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Qi S Zhang

Publications and source records attributed to Qi S Zhang.

14 recordsLinked to original sources
arXiv2026

On partial type I solutions to the Axially symmetric Navier-Stokes equations

Let $v= v_{r}e_{r} + v_þe_þ + v_{3}e_{3}$ be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations (ASNS). We call it a partial type I solution if $v_r(x, t) \ge -C/\sqrt{T-t}$ for some constant $C>0$ and $(x, t) \in \mathbf{R}^3 \times [0, T)$. In this paper, it is proven that such solution does not blow up at time $T$ under the extra mild assumption that $|v_θ(x, 0)| |x'|$ is bounded. This extends a well known result by two groups of people who proved the no blowup conclusion under the full type I condition: $|v(x, t)| \le C/\sqrt{T-t}$. The result also confirms the physical intuition that potential blow ups for ASNS are caused by super-critical inward radial velocity.

math.AP↗
arXiv2020

A note on convergence of noncompact nonsingular solutions of the Ricci flow

We extend some convergence results on nonsingular compact Ricci flows in the papers \cite{Ha:1}, \cite{Se:1} and \cite{FZZ:2} to certain infinite volume noncompact cases which are "partially" nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.

math.DG↗
arXiv2020

A formula for backward and control problems of the heat equation

(a). Using time analyticity result, we address a basic question for a nonhomogeneous backward heat equation (exact control problem) in the setting of smooth domains and compact manifolds, namely: when is essentially time independent control possible? i.e. The control function is 0 on one time interval and stationary on the other. For general $L^2$ initial values, the answer is: if and only if the full space domain is used for the control function. Also an explicit formula for the control function is found in the form of an infinite series involving the heat kernel, which converges rapidly. (b). A formal exact formula for a time dependent control function supported in a proper subdomain is also obtained via eigenfunctions of the Laplacian. The formula is rigorous on any finite dimensional space spanned by the eigenfunctions and there is no smoothness assumption on the whole domain, making partial progress on a problem on p74 \cite{Zu:1}. (c). A byproduct is an inversion formula for the heat kernel.

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arXiv2020

Time analyticity for the heat equation and Navier-Stokes equations

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pertains bounded mild solutions of the incompressible Navier-Stokes equations in the whole space. There are many long established analyticity results for the Navier-Stokes equations. See for example \cite{Ka:1} and \cite{FT:1} for spatial and time analyticity in certain integral sense, \cite{CN:1} for pointwise space-time analyticity of 3 dimensional solutions to the Cauchy problem, and also the pointwise time analyticity results of \cite{Ma:1} and \cite{Gi:1} under zero boundary condition. Our result seems to be the first general pointwise time analyticity result for the Cauchy problem for all dimensions, whose proof involves only real variable method. The proof involves a method of algebraically manipulating the integral kernel, which appears applicable to other evolution equations.

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arXiv2019

Local estimates on two linear parabolic equations with singular coefficients

We treat the heat equation with singular drift terms and its generalization: the linearized Navier-Stokes system. In the first case, we obtain boundedness of weak solutions for highly singular, "supercritical" data. In the second case, we obtain regularity result for weak solutions with mildly singular data ( those in the Kato class). This not only extends some of the classical regularity theory in [AS], [CrZ] and others from the case of elliptic and heat equations to that of linearized Navier-Stokes equations but also proves an unexpected gradient estimate, which extends the recent interesting boundedness result [O]. In the addendum in May 2019, a missing term in Theorem 1.7 and Lemma 3.3 is added. This is due to the use of a formula in a cited reference, which omitted a term. The main conclusion that local solutions of certain linearized Navier-Stokes equation have bounded spatial gradient is intact. This includes bounded local Leray-Hopf solutions to the Navier Stokes equation without condition on the pressure.

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arXiv2019

On ancient periodic solutions to Axially-Symmetric Navier-Stokes Equations

An old problem asks whether bounded mild ancient solutions of the 3 dimensional Navier-Stokes equations are constants. While the full 3 dimensional problem seems out of reach, in the works \cite{KNSS, SS09}, the authors expressed their belief that the following conjecture should be true. For incompressible axially-symmetric Navier-Stokes equations (ASNS) in three dimensions: \textit{bounded mild ancient solutions are constant}. Understanding of such solutions could play useful roles in the study of global regularity of solutions to the ASNS. In this article, we essentially prove this conjecture in the special case that $u$ is periodic in $z$. To the best of our knowledge, this seems to be the first result on this conjecture without unverified decay condition. It also shows that periodic solutions are not models of possible singularity or high velocity region. Some partial result in the non-periodic case is also given.

