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Qi S. Zhang

Publications and source records attributed to Qi S. Zhang.

50 records · Page 3Linked to original sources
arXiv2011

Extremal of Log Sobolev inequality and $W$ entropy on noncompact manifolds

Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This result has the following consequences. 1. It seems to give the first example of connected, complete manifolds with bounded geometry where a standard Log Sobolev inequality does not have an extremal. 2. It gives a negative answer to the open question on the existence of extremal of Perelman's $W$ entropy in the noncompact case, which was stipulated by Perelman \cite{P:1} p9, 3.2 Remark. 3. It helps to prove, in some cases, that noncompact shrinking breathers of Ricci flow are gradient shrinking solitons.

math.DG↗
arXiv2010

A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations

Let $v(x, t)= v^r e_r + v^θe_θ+ v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $Γ= r v^θ$ is Hölder continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$, then $v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.

math.AP↗
arXiv2010

Structure of Singularities of 3D Axi-symmetric Navier-Stokes Equations

Let $v$ be a solution of the axially symmetric Navier-Stokes equation. We determine the structure of certain (possible) maximal singularity of $v$ in the following sense. Let $(x_0, t_0)$ be a point where the flow speed $Q_0 = |v(x_0, t_0)|$ is comparable with the maximum flow speed at and before time $t_0$. We show after a space-time scaling with the factor $Q_0$ and the center $(x_0, t_0)$, the solution is arbitrarily close in $C^{2, 1, α}_{\rm local}$ norm to a nonzero constant vector in a fixed parabolic cube, provided that $r_0 Q_0$ is sufficiently large. Here $r_0$ is the distance from $x_0$ to the $z$ axis. Similar results are also shown to be valid if $|r_0v(x_0, t_0)|$ is comparable with the maximum of $|rv(x, t)|$ at and before time $t_0$.

math.AP↗
arXiv2010

The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $κ$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.

math.DG↗
arXiv2009

Heat kernel bounds, ancient $κ$ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $κ$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \cite{ChZ:1} lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.

math.DG↗
arXiv2007

Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries

We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.

math.DG↗
arXiv2007

A uniform Sobolev inequality under Ricci flow

Let ${\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\bf M}, g(t))$ for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results. An application to 3-d Ricci flow with surgery is also presented.

math.DG↗
arXiv2006

A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

We establish a point-wise gradient estimate for $all$ positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li-Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A generalization of Perelman's W-entropy is defined in both the Ricci flow and fixed metric case. We also find a new family of heat kernel estimates.

math.DG↗
arXiv2006

Some gradient estimates for the heat equation on domains and for an equation by Perelman

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

math.DG↗
arXiv2006

When does a Schrödinger heat equation permit positive solutions

We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are much more singular than those allowed under the current theory. In the case of $L^2$ potentials and $L^2$ solutions, we give a characterization of potentials which allow the Schrödinger heat equation to have a positive solution. This provides a new result on the long running problem of identifying potentials permitting a positive solution to the Schrödinger equation. We also establish a nearly necessary and sufficient condition on certain sign changing potentials such that the corresponding heat kernel has Gaussian upper and lower bound. Some applications to the Navier-Stokes equations are given. In particular, we derive a new type of a priori estimate for solutions of Navier-Stokes equations. The point is that the gap between this estimate and a sufficient condition for all time smoothness of the solution is logarithmic.

math.AP↗
arXiv2005

Large time behavior of heat kernels on forms

We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions.

math.DG↗
arXiv2005

Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive eternal solutions of the heat equation, under certain growth condition. Surprisingly, this Liouville theorem for the heat equation does not hold even in ${\bf R}^n$ without such a condition. We also prove a sharpened long time gradient estimate for the log of heat kernel on noncompact manifolds. This has been an open problem in view of the well known estimates in the compact, short time case.

math.DG↗
arXiv2004

Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation below can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous blow up conjecture about critical semilinear wave equations of the form $Δu - \partial^2_t u + |u|^p = 0$ in dimensions $n \ge 4$. The lower dimensional case $n \le 3$ was settled many years earlier.

math.AP↗