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We give an elementary proof for the double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u \in C^{0,\gamma}(\Omega;\mathbb{R}^d)$ with some $0<\gamma<1/2$, we show that the pressure $p \in C^{0,2\gamma}(\Omega)$. This is motivated by the studies of turbulence and anolalous dissipation in mathematical hydrodynamics and, recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int. Math. Res. Not. 2024.3 (2024), 2511-2560] over $C^{2,\alpha}$-domains by means of pseudodifferential calculus. Our approach involves only standard elliptic PDE techniques, and relies crucially on the modified pressure introduced in [C. W. Bardos, D. W. Boutros, and E. S. Titi, H\"{o}lder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains, arXiv: 2304.01952] and the potential estimates in [L. Silvestre, unpublished notes]. The key novel ingredient of our proof is the introduction of two cutoff functions whose localisation parameters are carefully chosen as a power of the distance to $\partial\Omega$., Comment: 23 pages

We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges., Comment: 18 pages. Survey article dedicated to Prof. Gui-Qiang Chen's 60th birthday

We are concerned with the existence of asymptotic directions for the group of volume-preserving diffeomorphisms of a closed 2-dimensional surface $(\Sigma,g)$ within the full diffeomorphism group, described by the Bao--Ratiu equations, a system of second-order PDEs introduced in [On a non-linear equation related to the geometry of the diffeomorphism group, Pacific J. Math. 158 (1993); On a non-linear equation related to the geometry of the diffeomorphism group, Pacific J. Math. 158 (1993)]. It is known [The Bao--Ratiu equations on surfaces, Proc. R. Soc. Lond. A 449 (1995)] that asymptotic directions cannot exist globally on any $\Sigma$ with positive curvature. To complement this result, we prove that asymptotic directions always exist locally about a point $x_0 \in \Sigma$ in either of the following cases (where $K$ is the Gaussian curvature on $\Sigma$): (a), $K(x_0)>0$; (b) $K(x_0)<0$; or (c), $K$ changes sign cleanly at $x_0$, i.e., $K(x_0)=0$ and $\nabla K(x_0) \neq 0$. The key ingredient of the proof is the analysis following Han [On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math. 58 (2005)] of a degenerate Monge--Amp\`{e}re equation -- which is of the elliptic, hyperbolic, and mixed types in cases (a), (b), and (c), respectively -- locally equivalent to the Bao--Ratiu equations.

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric immersion $\iota$ of $(\mathcal{M}^n,g)$ into the Euclidean space $\mathbb{R}^{n+k}$ whose extrinsic geometry coincides with $\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\mathfrak{S}$ and $\iota$. The best result up to date is $\mathfrak{S} \in L^p$ and $\iota \in W^{2,p}$ for $p>n \geq 3$ or $p=n=2$. In this paper, we extend the above result to $\iota \in \mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n \geq 3$. Indeed, $\mathcal{X}$ is the weak Morrey space $L^{p, n-p}_{2,w}$ with arbitrary $p \in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss--Codazzi--Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges -- in particular, Rivi\`{e}re--Struwe's work [Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008)] on harmonic maps in arbitrary dimensions and codimensions -- and compensated compactness., Comment: This paper corrects and supercedes 2003.05595

A central question in rough path theory is characterising the law of stochastic processes. It is established in [I. Chevyrev $\&$ T. Lyons, Characteristic functions of measures on geometric rough paths, $\textit{Ann. Probab.}$ $\textbf{44}$ (2016), 4049--4082] that the path characteristic function (PCF), $\textit{i.e.}$, the expectation of the unitary development of the path, uniquely determines the law of the unparametrised path. We show that PCF restricted to certain subspaces of sparse matrices is sufficient to achieve this goal. The key to our arguments is an explicit algorithm -- as opposed to the nonconstructive approach in [I. Chevyrev $ \&$ T. Lyons, $\textit{op. cit.}$] -- for determining a generic element $X$ of the tensor algebra $\mathcal{T}\left(\mathbb{R}^d\right) = \bigoplus_{n=0}^\infty\left(\mathbb{R}^d\right)^{\otimes n}$ from its moment generating function. Our only assumption is that $X$ has a nonzero radius of convergence, which relaxes the condition of having an infinite radius of convergence in the literature. As applications of the above theoretical findings, we propose the restricted path characteristic function distance (RPCFD), a novel distance function for probability measures on the path space that offers enormous advantages for dimension reduction. Its effectiveness is validated via hypothesis testing on fractional Brownian motions, thus demonstrating the potential of RPCFD in generative modeling for synthetic time series generation., Comment: 26 pages, 8 figures, 3 tables

