arXiv subjects

Siran Li

Publications and source records attributed to Siran Li.

66 records · Page 4Linked to original sources

arXiv2019

A Note on Generic Transversality of Euclidean Submanifolds

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

math.CA↗

arXiv2019

Towards a Theory of Multi-Parameter Geometrical Variational Problems: Fibre Bundles, Differential Forms and Riemannian Quasiconvexity

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}.

math.AP↗

arXiv2019

Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka

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 à la} F.~H. Clarke and the existence of special systems of Whitney flat $1$-forms with Sobolev regularity on certain families of homology manifolds.

math.DG↗

arXiv2019

Counterexamples to the $L^p$-Calderón--Zygmund Estimate on Open Manifolds

Based on a construction due to B. Güneysu 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 $L^p$-Calderón--Zygmund estimate \begin{equation*} \|\nabla \nabla f\|^p_{L^p} \leq C_1 \|Δf\|_{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 Calderón--Zygmund estimate.

math.AP↗

arXiv2018

The Inviscid Limit of the Navier-Stokes Equations with Kinematic and Navier Boundary Conditions

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 $C([0,T_\star); H^r(Ω; \mathbb{R}^3)) \cap C^1([0,T_\star); H^{r-2}(Ω;\mathbb{R}^3))$ with some $T_\star>0$ for the initial-boundary value problem with the kinematic and Navier boundary conditions on $\partial Ω$ and divergence-free initial data in the Sobolev space $H^r(Ω; \mathbb{R}^3)$ for $r\geq 2$. Then, for the strong solution with $H^{r+1}$--regularity in the spatial variables, we establish the inviscid limit in $H^r(Ω; \mathbb{R}^3)$ 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 $H^{r}(Ω;\mathbb{R}^3)$ 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.

math.AP↗

arXiv2018

Compensated Compactness in Banach Spaces and Weak Rigidity of Isometric Immersions of Manifolds

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.

math.DG↗

arXiv2018

On Vortex Alignment and Boundedness of $L^q$ Norm of Vorticity

We show that the spatial $L^q$ ($q > 5/3$) norm of the vorticity of an incompressible viscous fluid in $\mathbb{R}^3$ or $\mathbb{T}^3$ remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in the space variable, and that the space--time $L^q$ norm of the vorticity is finite. The Hölder index depends only on $q$. This serves as a variant of the classical result by P. Constantin and Ch. Fefferman (Direction of vorticity and the problem of global regularity for the Navier--Stokes equations, Indiana Univ. J. Math., 42 (1993), 775--789).

math.AP↗

arXiv2018

On the Existence of $C^{1,1}$ Isometric Immersions of Several Classes of Negatively Curved Surfaces Into $\mathbb{R}^3$

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These include the standard hyperbolic plane, generalised helicoid-type metrics and generalised Enneper metrics. Our proof is based on the method of compensated compactness and invariant regions in hyperbolic conservation laws, together with several observations on the geometric quantities (Gauss curvature, metric components etc.) of negatively curved surfaces.

math.AP↗

arXiv2017

Dimension Reduction of Compressible Fluid Models over Product Manifolds

In this paper we study the dimension reduction limits of the compressible Navier--Stokes equations over product Riemannian manifolds $\mathcal{O}_ε\cong \mathcal{M} \times ε\mathcal{F}$, such that $\dim\,(\mathcal{M})=n$ and $\dim\,(\mathcal{F})=d$ are arbitrary. Using the method of relative entropies, we establish the convergence of the suitable weak solutions of the Navier--Stokes equations on $\mathcal{O}_ε$ to the classical solution of the limiting equations on $\mathcal{M}$ as $ε\rightarrow 0^+$, provided the latter exists. In addition, we also deduce the vanishing viscosity limit. The limiting equations identified through our analysis contain the weight function $A:\mathcal{M} \rightarrow \mathbb{R}^+$ as a parameter, where $A(x)$ = area of fibre $\mathcal{F}_x$. Our work is based on and generalises the results in P. Bella, E. Feireisl, M. Lewicka and A. Novotný, A rigorous justification of the Euler and Navier--Stokes equations with geometric effects, \textit{SIAM J. Math. Anal.}, \textbf{48} (2016), 3907--3930, and it contains as special cases the physical models of circular nozzles, thin plate limits and finite-length longitudinal nozzles.

math.AP↗

arXiv2017

Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.

math.AP↗

arXiv2017

Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with unfixed metrics.

math.AP↗

arXiv2016

On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains

In this paper we consider the one-dimensional Navier-Stokes system for a heat-conducting, compressible reacting mixture which describes the dynamic combustion of fluids of mixed kinds on unbounded domains. This model has been discussed on bounded domains by Chen (SIAM Jour. Math. Anal., 23 (1992), 609--634) and Chen-Hoff-Trivisa (Arch. Rat. Mech. Anal. 166 (2003), 321--358) among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius Law. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in $H^1$. Moreover, the large-time behaviour of the weak solution is identified and proved. In particular, the uniform-in-time bounds for the temperature and specific volume have been established via energy estimates. For this purpose we utilise techniques developed by Kazhikhov and coauthors ({\it cf.} Siber. Math. Jour. 23 (1982), 44--49; Jour. Appl. Math. Mech., 41 (1977), 273--282), as well as a crucial estimate in the recent work by Li-Liang (Arch. Rat. Mech. Anal. 220 (2016), 1195--1208). Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.

math.AP↗