Hasan Inci, Thomas Kappeler, Peter Topalov
For $M$ being a closed manifold or the Euclidean space we present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > 1/2\dim M + 1$.
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Negative regularity mixing for random volume preserving diffeomorphisms
本文研究随机保体积微分同胚的负正则性指数混合，给出通用判据并应用于随机微分同胚、SDE 及随机不可压缩 Navier-Stokes 平流扩散，证明理想标量湍流存在唯一平稳测度。
We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.
On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity
本文研究固定原点的实直线非平坦微分同胚芽的共轭类，基于Takens和Yoccoz工作，证明低正则共轭下留数不变性，并推广Schwarzian导数值在$C^2$抛物共轭下不变而在$C^1$下可变的结果。
We study the weak regularity of mappings inverse to weighted Sobolev homeomorphisms $φ:Ω\to\widetildeΩ$, where $Ω$ and $\widetildeΩ$ are domains in $\mathbb R^n$. Using the weak regularity of inverse mappings we obtain the composition duality property of composition operators on weighted Sobolev spaces.
For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space $H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In…
This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [J.Fournier '17], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note firs…
We give sufficient conditions on initial and target measures supported on the sphere $§^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.
It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.
Regularity and non-degeneracy of $Φ^*[I]$ implies regularity of the fixed boundary $\partialΩ$
研究满足特定非退化条件的微分同胚 Φ，证明其推前 Φ^*[I] 的局部正则性可推出固定边界 ∂Ω 的局部正则性，补充了非散射非均匀介质的正则性结果。
$L^p$-Extremal Teichmüller mappings between Riemann surfaces are diffeomorphisms
研究解析有限黎曼曲面间同胚类中 \(L^p\) 共形能量最小化问题，证明 \(1\leq p<\infty\) 时最小化映射存在、唯一且为微分同胚，\(p = \infty\) 时为准共形但非微分同胚。
Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations
本文研究非齐次p-拉普拉斯方程解的梯度的齐次函数的更高正则性，利用内在尺度、Schauder估计等方法，得到了相关函数的正则性结果及多孔介质方程的正则性准则。
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harm…
Composition and substitution of Regularity Structures B-series
本研究提出用于描述奇异随机偏微分方程解的正则结构B-级数，定义其合成与替换运算并通过乘积和Hopf代数重述，为相关研究开辟新方向。
We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.
In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimension $4$ conjecture, namely that $X$ is smooth away from a closed subset of codimension $4$. We combine this result with the ideas of quantitative stratification to prove a priori $L^q$ estimates on the full curvature $|Rm|$ for all $q<2$. In the case of Einstein manifolds, we im…
Propagation of regularity along unstable manifolds
该研究证明了在不稳定流形上光滑的共振态在整个流形上也是光滑的，并展示了叶状空间伪微分算子理论的应用。
A new proof is given of the fact that the particle trajectories of the ideal incompressible fluid are analytic curves, though the solutions of the Euler equations may have a finite regularity. This is a consequence of a general fact that the geodesic exponential map on the group of volume preserving diffeomorphisms belonging to the Sobolev space is real-analytic. The proof is based on the general properties of holomorphic maps in complex Banach spaces.
The Stokes problem with Navier slip boundary condition: Minimal fractional Sobolev regularity of the domain
本文研究了带有滑移边界条件的 stationary 斯托克斯问题的弱解在反射索伯列夫空间中的适定性，证明了域的最小分数正则性为 $W^{2-1/s}_s$，并改进了 Amrouche-Seloula 的结果。
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