arXiv subjects
Publications and source records attributed to Feng Shao.
Let $\Omega\subset\mathbb{R}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson--Shapiro conjecture in this setting: if every polynomial on $\mathbb{R}^2$ agrees on $\partial\Omega$ with a harmonic polynomial, then $\partial\Omega$ is an ellipse and $\Omega$ is its bounded interior.
In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(\rho_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $\rho_0$ has a positive lower bound. Furthermore, if $\rho_0 \in C^2$ and $||\rho_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.
We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d\ge 3$. For any $\alpha\in(0, \alpha_d)$ with $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity field satisfies $u_0\in C^{1,\alpha}_{\rm loc}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile equations, which form a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, while the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge consists in choosing appropriate function spaces that remain invariant under such nonlinear compositions and that simultaneously capture the exact singular behavior near the origin and the symmetry axis. Furthermore, we establish a finite-codimensional stability result for the self-similar profiles obtained above. As a consequence, after suitable truncation and correction of finitely many unstable modes, we obtain finite-energy blow-up solutions with initial velocity in $C^{1,\alpha}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})\cap L^2(\mathbb R^d)$ and compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.
In this paper, we consider the Zakharov-Ito equation \begin{equation*} \begin{cases} u_t+u_{xxx}+3uu_x+\rho\rho_x=0,\\ \rho_t+{(u\rho)}_x=0. \end{cases} \end{equation*} We prove the local well-posedness in $H^s\times H^s$ for $s>3/2$ and global well-posedness in $H^s\times H^s$ for $s\geq2$. When $\rho=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\in(0,+\infty)$. We prove the orbital stability of these solitary waves in $H^1\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \cite{GSS1987}.
This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets after the formation of curvature singularities. The most challenging part of this paper is to handle the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.
In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $\gamma=5/3$, which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Rapha\"el, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for $\gamma=5/3$. Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.
In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $\rho_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $\rho_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ and $\rho_0$ is close enough to a positive constant in $L^\infty$, we establish the global existence of strong solution to (INS) for $3<p<\infty$ and provide the uniqueness of the solution for $3<p<6$. This result corresponds to Cannone-Meyer-Planchon solution of the classical Navier-Stokes system. Furthermore, with the additional assumption that $u_0\in L^2(\mathbb{R}^3)$, we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). Finally, we prove the global well-posedness of (INS) with $u_0\in \dot{B}^{\frac{1}{2}}_{2,\infty}(\mathbb{R}^3)$ being small and only an upper bound on the density. This gives the first existence result of the forward self-similar solution for (INS).
For a sequence of vectors $\{v_n\}_{n\in\mathbb{N}}$ in the uniformly convex Banach space $X$ which for all $n, m\in \mathbb{N}$ satisfy $\|v_{n+m}\|\le \|v_n + v_m\|$ we show the existence of the limit $\lim_{n\to \infty} \frac{v_n}{n}$. This extends the classical Fekete's subadditivite lemma to Banach space-valued sequences.
Three-dimensional (3D) point cloud, as an emerging visual media format, is increasingly favored by consumers as it can provide more realistic visual information than two-dimensional (2D) data. Similar to 2D plane images and videos, point clouds inevitably suffer from quality degradation and information loss through multimedia communication systems. Therefore, automatic point cloud quality assessment (PCQA) is of critical importance. In this work, we propose a novel no-reference PCQA method by using a graph convolutional network (GCN) to characterize the mutual dependencies of multi-view 2D projected image contents. The proposed GCN-based PCQA (GC-PCQA) method contains three modules, i.e., multi-view projection, graph construction, and GCN-based quality prediction. First, multi-view projection is performed on the test point cloud to obtain a set of horizontally and vertically projected images. Then, a perception-consistent graph is constructed based on the spatial relations among different projected images. Finally, reasoning on the constructed graph is performed by GCN to characterize the mutual dependencies and interactions between different projected images, and aggregate feature information of multi-view projected images for final quality prediction. Experimental results on two publicly available benchmark databases show that our proposed GC-PCQA can achieve superior performance than state-of-the-art quality assessment metrics. The code will be available at: https://github.com/chenwuwq/GC-PCQA.
In this paper, we solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our weak solution equals to Lions' weak solution, which in particular implies the uniqueness of Lions' weak solution. We also extend a celebrated result by Fujita and Kato on the 3-D incompressible Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in $\dot H^{1/2}(\mathbb R^3)$}. The proof of the uniqueness is based on a surprising finding that the estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of $\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the uniqueness of the solution.
In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $\rho_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.}
In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.
Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.
Let $f\colon X\to Y$ be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle $T_X$ is big. We show that $f$ is an isomorphism unless $Y$ is a projective space. As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and $\mathring{\omega}\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$) and a dominant condition. As an important application, we prove the existence of weak solution when $\mathring{\omega}$ is a Radon measure on $\mathbb T$ with $m$-fold symmetry, which is related to the vortex sheet solution.
Arbitrary neural style transfer is a vital topic with great research value and wide industrial application, which strives to render the structure of one image using the style of another. Recent researches have devoted great efforts on the task of arbitrary style transfer (AST) for improving the stylization quality. However, there are very few explorations about the quality evaluation of AST images, even it can potentially guide the design of different algorithms. In this paper, we first construct a new AST images quality assessment database (AST-IQAD), which consists 150 content-style image pairs and the corresponding 1200 stylized images produced by eight typical AST algorithms. Then, a subjective study is conducted on our AST-IQAD database, which obtains the subjective rating scores of all stylized images on the three subjective evaluations, i.e., content preservation (CP), style resemblance (SR), and overall vision (OV). To quantitatively measure the quality of AST image, we propose a new sparse representation-based method, which computes the quality according to the sparse feature similarity. Experimental results on our AST-IQAD have demonstrated the superiority of the proposed method. The dataset and source code will be released at https://github.com/Hangwei-Chen/AST-IQAD-SRQE
Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such that the Fano index of $X$ is greater than 1. On the one hand, when both $X$ and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is bounded in terms of $X$ and $Y$ unless $X\cong\mathbb{P}^4$; hence, such $X$ does not admit any non-isomorphic surjective endomorphism. On the other hand, when $X=Y$ is either a fourfold or a del Pezzo manifold, we prove that, if $f$ is an int-amplified endomorphism, then $X$ is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms.
Just noticeable difference (JND) of natural images refers to the maximum pixel intensity change magnitude that typical human visual system (HVS) cannot perceive. Existing efforts on JND estimation mainly dedicate to modeling the diverse masking effects in either/both spatial or/and frequency domains, and then fusing them into an overall JND estimate. In this work, we turn to a dramatically different way to address this problem with a top-down design philosophy. Instead of explicitly formulating and fusing different masking effects in a bottom-up way, the proposed JND estimation model dedicates to first predicting a critical perceptual lossless (CPL) counterpart of the original image and then calculating the difference map between the original image and the predicted CPL image as the JND map. We conduct subjective experiments to determine the critical points of 500 images and find that the distribution of cumulative normalized KLT coefficient energy values over all 500 images at these critical points can be well characterized by a Weibull distribution. Given a testing image, its corresponding critical point is determined by a simple weighted average scheme where the weights are determined by a fitted Weibull distribution function. The performance of the proposed JND model is evaluated explicitly with direct JND prediction and implicitly with two applications including JND-guided noise injection and JND-guided image compression. Experimental results have demonstrated that our proposed JND model can achieve better performance than several latest JND models. In addition, we also compare the proposed JND model with existing visual difference predicator (VDP) metrics in terms of the capability in distortion detection and discrimination. The results indicate that our JND model also has a good performance in this task.