The entropy formula for linear heat equation
نویسندگان
چکیده
منابع مشابه
The Entropy Formula for Linear Heat Equation
§0 Introduction. In a recent paper of Perelman[P1], an entropy formula for Ricci flow was derived. The formula turns out being of fundamental importance in the study of Ricci flow (cf. [P1, Sections 3, 4, 10]) as well as the Kähler-Ricci flow [P2]. The derivation of the entropy formula in [P1, Section 9] resembles the gradient estimate for the linear heat equation proved by Li-Yau in another fu...
متن کاملAddenda to “ The Entropy Formula for Linear Heat Equation ”
We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equalit...
متن کاملThe Entropy Formula for Linear Heat Equation 87 Corollary 1
We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on 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 on volume non-collaps...
متن کاملNon-linear Rough Heat Equation
This article is devoted to define and solve an evolution equation of the form dyt = ∆yt dt + dXt(yt), where ∆ stands for the Laplace operator on a space of the form L(R), and X is a finite dimensional noisy nonlinearity whose typical form is given by Xt(φ) = ∑N i=1 x i tfi(φ), where each x = (x , . . . , x) is a γ-Hölder function generating a rough path and each fi is a smooth enough function d...
متن کاملThe 1-D Heat Equation 18.303 Linear Partial Differential Equations
Rate of heat transfer ∂u = (1) −K0 area ∂x where K0 is the thermal conductivity, units [K0] = MLT U . In other words, heat is transferred from areas of high temp to low temp. 3. Conservation of energy. Consider a uniform rod of length l with non-uniform temperature lying on the x-axis from x = 0 to x = l. By uniform rod, we mean the density ρ, specific heat c, thermal conductivity K0, cross-sec...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Geometric Analysis
سال: 2004
ISSN: 1050-6926,1559-002X
DOI: 10.1007/bf02921867