We construct an explicit example of a smooth isotopy $\{\xi_t\}_{t \in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1]^n$ ($n \geq 3$) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of $(M,g)$, a "topologically complicated" Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with $\{\xi_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy., Comment: 13 pages; supercedes ArXiv 2006.15423. arXiv admin note: text overlap with arXiv:2006.15423

In this paper, a novel observation is made on a one-dimensional compressible Navier--Stokes model for the dynamic combustion of a reacting mixture of $\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction rate function, on both bounded and unbounded domains. We show that the mass fraction of the reactant (denoted as $Z$) satisfies a weighted gradient estimate $Z_y/ \sqrt{Z} \in L^\infty_t L^2_y$, provided that at time zero the density is Lipschitz continuous and bounded strictly away from zero and infinity. Consequently, the graph of $Z$ cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any instant, and the entropy of $Z$ is bounded from above. The key ingredient of the proof is a new estimate based on the Fisher information, first exploited by [2, 7] with applications to PDEs in chemorepulsion and thermoelasticity. Along the way, we also establish a Lipschitz estimate for the density., Comment: 18 pages

We formulate and prove compensated compactness theorems concerning the limiting behaviour of wedge products of weakly convergent differential forms on closed Riemannian manifolds. The case of critical regularity exponents is considered, which goes beyond the regularity regime entailed by H\"{o}lder's inequality. Implications on the weak continuity of $L^p$-extrinsic geometry of isometric immersions of Riemannian manifolds are discussed., Comment: 25 pages

Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. Towards this goal, a key step is the development of an effective discriminator to distinguish between time series distributions. We propose the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance. On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality. Code is available at https://github.com/DeepIntoStreams/PCF-GAN.

The model of physical Brownian motion describes the dynamics of a Brownian particle experiencing both friction and external magnetic field. It was investigated as a physically meaningful approach to realising the standard ''mathematical'' Brownian motion, via sending the mass $m \to 0^+$ and performing natural scaling. In this paper, we are concerned with the singular limit analysis of a generalised stochastic differential equation (SDE) model, motivated by and encompassing the physical Brownian motion. We show that the expected signature of the solution $\left\{P_t\right\}_{t \geq 0}$ for the generalised SDE converges in the limit $m \to 0^+$ to a nontrivial tensor, at each degree of tensors and on each compact time interval $[0,T]$. The solution $\left\{P_t\right\}_{t \geq 0}$, viewed as a rough path, generalises the momentum of particle in classical physical Brownian motion. This singular limit is identified through a delicate convergence analysis based on the graded PDE system for the expected signature of It\^{o} diffusion processes. Explicit closed-form solutions exhibiting intriguing combinatorial patterns are obtained when the coefficient matrix $\mathbf{M}$ in our SDE, which generalises the stress tensor in physical Brownian motion, is diagonalisable. Our work appears among the very first endeavours to study the singular limit of expected signature of diffusion processes, in addition to the literature on the analysis of the expected signature of stochastic processes.

