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Qi S. Zhang

Publications and source records attributed to Qi S. Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Concavity and other properties of the entropy on manifolds

In this paper, we establish systematic estimates for the entropy and its density on Riemannian manifolds, focusing on the more challenging cases where the Ricci curvature changes sign or the boundary is nonconvex. For example, the refined second law of thermodynamics states that the entropy in a compact domain in $\mathbb R^n$ is increasing in time, furthermore, it is concave if the domain is convex. The concavity property is equivalent to the property that the Fisher information is decreasing, which also holds for convex domains in a Riemannian manifold with nonnegative Ricci curvature (cf. \cite{NiLei}). In view of the wide application of entropy in mathematics, information theory, physics, etc., there is certain desire in the community to extend the property to broader settings, especially to the case with nonconvex boundary (see e.g. \cite[p. 3]{CFM}). Here, we prove that the refined second law still holds if the domain is not too far from convex and the negative part of the Ricci curvature is not too large, in an explicit, nonperturbative sense, thus realizing some of the expectations. The proof is based on a new second order log Poincaré inequality that does not require explicit curvature conditions of the manifold. If the negative part of the Ricci curvature is too large, a counterexample to the concavity is given. Some other related estimates for the entropy density (Hamilton type estimates) are also proven.

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arXiv2026

On the axisymmetric Navier-Stokes flow passing a cone with the total-slip boundary condition

(A) It is known that among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,θ}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,θ}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. Note that, for the initial velocity, there is neither a size restriction on other components, nor a parity assumption. There are four key ingredients in the proof. (1) Three new good unknowns are introduced, and a self-closed energy estimate for them is derived. (2) An elliptic estimate for pressure is established to control boundary terms arising from the boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of $rv_θ$. (4) A new anisotropic Hardy's inequality is derived for weighted mean-zero functions to overcome the lack of parity of $\boldsymbol{v}$. (B) Based on (A), we introduce and prove the so-called controlled regularity for the above problem, i.e. for suitable initial data without any smallness assumption, there exists an external force supported away from the axis of symmetry such that the corresponding problem admits a global strong solution. This seems to add a little weight to the regularity scenario for ASNS, since the force is supported away from the axis which is the only place regularity may break down. We also prove that if there exists a solution that blows up in finite time, an unstable blow-up solution must exist.

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arXiv2025

Laplace comparison on Kähler Ricci flow and convergence

We first prove a uniform integral Laplace comparison result for the Kähler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.

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arXiv2025

Backward Harnack inequality and Hamilton estimates for heat type equations

Based on gradient estimates for the heat equation by Hamilton, we discover a backward in time Harnack inequality for positive solutions on compact manifolds without further restrictions such as boundedness or vanishing boundary value for solutions. Contrary to the usual Harnack inequality, it allows comparison of the values of a solution at two different space time points, in both directions of time. In view of the importance of the usual Harnack inequality, further application is expected.

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arXiv2025

Space of ancient caloric functions on some manifolds beyond volume doubling

Under a condition that breaks the volume doubling barrier, we obtain a time polynomial structure result on the space of ancient caloric functions with polynomial growth on manifolds. As a byproduct, it is shown that the finiteness result for the space of harmonic functions with polynomial growth on manifolds in \cite{CM97} and \cite{Li97} are essentially sharp, except for the multi-end cases, addressing an issue raised in \cite{CM98} and removing all {\it local} topological or geometric conditions on the manifold with respect to a reference point.

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arXiv2024

A Sharp Li-Yau gradient bound on Compact Manifolds

Let $(\M^n, g)$ be a $n$ dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant $-K \le 0$. Let $u$ be a positive solution of the heat equation on $\M^n \times (0, \infty)$. The well known Li-Yau gradient bound states that $$ t \left(\frac{|\nabla u|^2}{u^2} - α\frac{\pa_t u}{u}\right) \leq \frac{nα^2}{2} + t \frac{nα^2K}{2(α-1)},\quad \forall α>1, t>0. $$ The bound with $α=1$ is sharp if $K=0$. If $-K < 0$, the bound tends to infinity if $α=1$. In over 30 years, several sharpening of the bounds have been obtained with $α$ replaced by several functions $α=α(t)>1$ but not equal to $1$. An open question (\cite{CLN}, \citeLX} etc) asks if a sharp bound can be reached. In this short note, we observe that for all complete compact manifolds one can take $α=1$. Thus a sharp bound, up to computable constants, is found in the compact case. This result also seems to sharpen Theorem 1.4 in \cite{LY} for compact manifolds with convex boundaries. In the noncompact case one can not take $α=1$ even for the hyperbolic space. An example is also given, which shows that there does not exist an optimal function of time only $α=α(t)$ for all noncompact manifolds with Ricci lower bound, giving a negative answer to the open question in the noncompact case.

