arXiv subjects
Publications and source records attributed to De Huang.
For the 2-D incompressible Euler equation, we prove the existence of $m$-fold symmetric uniformly rotating vortex patches with 90-degree corners for all $m\geq 12$. Properties of these patches and the induced stationary flows in the co-rotating frame are characterized. Our construction is based on a novel fixed-point method.
We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.
We investigate novel scenarios of self-similar finite-time blowups of the generalized Constantin-Lax-Majda model with a parameter $a$, which are induced by a new setting where the smooth initial data satisfy certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.
We establish non-asymptotic error bounds for the classical Maximal Likelihood Estimation of the transition matrix of a given Markov chain. Meanwhile, in the reversible case, we propose a new reversibility-preserving online Symmetric Counting Estimation of the transition matrix with non-asymptotic deviation bounds. Our analysis is based on a convergence study of certain Markov chains on the length-2 path spaces induced by the original Markov chain.
We establish a new Bernstein-type deviation inequality for general (non-reversible) discrete-time Markov chains via an elementary approach. More robust than existing works in the literature, our result only requires the Markov chain to satisfy an iterated Poincar\'e inequality. Moreover, our method can be readily generalized to continuous-time Markov processes.
We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb-Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.
We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the 1D Constantin-Lax-Majda model. The larger spatial scale measures the decreasing distance between the bulk of the solution and the eventual blowup point, while the smaller scale measures the shrinking size of the bulk of the solution. Similar multi-scale blowup phenomena have recently been discovered for many higher dimensional equations. Our study may provide some understanding of the common mechanism behind these multi-scale blowups.
We show that the 1D Hou-Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We also prove that the profiles satisfy some monotonicity and convexity properties that were unknown before, and we give rigorous estimates on the algebraic decay rates of the profiles in the far field. Our result supplements the previous computer-assisted proof of self-similar finite-time blowup for the Hou-Luo model with finer characterizations of the profiles.
We show that the $a$-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an $a$-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of $a$ and also explains previous numerical observations for a wide range of $a$.
We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator.
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in \cite{luo2014potentially,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in \cite{luo2014potentially,luo2013potentially-2} share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in \cite{chen2019finite} to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $C^γ$ norm of the density $θ$ with $γ\approx 1/3$ is uniformly bounded up to the singularity time.
In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thickness of the ring collapses at a rate $O(T-t)$. The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the $3$D Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier-Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.
We analyze Oja's algorithm for streaming $k$-PCA and prove that it achieves performance nearly matching that of an optimal offline algorithm. Given access to a sequence of i.i.d. $d \times d$ symmetric matrices, we show that Oja's algorithm can obtain an accurate approximation to the subspace of the top $k$ eigenvectors of their expectation using a number of samples that scales polylogarithmically with $d$. Previously, such a result was only known in the case where the updates have rank one. Our analysis is based on recently developed matrix concentration tools, which allow us to prove strong bounds on the tails of the random matrices which arise in the course of the algorithm's execution.
We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for Hölder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations.
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. In particular, it is shown that the classic Bakry-Émery curvature criterion implies subgaussian concentration for "matrix Lipschitz" functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron-Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.
We introduce the notion of $k$-trace and use interpolation of operators to prove the joint concavity of the function $(A,B)\mapsto\text{Tr}_k\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big]^\frac{1}{k}$, which generalizes Lieb's concavity theorem from trace to a class of homogeneous functions $\text{Tr}_k[\cdot]^\frac{1}{k}$. Here $\text{Tr}_k[A]$ denotes the $k_{\text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $A\mapsto\text{Tr}_k\big[\exp(H+\log A)\big]^\frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.
This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen-Ward, and they are similar in spirit to the results of Ahlswede-Winter and of Tropp for a sum of independent random matrices. The argument relies on the uniform smoothness properties of the Schatten trace classes.