TIME ANALYTICITY FOR THE HEAT EQUATION UNDER BAKRY-ÉMERY RICCI CURVATURE CONDITION
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초록
Inspired by Hongjie Dong and Qi S. Zhang's article \cite{ZQ2}, we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain $L^p$ spaces with $p\in[2,+\infty)$ and prove its analyticity with respect to time.
Abstract
Inspired by Hongjie Dong and Qi S. Zhang's article \cite{ZQ2}, we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain $L^p$ spaces with $p\in[2,+\infty)$ and prove its analyticity with respect to time.
참고문헌 (19)
- D. Bakry, "Diffusions hypercontractives", Lecture Notes in Mathematics, pp.177~206 (1985), Springer Berlin Heidelberg[DOI]
- B. -L. Chen, "Strong uniqueness of the Ricci flow", J. Differential Geom, 82(2), pp.363~382 (2009)
- Hongjie Dong, "Time analyticity for the heat equation and Navier-Stokes equations", Journal of Functional Analysis, 279(4), pp.108563~ (2020), Elsevier BV[DOI]
- Luis Escauriaza, "Analyticity of Solutions to Parabolic Evolutions and Applications", SIAM Journal on Mathematical Analysis, 49(5), pp.4064~4092 (2017), Society for Industrial & Applied Mathematics (SIAM)[DOI]
- Yoshikazu Giga, "Time and spartial analyticity of solutions of the navier-stokes equations", Communications in Partial Differential Equations, 8(8), pp.929~948 (1983), Informa UK Limited[DOI]
- R. S. Hamilton, "Surveys in differential geometry, Vol. II", pp.7~136 (1995), Int. Press
- Fengwen Han, "Time analyticity of solutions to the heat equation on graphs", Proceedings of the American Mathematical Society, 149(6), pp.2279~2290 (2021), American Mathematical Society (AMS)[DOI]
- John Lott, "Ricci curvature for metric-measure spaces via optimal transport", Annals of Mathematics, 169(3), pp.903~991 (2009), Annals of Mathematics[DOI]
- Ovidiu Munteanu, "Smooth metric measure spaces with non-negative curvature", Communications in Analysis and Geometry, 19(3), pp.451~486 (2011), International Press of Boston[DOI]
- G. Perelman, "The entropy formula for the Ricci flow and its geometric applications", arXiv:math
- Stefano Pigola, "Remarks on non-compact gradient Ricci solitons", Mathematische Zeitschrift, 268(3-4), pp.777~790 (2010), Springer Science and Business Media LLC[DOI]
- L. Saloff-Coste, "London Mathematical Society Lecture Note Series", pp.289~ (2002), Cambridge Univ. Press
- X. Y. Song, "Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Emery Ricci curvature condition and applications", arXiv:2306.13261, ~
- Karl-Theodor Sturm, "On the geometry of metric measure spaces", Acta Mathematica, 196(1), pp.65~131 (2006), International Press of Boston[DOI]
- Karl-Theodor Sturm, "On the geometry of metric measure spaces. II", Acta Mathematica, 196(1), pp.133~177 (2006), International Press of Boston[DOI]
- D. V. Widder, "Analytic solutions of the heat equation", Duke Math. J, 29, pp.497~503 (1962)[DOI]
- Jiayong Wu, "Time Analyticity for the Heat Equation on Gradient Shrinking Ricci Solitons", Acta Mathematica Scientia, 42(4), pp.1690~1700 (2022), Springer Science and Business Media LLC[DOI]
- Qi S. Zhang, "A note on time analyticity for ancient solutions of the heat equation", Proceedings of the American Mathematical Society, 148(4), pp.1665~1670 (2019), American Mathematical Society (AMS)[DOI]
- Shi Jin Zhang, "On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below", Acta Mathematica Sinica, English Series, 27(5), pp.871~882 (2011), Springer Science and Business Media LLC[DOI]