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arXiv2024

A blow up solution of the Navier-Stokes equations with a critical force

A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up moment, is in the energy space through out. Since most physical forces from a point source in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this result indicates possible singularity formation under these kind of forces. The result even holds for some log subcritical forces or some forces in the standard critical space $L^\infty_t L^{3/2}_x$, including the explicit force: $F=- δ\frac{e^{-|x|^2}}{(|x|^2 + T-t) \,[1+ | \ln (|x|^2 + T-t)|]} (1, 0, 0) $ for any small $δ>0$. The result can also be considered as a step in Scheffer's plan.

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arXiv2024

An improved Hamilton matrix estimates for the heat equation

In this paper, we remove the assumption on the gradient of the Ricci curvature in Hamilton's matrix Harnack estimate for the heat equation on all closed manifolds, answering a question which has been around since the 1990s. New ingredients include a recent sharp Li-Yau estimate, construction of a suitable vector field and various use of integral arguments, iteration and a little tensor algebra.

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arXiv2024

A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail. In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.

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arXiv2024

Log gradient estimates for heat type equations on manifolds

In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and extensions focusing on sharp constants and improved curvature conditions.

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arXiv2024

A rigidity result for ancient Ricci flows

Using a size condition of the sharp log Sobolev functional (log entropy) near infinity only, we prove a rigidity result for ancient Ricci flows without sign condition on the curvatures. The result is also related to the problem of identifying type II ancient Ricci flows and their backward limits.

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arXiv2023

Dimension reduction of axially symmetric Euler equations near maximal points off the axis

Let $v$ be a solution of the axially symmetric Euler equations (ASE) in a finite cylinder in $\mathbb{R}^3$. We show that suitable blow-up limits of possible velocity singularity and most self similar vorticity singularity near maximal points off the vertical axis are two dimensional ancient solutions of the Euler equation in either $\mathbb{R}^2 \times (-\infty, 0]$ or $\mathbb{R}^2_+ \times (-\infty, 0]$. This reduces the search of off-axis self-similar or other velocity blow-up solutions to a problem involving purely 2-dimensional Euler equations. Also, some asymptotic self-similar velocity blow-up and expected asymptotic self-similar vorticity blow up scenario at the boundary appear to be ruled out. On the other hand, this method may provide a path to velocity blow up if one can construct certain stable ancient solutions to the 2-d Euler equation in the half plane.

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arXiv2023

Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

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arXiv2023

Finite speed axially symmetric Navier-Stokes flows passing a cone

Let $D$ be the exterior of a cone inside a ball, with its altitude angle at most $π/6$ in $\mathbb{R}^3$, which touches the $x_3$ axis at the origin. For any initial value $v_0 = v_{0,r}e_{r} + v_{0,θ} e_θ + v_{0,3} e_{3}$ in a $C^2(\overline{D})$ class, which has the usual even-odd-odd symmetry in the $x_3$ variable and has the partial smallness only in the swirl direction: $ | r v_{0, θ} | \leq \frac{1}{100}$, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition has a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components $v_{0,r}$ and $v_{0,3}$. In a broad sense, this result appears to solve $2/3$ of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry. Equivalently, this result is connected to the general open question which asks that if an absolute smallness of one component of the initial velocity implies the global smoothness, see e.g. page 873 in \cite{CZZ17}. Our result seems to give a positive answer in a special setting. As a byproduct, we also construct an unbounded solution of the forced Navier Stokes equation in a special cusp domain that has finite energy. The forcing term, with the scaling factor of $-1$, is in the standard regularity class. This result confirms the intuition that if the channel of a fluid is very thin, arbitrarily high speed in the classical sense can be attained under a mildly singular force which is physically reasonable in view that Newtonian gravity and Coulomb force have scaling factor $-2$.

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arXiv2022

Time analyticity for nonlocal parabolic equations

In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-\mathrm{L}_α^κ u(t,x)=0$, where $\mathrm{L}_α^κ$ is a nonlocal operator of order $α$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|\leq C(1+|x|)^{α-ε}$ for any $(t,x)\in (0,1]\times \mathbb{R}^d$ and $ε\in(0,α)$. We also obtain pointwise estimates for $\partial_t^kp_α(t,x;y)$, where $p_α(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $\mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $\mathrm{M}$ satisfies the Poincaré inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $\mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $α\in (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x\neq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.

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arXiv2022

Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region

A domain in $\mathbb{R}^3$ that touches the $x_3$ axis at one point is found with the following property. For any initial value in a $C^2$ class, the axially symmetric Navier Stokes equations with Navier slip boundary condition has a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.

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arXiv2021

Smooth solutions to the heat equation which are nowhere analytic in time

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Crelle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown in \cite{Zha20, DZ20, DP20} that solutions to the heat equation in the whole space, or half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $δ>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+δ$, which is nowhere analytic in time.

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