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arXiv2018

Li-Yau gradient bounds on compact manifolds under nearly optimal curvature conditions

We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some $p>n/2$, or $\sup_\M \int_\M |Ric^-|^2(y)d^{2-n}(x,y)dy<\infty$, where $n$ is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li-Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below.

math.DG↗
arXiv2017

Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler \cite{W:1} on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy \cite{P:1} in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to $\R^n$. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $\R^n$. Comparing with the well known flatness results in \cite{An:1}, \cite{Ba:1} and \cite{BKN:1} on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in $C^1$ sense at infinity. No second order condition on the metric is needed.

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arXiv2017

Bounds on harmonic radius and limits of manifolds with bounded Bakry-Émery Ricci curvature

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^α W^{1, q}$ harmonic radius for manifolds with bounded Bakry-Émery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^αW^{1,q}$, where $q>2n$ and $n$ is the dimension of the manifolds. This is almost 1 order lower than that in the classical $C^{1,α} W^{2, p}$ harmonic coordinates under bounded Ricci curvature condition [And]. The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $W^{2, p}$ convergence in the classical case. Based on this lower bound and the techniques in [ChNa2] and [WZ], we extend Cheeger-Naber's Codimension 4 Theorem in [ChNa2] to the case where the manifolds have bounded Bakry-Émery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green's function argument and adopt a linear algebra argument in [Bam]. A new ingradient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.

math.DG↗
arXiv2017

Improved Liouville theorems for axially symmetric Navier-Stokes equations

In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. In 2D and the 3D axially symmetric case without swirl, we prove sharp Liouville theorems for smooth ancient mild solutions: velocity fields $v$ are constants if vorticity fields satisfy certain condition and $v$ are sublinear with respect to spatial variables, and we also give counterexamples when $v$ are linear with respect to spatial variables. The condition which vorticity fields need to satisfy is $\lim\limits_{|x|\rightarrow +\infty}|w(x,t)|=0$ and $\lim\limits_{r\rightarrow +\infty}\frac{|w|}{\sqrt{x_1^2+x_2^2}}=0$ uniformly for all $t\in(-\infty,0)$ in 2D and 3D axially symmetric case without swirl, respectively. In the case when solutions are axially symmetric with nontrivial swirl, we prove that if $Γ=rv_θ\in L^\infty_tL^p_x(\mathbb{R}^3\times(-\infty,0))$ where $1\leq p<\infty$, then bounded ancient mild solutions are constants.

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arXiv2016

Li-Yau gradient bound for collapsing manifolds under integral curvature condition

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough, certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\M$ with $0<r\leq 1$. Here the assumption that $k(p,1)$ being small allows the situation where the manifolds is collapsing. Recall that in \cite{ZZ}, certain Li-Yau gradient bounds was also obtained by the authors, assuming that $|Ric^-|\in L^p(\M)$ and the manifold is noncollaped. Therefore, to some extent, the results in this paper and in \cite{ZZ} complete the picture of Li-Yau gradient bound for the heat equation on manifolds with $|Ric^-|$ being $L^p$ integrable, modulo sharpness of constants.

math.DG↗
arXiv2016

New Volume Comparison results and Applications to degeneration of Riemannian metrics

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead. Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given. To this effect, in particular for the Bakry-Émery Ricci curvature condition, the gradient of the potential function is allowed to have singularity of order close to $1$ while the traditional method of weighted manifolds allows bounded gradient. This approach enables us to extend some of the results in the papers \cite{Co}, \cite{ChCo2}, \cite{zZh}, \cite{TZ} and \cite{WZ}. The condition also covers general Ricci solitons instead of just gradient Ricci solitons.

math.DG↗
arXiv2013

On the question of diameter bounds in Ricci flow

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman's $ν$ invariant. Here $n$ is the dimension. His result is sharp when Perelman's F entropy is positive. In this note, we give a direct proof that for all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and the Sobolev constants (or positive Yamabe constant). This bound seems directly computable in large time for some Ricci flows. In addition, since the result in its most general form is independent of Ricci flow, further applications may be possible. A generally sharp lower bound for the diameters is also given, which depends only on the initial metric, time and $L^\infty$ bound of the scalar curvature. These results imply that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows up.

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