We show that the Gaussian kernel $\exp\left\{-\lambda d_g^2(\bullet, \bullet)\right\}$ on any non-simply-connected closed Riemannian manifold $(\mathcal{M},g)$, where $d_g$ is the geodesic distance, is not positive definite for any $\lambda > 0$, combining analyses in the recent preprint~[9] by Da Costa--Mostajeran--Ortega and classical comparison theorems in Riemannian geometry., Comment: 9 pages

We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of Ebin--Misio\l{}ek--Preston: the exponential is a nonlinear Fredholm mapping of index zero. We extend Shnirelman's rigidity result for the exponential map from 2-dimensional flat torus to arbitrary orientable closed surfaces. That is, we prove that the exponential map is Fredholm quasiregular., Comment: 20 pages

Signature, lying at the heart of rough path theory, is a central tool for analysing controlled differential equations driven by irregular paths. Recently it has also found extensive applications in machine learning and data science as a mathematically principled, universal feature that boosts the performance of deep learning-based models in sequential data tasks. It, nevertheless, suffers from the curse of dimensionality when paths are high-dimensional. We propose a novel, trainable path development layer, which exploits representations of sequential data through finite-dimensional Lie groups, thus resulting in dimension reduction. Its backpropagation algorithm is designed via optimization on manifolds. Our proposed layer, analogous to recurrent neural networks (RNN), possesses an explicit, simple recurrent unit that alleviates the gradient issues. Our layer demonstrates its strength in irregular time series modelling. Empirical results on a range of datasets show that the development layer consistently and significantly outperforms signature features on accuracy and dimensionality. The compact hybrid model (stacking one-layer LSTM with the development layer) achieves state-of-the-art against various RNN and continuous time series models. Our layer also enhances the performance of modelling dynamics constrained to Lie groups. Code is available at https://github.com/PDevNet/DevNet.git.

Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from Riemannian manifolds to spheres (in the spirit of Friesecke-James-M\"{u}ller [23]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy-Green tensors and second fundamental forms, which extends the Ciarlet-Mardare theorem in [18] to arbitrary dimensions and co-dimensions., Comment: 25 pages, Journal de Math\'ematiques Pures et Appliqu\'ees (to appear)

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and H\"older regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,\frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions., Comment: 37 pages; improved presentation and made some minor corrections in v2. This version will be submitted

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $\Omega$, it is only known that the radius of convergence for the expected signature on sufficiently regular $\Omega$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $\Omega$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,\alpha}$-domain $\Omega \subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains., Comment: 35 pages, 1 figure. The results obtained in V1 have been extended from $d=2$ to $2\leq d \leq 8$

Let $(\mathcal{M},g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney's inequality for differential forms in boundary value spaces over $\mathcal{M}$, via the variational approach \`{a} la Kozono--Yanagisawa [$L^r$-variational inequality for vector fields and the Helmholtz--Weyl decomposition in bounded domains, Indiana Univ. Math. J. 58 (2009), 1853--1920] combined with global computations based on the Bochner's technique., Comment: 25 pages. Further references and historical accounts have been added in V2

We are concerned with two interrelated problems: smoothability of connection 1-forms with low regularity on bundles with prescribed smooth curvature 2-forms, and existence of isometric immersions with low regularity. We first show that if $\Omega$ is an $L^p$-connection $1$-form on a vector bundle over a closed Riemannian $n$-manifold $\mathcal{M}$ with small $L^p$-norm ($p>n$) and smooth curvature $2$-form $\mathscr{F}$, then $\Omega$ can be approximated in the $L^p_{\rm loc}$-topology by smooth connections of the same curvature (not necessarily gauge equivalent). Our proof, adapted from S. Mardare's work on the fundamental theory of surfaces with $L^p$-second fundamental form, is elementary in nature and uses only Hodge decomposition and fixed point theorems. This result is then applied to the study of isometric immersions of Riemannian manifolds with low regularity. We revisit the proof for the existence of $W^{2,p}$-isometric immersion $\mathcal{M}^n \to \mathbf{R}^{n+k}$ with arbitrary $n$ and $k$ given weak solutions to the Gauss--Codazzi--Ricci equations, aiming at elucidating some global vs. local issues, and also we provide a characterisation for metrics on $\mathcal{M}$ that admit $W^{2,p}$- but no $C^\infty$-isometric immersions., Comment: 27 pages. This note replaces ArXiv: 2003.05595; a weaker statement is proved (with $p>n=\dim \mathcal{M}$ assumed instead of $p \geq 2$, plus a smallness condition)

