Differential Harnack Inequalities Research Articles

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds II: Integral curvature condition

We present a unified method for deriving differential Harnack inequalities for positive solutions to semilinear parabolic equations on compact manifolds and complete Riemannian manifolds, subject to an integral curvature condition. Specifically, we obtain the differential Harnack inequalities by solving a related system of ordinary differential equations. In addition to the case of scalar equations, we also establish an elliptic estimate for the heat flow under the same condition, which is a novel result for both harmonic map and heat equations. Many of the results presented here are nearly sharp, meaning they are sharp under the assumption of Ricci nonnegativity.

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation∂tu=ΔVu+H(u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems.

Differential Harnack inequalities for semilinear parabolic equations on [formula omitted] metric measure spaces

We present a unified method to derive differential Harnark inequalities for locally positive weak solutions to semilinear parabolic equations on RCD∗(K,N)metric measure spaces, as introduced by Gigli (2015) and Erbar et al. (2015). As its application, on the one hand, we recover many known differential Harnack inequalities on heat equation, on the other hand, for logarithmic type equation and Yamabe type equation, we get some sharp differential Harnack inequalities on RCD∗(0,N)metric measure spaces. As its corollary, we get some sharp Harnack inequalities and Liouville theorems.

Differential Harnack Inequality for a Parabolic Equation Under the Finsler-Geometric Flow

Differential Harnack Inequality for a Parabolic Equation Under the Finsler-Geometric Flow

Differential Harnack inequalities via Concavity of the arrival time

Differential Harnack inequalities via Concavity of the arrival time

Differential Harnack inequalities on path space

Recall that if (Mn,g) satisfies Ric≥0, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative f:M→R+, with ft its heat flow, that Δftft−|∇ft|2ft2+n2t≥0. Our main result will be to generalize this to path space PxM of the manifold.A key point is that instead of considering infinite dimensional gradients and Laplacians on PxM we will consider, in a spirit similar to [13,8], a family of finite dimensional gradients and Laplace operators. Namely, for each H01-function φ:R+→R we will define the φ-gradient ∇φF:PxM→TxM and the φ-Laplacian ΔφF=trφHessF:PxM→R, where Hess F is the Markovian Hessian and both the gradient and the φ-trace are induced by n vector fields naturally associated to φ under stochastic parallel translation.Now let (Mn,g) satisfy Ric=0, then for each nonnegative F:PxM→R+ we will show the inequalityEx[ΔφF]Ex[F]−Ex[∇φF]2Ex[F]2+n2||φ||2≥0 for each φ, where Ex denotes the expectation with respect to the Wiener measure on PxM. By applying this to the simplest functions on path space, namely cylinder functions of one variable F(γ)≡f(γ(t)), we will see we recover the classical Li-Yau Harnack inequality exactly. We have similar estimates for Einstein manifolds, with errors depending only on the Einstein constant, as well as for general manifolds, with errors depending on the curvature. Finally, we derive generalizations of Hamilton's Matrix Harnack inequality on path space PxM. It is our understanding that these estimates are new even on the path space of Rn.

Discrete versions of the Li-Yau gradient estimate

We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new com- putation rules for differential operators on discrete spaces and introduce a re- laxation function that governs the time dependency in the differential Harnack estimate

Exponential convergence of parabolic optimal transport on bounded domains

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve the linearized problem. Using this Harnack inequality and certain techniques specific to mass transport, we control the oscillation in time of solutions to the parabolic equation, and obtain exponential convergence. Additionally, in the course of the proof, we present a connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport, which is interesting in its own right.

New Differential Harnack Inequalities for Nonlinear Heat Equations

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $\omega_t=\Delta\omega+a\omega\ln \omega$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $\omega_t=\Delta\omega-\omega\ln\omega+\varepsilon R\omega$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove a new differential Harnack inequality for the equation $\omega_t=\Delta\omega-\omega\ln\omega$ under the Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.

Li–Yau gradient estimates for curvature flows in positively curved manifolds

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.

Harnack estimate for mean curvature flow on the sphere

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the mean curvature flow on the sphere.

Ricci flow with surgery on manifolds with positive isotropic curvature

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$. In the second part, we develop a theory of ancient solutions which have bounded curvature; are $\kappa$-noncollapsed; are weakly PIC2; and are uniformly PIC. This is an extension of Perelman's work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1. In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman's surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ which do not contain non-trivial incompressible $(n-1)$-dimensional space forms.

Differential Harnack Estimates for a Semilinear Parabolic System

In this paper, we prove differential Harnack inequalities for positive solutions of a semilinear parabolic system on hyperbolic space. We use the inequalities to construct classical Harnack estimates by integrating along space-time.

Li-Yau estimates for a semilinear parabolic equation on an evolving manifold

Let (M,g(t)) be a complete Riemannian manifold of dimension n, we obtain Li-Yau type gradient estimates on positive bounded solutions to the following semilinear parabolic equation&#x0D; ∂u(t,x) ∕ ∂t = Δ u(t,x) + a(x) us(t,x) -λu(t,x),&#x0D; where (t,x) ∈ ([0,T] × M), T &lt; ∞, s &gt; 1, λ ∈ ℝ and a ∈ C2(M) on evolving Riemannian metrics g(t) with bounded below Ricci tensor. The application of our gradient estimates yields the classical differential Harnack inequality, which compares a solution at some time with those at previous time.

Harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces

We obtain a differential Harnack inequality for anisotropic curvature flow of convex hypersurfaces in Euclidean space with its speed given by a curvature function of homogeneity degree one in a certain class, and restrictions depending only on the initial data and the anisotropic factor which reflects the influence of the ambient space. Moreover, the pinching estimate for such flows is derived from the maximum principle for tensors.

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(−K,m)-condition, where m∈[n,∞) and K≥0 are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the CD(−K,m)-condition and on compact manifolds equipped with (−K,m)-super Ricci flows.

Heat Kernel Estimates Under the Ricci–Harmonic Map Flow

AbstractThis paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn.

Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities (Theorem 2.1) for the positive solutions to the Schrödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality (Theorem 4.1). Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity (Theorem 5.3).

Harnack estimate for a semilinear parabolic equation

We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blowup of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation $$ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$$ on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.