The Experts below are selected from a list of 116190 Experts worldwide ranked by ideXlab platform
Qi S Zhang - One of the best experts on this subject based on the ideXlab platform.
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time analyticity for the Heat Equation and navier stokes Equations
arXiv: Analysis of PDEs, 2019Co-Authors: Hongjie Dong, Qi S ZhangAbstract:We prove the analyticity in time for non-decaying solutions of two parabolic Equations in the whole space. One of them involves solutions to the Heat Equation of double exponential growth on $\M$. Here $\M$ is $\R^n$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. The other pertains bounded mild solutions of the incompressible Navier-Stokes Equations. An implication is a sharp solvability condition for the backward Heat Equation.
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a note on time analyticity for ancient solutions of the Heat Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Qi S ZhangAbstract:It is well known that generic solutions of the Heat Equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and sufficient condition is found on the solvability of backward Heat Equation in the class of functions with exponential growth.
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the conjugate Heat Equation and ancient solutions of the ricci flow
Advances in Mathematics, 2011Co-Authors: Xiaodong Cao, Qi S ZhangAbstract:Abstract 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 Perelmanʼs previous result on backward limits of κ -solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23] , where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate Heat Equation under evolving metric might be of independent interest.
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the conjugate Heat Equation and ancient solutions of the ricci flow
arXiv: Differential Geometry, 2010Co-Authors: Xiaodong Cao, Qi S ZhangAbstract: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 $\kappa$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $\kappa$-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.
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Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
Bulletin of the London Mathematical Society, 2006Co-Authors: Philippe Souplet, Qi S ZhangAbstract:We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the Heat Equation. These estimates are related 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 ancient (including eternal) solutions of the Heat Equation, under certain growth conditions. Surprisingly this Liouville theorem for the Heat Equation does not hold even in without such a condition. We also prove a sharpened long-time gradient estimate for the log of the Heat kernel on noncompact manifolds.
Artem Pulemotov - One of the best experts on this subject based on the ideXlab platform.
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gradient estimates for the Heat Equation under the ricci flow
Journal of Functional Analysis, 2010Co-Authors: Mihai Bailesteanu, Xiaodong Cao, Artem PulemotovAbstract:The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the Heat Equation on M.
Xiaodong Cao - One of the best experts on this subject based on the ideXlab platform.
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the conjugate Heat Equation and ancient solutions of the ricci flow
Advances in Mathematics, 2011Co-Authors: Xiaodong Cao, Qi S ZhangAbstract:Abstract 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 Perelmanʼs previous result on backward limits of κ -solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23] , where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate Heat Equation under evolving metric might be of independent interest.
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the conjugate Heat Equation and ancient solutions of the ricci flow
arXiv: Differential Geometry, 2010Co-Authors: Xiaodong Cao, Qi S ZhangAbstract: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 $\kappa$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $\kappa$-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.
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gradient estimates for the Heat Equation under the ricci flow
Journal of Functional Analysis, 2010Co-Authors: Mihai Bailesteanu, Xiaodong Cao, Artem PulemotovAbstract:The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the Heat Equation on M.
Ofer Zeitouni - One of the best experts on this subject based on the ideXlab platform.
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the edwards wilkinson limit of the random Heat Equation in dimensions three and higher
Communications in Mathematical Physics, 2018Co-Authors: Lenya Ryzhik, Ofer ZeitouniAbstract:We consider the Heat Equation with a multiplicative Gaussian potential in dimensions d ≥ 3. We show that the renormalized solution converges to the solution of a deterministic diffusion Equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards–Wilkinson model, that is, the stochastic Heat Equation (SHE) with additive white noise, with an effective variance.
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the edwards wilkinson limit of the random Heat Equation in dimensions three and higher
arXiv: Probability, 2017Co-Authors: Lenya Ryzhik, Ofer ZeitouniAbstract:We consider the Heat Equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion Equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic Heat Equation (SHE) with additive white noise, with an effective variance.
Oscar Orellana - One of the best experts on this subject based on the ideXlab platform.
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on an inverse problem recovery of non smooth solutions to backward Heat Equation
Applied Mathematical Modelling, 2012Co-Authors: Fabien Ternat, Prabir Daripa, Oscar OrellanaAbstract:We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward Heat Equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward Heat Equation. In particular, we numerically integrate the backward Heat Equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-L- and U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.