We obtain a dynamical--topological obstruction for the existence of isometric embedding of a Riemannian manifold-with-boundary $(M,g)$: if the first real homology of $M$ is nontrivial, if the centre of the fundamental group is trivial, and if $M$ is isometrically embedded into a Euclidean space of dimension at least $3$, then the isometric embedding must violate a certain dynamical, kinetic energy-related condition (the "rigid isotopy extension property" in Definition 1.1). The arguments are motivated by the incompressible Euler equations with prescribed initial and terminal configurations in hydrodynamics., Comment: The main Theorem 1.4 of this manuscript is flawed --- in fact, it is vacuously true, for the "rigid isotopy extension property'' can never hold for any isometric embedding. We shall correct this issue in a new preprint entitled "A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions'' (to be uploaded to ArXiv shortly)

It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow \infty$. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations., Comment: 5 pages; new (fun) proofs for elementary results

We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework \`{a} la Labourie (Immersions isom\'{e}triques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov., Comment: 11 pages

In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by a spatially uniform dislocation density field in a body comprising a nonlinear elastic material may fail to vanish under no loads. The class of counterexamples constructed in [1] is essentially $2$-dimensional: it works with the subgroup $\mathcal{O}(2) \oplus \langle{\bf Id}\rangle \subset \mathcal{O}(3)$. The objective of this note is to extend Acharya's result in [1] to $\mathcal{O}(3)$, subject to an additional structural assumption and less regularity requirements., Comment: 5 pages

We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of $W^{2,2}$-isometric immersions and the weak solubility of the Gauss--Codazzi--Ricci equations on simply-connected domains. The regularity assumptions of these results are sharp. As an application, we deduce a weak compactness theorem for $W^{2,2}_{\rm loc}$-immersions., Comment: Equation (24) was wrong: algebraic cancellations of this type are invalid in general

Forecasts of multivariate probability distributions are required for a variety of applications. Scoring rules enable the evaluation of forecast accuracy, and comparison between forecasting methods. We propose a theoretical framework for scoring rules for multivariate distributions, which encompasses the existing quadratic score and multivariate continuous ranked probability score. We demonstrate how this framework can be used to generate new scoring rules. In some multivariate contexts, it is a forecast of a level set that is needed, such as a density level set for anomaly detection or the level set of the cumulative distribution as a measure of risk. This motivates consideration of scoring functions for such level sets. For univariate distributions, it is well-established that the continuous ranked probability score can be expressed as the integral over a quantile score. We show that, in a similar way, scoring rules for multivariate distributions can be decomposed to obtain scoring functions for level sets. Using this, we present scoring functions for different types of level set, including density level sets and level sets for cumulative distributions. To compute the scores, we propose a simple numerical algorithm. We perform a simulation study to support our proposals, and we use real data to illustrate usefulness for forecast combining and CoVaR estimation.

We give a "soft" proof of Alberti's Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed., Comment: 8 pages. Ricerche di Mat. (2020)

We extend the well-known result that any $f \in W^{1,n}(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\Omega)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of compensated compactness. We then deduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: For a family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on a semi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformly bounded in $L^p$ and satisfies the Cartan structural system, then any weak $L^p$ limit of $\{\mathcal{W}_\varepsilon\}$ is also a solution of the Cartan structural system. Moreover, it is proved that isometric immersions of semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci system (Theorem 5.1), which leads to the $L^p$ weak continuity of the Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further applications, the weak continuity of Einstein's constraint equations, general immersed hypersurfaces, and the quasilinear wave equations is also established., Comment: 64 pages

We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to a relaxation theory. Moreover, we propose the notion of ``Riemannian quasiconvexity'' for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been established. This work may serve as a tentative generalisation of the framework developed in the recent paper: B. Dacorogna and W. Gangbo, Quasiconvexity and relaxation in optimal transportation of closed differential forms, \textit{Arch. Ration. Mech. Anal.} (2019), to appear. DOI: \texttt{https://doi.org/10.1007/s00205-019-01390-9}., Comment: 14 pages

