Searcharxiv⌕ Search

arXiv subjects

Qi S. Zhang

Publications and source records attributed to Qi S. Zhang.

At least 37 records · Page 2Linked to original sources
arXiv2020

Gradient and Eigenvalue Estimates on the canonical bundle of Kähler manifolds

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of the canonical bundle of Kähler manifolds, where $m$ is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of $(m, 0)$ forms, which involves only the Ricci curvature and the gradient of the scalar curvature.

math.DG↗
arXiv2020

Decay and vanishing of some D-solutions of the Navier-Stokes equations

An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two special cases: (1) when the solution is axially symmetric and periodic in the vertical variable; (2) full 3 dimensional slab case $\mathbb{R}^2 \times [0, 1]$ with Dirichlet boundary condition. Other partial results are also presented. The paper is self contained comparing with the first part \cite{CPZ:1} although the general idea is related.

math.AP↗
arXiv2020

Decay and vanishing of some axially symmetric D-solutions of the Navier-Stokes equations

We study axially symmetric D-solutions of the 3 dimensional Navier-Stokes equations. The first result is an a priori decay estimate of the velocity for general domains. The second is an a priori decay estimate of the vorticity in $\bR^3$, which improves the corresponding results in the literature. In addition, we prove a similar decay of full 3d solutions except for a small set of angles. Next we turn to D-solutions which are periodic in the third variable and prove vanishing result under a reasonable condition. As a corollary we prove that axially symmetric D-solutions in the slab $\bR^2 \times I$ with suitable boundary condition is $0$. Here $I$ is any finite interval. To the best of our knowledge, this seems to be the first vanishing result on a 3 dimensional D-solution without extra integral or decay or smallness assumption on the solution. The tools used include Brezis-Gallouet inequality, dimension reduction, scaling, Green's function bound and Liouville theorems for Navier-Stokes equations.

math.AP↗
arXiv2019

A Liouville Theorem for Axi-symmetric Navier-Stokes Equations on $\mathbb{R}^2 \times \mathbb{T}^1$

We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. This is a step forward to completely solve the conjecture on $(-\infty, 0] \times \mathbb{R}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.

math.AP↗
arXiv2019

A note on time analyticity for ancient solutions of the heat equation

It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and sufficient condition is found on the solvability of backward heat equation in the class of functions with exponential growth.

math.AP↗
arXiv2018

On ancient solutions of the heat equation

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of positive solutions of the family of elliptic operator $Δ- s$ with $s>0$. Further relaxation of the curvature assumption is also possible. It is also shown that the linear space of ancient solutions of polynomial growth has finite dimension and these solutions are polynomials in time.

math.AP↗
arXiv2016

Regularity of Weak Solutions of Elliptic and Parabolic Equations with Some Critical or Supercritical Potentials

We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations %$Δu-\frac{A}{|x|^{2+β}}u=0,\,\,(β\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01} \partial_{i} (a_{ij}(x) \partial_{j}u(x)) - \frac{A}{|x|^{2+β}} u(x) =0\quad (A>0,\quadβ\geq 0), \end{equation} \begin{equation}\label{02} \partial_{i} (a_{ij}(x,t) \partial_{j}u(x,t)) - \frac{A}{|x|^{2+β}} u(x,t)-\partial_{t}u(x,t) =0\quad (A>0,\quadβ\geq 0), \end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. We also prove, in some special cases, weak solutions are even differentiable. Previously P. Baras and J. A. Goldstein \cite{Baras1984} treated the case when $A<0$, $(a_{ij})=I$ and $β=0$ for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of $|A|$. When $A>0$, $β=0$ and $(a_{ij})=I$, P. D. Milman and Y. A. Semenov \cite{Milman2003}\cite{Milman2004} obtain bounds for the heat kernel.

math.AP↗
arXiv2015

Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform $L^2$ curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger $\varepsilon$-regularity theorem for Ricci flows. This result is particularly useful in the study of Kähler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.

math.DG↗
arXiv2015

Criticality of the Axially Symmetric Navier-Stokes Equations

Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\sup_{t\geq 0}\|rv^θ(t, \cdot)\|_{L^\infty} \leq \|rv^θ(0, \cdot)\|_{L^\infty}.$$ We prove that all solutions with initial data in $H^{\frac{1}{2}}$ is smooth globally in time if $rv^θ$ satisfies a kind of Form Boundedness Condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if $rv^θ$ satisfies \begin{equation}\nonumber \sup_{t \geq 0}|rv^θ(t, r, z)| \leq C_\ast|\ln r|^{- 2},\ \ r \leq δ_0 \in (0, \frac{1}{2}),\ C_\ast < \infty, \end{equation} then our FBC is satisfied. Here $δ_0$ and $C_\ast$ are independent of neither the profile nor the norm of the initial data. So the gap from regularity is logarithmic in nature. We also prove the global regularity of solutions if $\|rv^θ(0, \cdot)\|_{L^\infty}$ or $\sup_{t \geq 0}\|rv^θ(t, \cdot)\|_{L^\infty(r \leq r_0)}$ is small but the smallness depends on certain dimensionless quantity of the initial data.

math.AP↗
arXiv2015

A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations

Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that $v$ is regular if the angular velocity $v_θ$ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying \[ |v_θ(x, t)| \le \frac{C}{r |\ln r|^{2+ε}}, \quad r<1/2, \] where $r$ is the distance from $x$ to the axis, $C$ and $ε$ are any positive constants. Comparing with the critical a priori bound \[ |v_θ(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2, \]our condition is off by the log factor $|\ln r|^{2+ε}$ at worst. This is inspired by the recent interesting paper \cite{CFZ:1} where H. Chen, D. Y. Fang and T. Zhang establish, among other things, an almost critical regularity condition on the angular velocity. Previous regularity conditions are off by a factor $r^{-1}$. The proof is based on the new observation that, when viewed differently, all the vortex stretching terms in the 3 dimensional axially symmetric Navier-Stokes equations are critical instead of supercritical as commonly believed.

math.AP↗
arXiv2015

A priori bound on the velocity in axially symmetric Navier-Stokes equations

Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound \[ |v(x, t)| \le \frac{C}{r^2} |\ln r|^{1/2}, \]where $r \in (0, 1/2)$ is the distance from $x$ to the z axis, and $C$ is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \cite{CSTY2} and \cite{KNSS} gave a lower bound. The gap is polynomial order 1 modulo a half log term.

math.AP↗
arXiv2014

A compactness result for Fano manifolds and Kähler Ricci flows

We obtain a compactness result for Fano manifolds and Kähler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the Kähler Ricci flows. One assumption is the boundedness of the Ricci potential and the other is the smallness of Perelman's entropy. As one application, we obtain a new local regularity criteria and structure result for Kähler Ricci flows. The proof is based on a Hölder estimate for the gradient of harmonic functions, which may be of independent interest.

math.DG↗
arXiv2013

No breathers theorem for some noncompact Ricci flows

Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature. Since the method for the compact case faces a difficulty, the proof involves solving a new non-local elliptic equation which is the Euler-Lagrange equation of a scaling invariant log Sobolev inequality.

math.DG↗
arXiv2013

Isoperimetric inequality under Kähler Ricci flow

Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bounded from below.

math.DG↗
arXiv2011

Bounds on volume growth of geodesic balls under Ricci flow

We prove a so called $κ$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $κ$ non-collapsing property for Ricci flow. These two results together imply volume doubling property for Ricci flow without assuming Ricci curvature lower bound.

math.DG↗