New entropy formulae for heat equation on manifolds.
problem Entropy formulae for linear heat equation on Riemannian manifolds.
method Proved new entropy formulae for linear heat equation on static Riemannian manifolds with nonnegative Ricci curvature.
result Results are analogies of Cao and Hamilton's entropies for Ricci flow.
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. Thi…
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the ε-Ricci flow on a closed surface. These new Harnac…
Article provides Bernstein gradient estimates for heat equations with potential terms.
problem Gradient estimates for heat equations with potential terms on weighted Riemannian manifolds.
method Derived Bernstein type gradient estimates for two systems of heat equations with linear, exponential, and combined potentials.
result Resolves part of the problem raised by Bhattacharyya et al. in \cite{SB-1}.
We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.
In this paper we introduce a new logarithmic entropy functional for the linear heat equation on complete Riemannian manifolds and prove that it is monotone decreasing on complete Riemannian manifolds with nonnegative Ricci curvature. Our results are simpler version, without Ricci flow, of R.-G. Ye's recent result (arXi…
New gradient estimates for heat equation on Riemannian manifolds.
problem Improving gradient estimates for heat equations on manifolds.
method Provided a new version of Li-Yau gradient estimate for the linear heat equation.
result Generalizes and provides new gradient estimates for heat equations.
Second part of a study on heat equations on special manifolds, focusing on parametrix construction.
problem Analysis of heat-type equations on manifolds with fibered boundaries.
method Construction of parametrix for heat-type equations.
result Inference of existence and regularity of certain parabolic equations.
Graphs prove curvature condition with modified heat equation.
problem Proving curvature condition for infinite graphs.
method Establishing existence and uniqueness of modified heat equation solutions.
result Explicit examples of graphs satisfying assumptions.
The paper establishes new inequalities for Finsler measure spaces.
problem Developing inequalities for Finsler measure spaces.
method Study of linearized heat semigroup and application of Li-Yau's inequalities.
result Established new Li-Yau's type inequalities for Finsler measure spaces.
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 …
Explicit formulas found for ancient solutions of heat equation.
problem Finding explicit formulas for ancient solutions of the heat equation.
method Explicit representation formulas for positive ancient solutions in Euclidean and Riemannian cases.
result Ancient solutions are the Laplace transform of positive solutions of a family of elliptic operators.
Develops heat kernel and Green's function estimates for manifolds.
problem Solving Poisson equation on manifolds with Ricci curvature bounds.
method Heat kernel and Green's function estimates for manifolds with positive spectrum.
result Existence and sharp estimates of Poisson equation solutions on manifolds with Ricci curvature bounds.
The paper establishes estimates for heat equations and harmonic functions on metric spaces with curvature-dimension condition.
problem Analyzing geometric properties of metric measure spaces with curvature-dimension condition.
method Establishing local Li-Yau estimates and proving sharp Yau's gradient estimates for heat equations and harmonic functions.
result Sharp Li-Yau and gradient estimates for weak solutions of heat equations and harmonic functions on RCD∗(K,N) spaces. Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results o…
In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with Ricci(M)≥−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci…
We derive a Harnack inequality for positive solutions of the f-heat equation and Gaussian upper and lower bounds for the f-heat kernel on complete smooth metric measure spaces (M,g,e−fdv) with Bakry-Émery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser t…
Study asymptotically almost periodic solutions on real hyperbolic manifolds.
problem Existence and asymptotic behavior of solutions to parabolic equations.
method Dispersion and smoothing estimates, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic solutions.
Optimal gap theorem proven for CR manifolds with curvature conditions.
problem Proving optimal gap theorem for CR manifolds with specific curvature conditions.
method Applying Li-Yau-Hamilton inequality and heat equation deformation monotonicity.
result Proves flatness of CR manifold under certain curvature decay conditions.
W-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these t…
Researchers found a new type of singularity in surface evolution equations.
problem Finite-time singularity formation in surface evolution equations.
method Constructed first example of finite time blow-up solutions for the heat flow of the H-system.
result Singularity forms as a scaled least energy H-bubble with decoupled linearized operators.
Study describes heat kernel expansion for hypoelliptic operators.
problem Characterize coefficients in small time heat kernel expansion.
method Geometric characterization of coefficients using drift field and curvature-like invariants.
result Geometric characterization of coefficients in terms of drift field and curvature-like invariants.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, S−curvature and Hessian) and suitable Ricci curvature bounds throug…
Study fast diffusion equation under Ricci flow with estimates.
problem Analyzing fast diffusion equation under Ricci flow.
method Proved Aronson-Bénilan and Li-Yau-Hamilton type estimates.
result Extended Li-Yau-Hamilton estimates to noncompact settings.
Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure of the flow, this partial differential equation problem can be resolved using a…
Quaternionic contact heat equation studied on compact manifolds.
problem Heat equation on quaternionic contact manifolds.
method Introduced quaternionic contact heat equation and energy functional.
result Monotonicity of qc energy functional along the heat equation.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in Bk×(0,T) to the minimal fundamental solut…
In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…
The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
problem Gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
method Gradient estimates and Harnack inequalities for heat equations under the Laplacian G_2 flow.
result Monotonicity of parabolic frequency and backward uniqueness for positive solutions.
Bounds on Hessian of heat equation coupled with Ricci flow.
problem Estimating the Hessian of a solution to the conjugate heat equation coupled with Ricci flow.
method Obtained upper bounds for the Hessian.
result Local and global upper bounds for the Hessian of a positive solution.
Sharp time analyticity proved for heat equation on Ricci solitons.
problem Analyticity of heat equation solutions on gradient shrinking Ricci solitons.
method Proved analyticity for solutions with quadratic exponential growth.
result Sharp growth condition for analyticity is established.
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
Study nonlinear heat equation gradient estimates and applications.
problem Gradient estimates for nonlinear heat equation.
method Elliptic gradient estimates for a nonlinear f-heat equation. result Obtain gradient estimates for positive solutions.
In this paper, we study the Poisson equation and heat equation in a model matrix geometry Mn. Our main results are about the Poisson equation and global behavior of the heat equation on Mn. We can show that if c0 is the initial positive definite matrix in Mn, then c(t) exists for all time and is positive …
In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem ut−Δu=aulogu, u>0 on the compact Riemannian manifold (M,g) of dimension n and with non-negative (Bakry-Emery)-Ricci curvature. Here…
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
This paper proposes an unsupervised learning method to solve heat equations on chips.
problem Critical need for solving heat transfer equations on chips for 5G and AI.
method Hybrid framework of Auto Encoder and Image Gradient for unsupervised learning.
result Framework can solve heat transfer problems with a single training process and predict unseen cases.
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Ancient solutions to heat equations on graphs are shown to be analytic in time.
problem Analyzing the analyticity of ancient solutions to heat equations on graphs.
method Proving time analyticity under a sharp growth condition.
result Ancient solutions to heat equations on graphs are time analytic under certain conditions.
Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H1. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Mar…
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
Proves uniqueness of heat equation solutions on Riemannian manifolds.
problem Proving uniqueness of solutions to the heat equation on Riemannian manifolds.
method Analyzes Lp solutions for 0<p<1 and improves L1 uniqueness result. result Improves curvature assumption for L1 uniqueness result.