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statement. It explores the connections between the theory of non-smooth analysis {\it \`{a} la} F.~H. Clarke and the existence of special systems of Whitney flat $1$-forms with Sobolev regularity on certain families of homology manifolds., Comment: 9 pages

Based on a construction due to B. G\"{u}neysu and S. Pigola (\textit{Adv. Math.} \textbf{281} (2015), pp.353--393), for each $p \in [1,\infty]$ and $m \in \mathbb{Z}_{\geq 2}$, we exhibit an $m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calder\'{o}n--Zygmund estimate \begin{equation*} \|\nabla \nabla f\|^p_{L^p} \leq C_1 \|\Delta f\|_{L^p}^p + C_2 \| f\|_{L^p}^p \qquad \text{ for all } f \in C^\infty_c(\mathcal{M}) \end{equation*} is false for any $C_1, C_2$ depending on $m$ and $p$. Therefore, one must impose further geometric conditions on the manifold to ensure the validity of the Calder\'{o}n--Zygmund estimate., Comment: 9 pages. Comments are welcome!

We are concerned with the inviscid limit of the Navier-Stokes equations on bounded regular domains in $\mathbb{R}^3$ with the kinematic and Navier boundary conditions. We first establish the existence and uniqueness of strong solutions in the class $C([0,T_\star); H^r(\Omega; \mathbb{R}^3)) \cap C^1([0,T_\star); H^{r-2}(\Omega;\mathbb{R}^3))$ with some $T_\star>0$ for the initial-boundary value problem with the kinematic and Navier boundary conditions on $\partial \Omega$ and divergence-free initial data in the Sobolev space $H^r(\Omega; \mathbb{R}^3)$ for $r\geq 2$. Then, for the strong solution with $H^{r+1}$--regularity in the spatial variables, we establish the inviscid limit in $H^r(\Omega; \mathbb{R}^3)$ uniformly on $[0,T_\star)$ for $r > \frac{5}{2}$. This shows that the boundary layers do not develop up to the highest order Sobolev norm in $H^{r}(\Omega;\mathbb{R}^3)$ in the inviscid limit. Furthermore, we present an intrinsic geometric proof for the failure of the strong inviscid limit under a non-Navier slip-type boundary condition., Comment: 30 pages

Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface $(M,g)$ isometrically embedded in $\mathbb{R}^3$, we construct a mapping which sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on $(M,g)$. From the PDE perspectives, this mapping sends solutions to the Gauss--Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori, as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multi-dimensions., Comment: 35 pages

In this short note, we establish a quantitative description of the genericity of transversality of $C^1$-submanifolds in $\mathbb{R}^n$: Let $\Sigma \subset \mathbb{R}^n$ be a $d$-dimensional $C^1$-embedded submanifold where $n \geq d+1$. Denote by \begin{equation} \mathscr{A}(\Sigma) := \bigg\{ a \in \mathbb{R}^n: {\rm volume}\,\Big\{ p\in\Sigma : \partial\mathbb{B}(a, |a-p|) \text{ is not transversal to $\Sigma$ at $p$} \Big\} > 0 \bigg\}. \end{equation} Then $\mathscr{A}(\Sigma)$ is contained in a countable union of $(n-d-1)$-dimensional affine planes., Comment: 6 pages. To appear in Manuscripta Math., 2019

Let $\Omega \subset \mathbb{R}^3$ be a Lipschitz domain, and consider a harmonic map $v: \Omega \rightarrow \mathbb{S}^2$ with boundary data $v|\partial\Omega = \varphi$ which minimises the Dirichlet energy. For $p\geq 2$, we show that any energy minimiser $u$ whose boundary map $\psi$ has a small $W^{1,p}$-distance to $\varphi$ is close to $v$ in H\"{o}lder norm modulo bi-Lipschitz homeomorphisms, provided that $v$ is the unique minimiser attaining the boundary data. The index $p=2$ is sharp: the above stability result fails for $p<2$ due to the constructions by Almgren--Lieb \cite{al} and Mazowiecka--Strzelecki \cite{ms}., Comment: V3 contains more elaboration in Section 3, p.7 on the usage of L. Simon's asymptotic theory than the published version (to appear in JDE, 2021)