，证明了在特定条件下解的存在性并给出了反例。
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $Ω\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary a…
In this paper we consider extended stationary mean field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean field games in [GPSM12]. In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in [LL06a], [GSM11] or [GPSM12] this paper yields various new estimates and regularity prop…
An Lp-theory for fractional stationary Navier-Stokes equations
该研究在Lebesgue与Lorentz空间框架下，建立含分数阶拉普拉斯算子的三维定常Navier-Stokes方程的Lp理论，给出解存在/不存在的充分条件，得到解的逐点衰减速率、渐近轮廓与全局正则性，还得到Liouville型唯一性结果及经典方程的新正则性准则。
Well-Posedness and regularity properties of 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel
本文首次在数学上严格处理了周期通道中同时含随机噪声和科里奥利力的二维β平面Navier-Stokes方程，证明了适定性与平稳测度的存在性，并给出了支撑集的正则性估计，为级联现象研究奠定基础。
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