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Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and...

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Main Authors: Antoniouk, A.Val., Antoniouk, A.Vict.
Format: Article
Language:English
Published: Інститут математики НАН України 2008
Series:Український математичний журнал
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Online Access:http://dspace.nbuv.gov.ua/handle/123456789/164759
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spelling irk-123456789-1647592020-02-11T01:26:12Z Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds Antoniouk, A.Val. Antoniouk, A.Vict. Статті We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of manifold there are estimates exponential in time on the continuity of diffusion process with respect to the initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to the initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent space, nor uses embeddings of manifold to linear spaces of higher dimensions. Досліджено залежність за початковими умовами для розв'язків дифузійних рівнянь з глобально неліпшицевими коефіцієнтами на некомпактних багатовидах. Хоча функція метричної відстані може бути не скрізь двічі диференційовною, показано, що за певних умов монотонності на коефіцієнти та кривину багатовиду існують експоненціальні за часом оцінки на неперервність дифузійного процесу за початковими умовами. У поєднанні з методами теорії абсолютно неперервних функцій ці оцінки приводять до першого порядку регулярності розв'язків за початковими умовами. Запропонований підхід не використовує техніку моментів часу виходу процесу з локальних координатних околів, а також експоненціальних відображень з дотичного простору або вкладення багатовиду до лінійного простору більшої розмірності. 2008 Article Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2008. — Т. 60, № 10. — С. 1299–1316. — Бібліогр.: 14 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164759 519.217.4, 517.955.4, 517.956.4, 517.958:536.2 en Український математичний журнал Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Статті
Статті
spellingShingle Статті
Статті
Antoniouk, A.Val.
Antoniouk, A.Vict.
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
Український математичний журнал
description We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of manifold there are estimates exponential in time on the continuity of diffusion process with respect to the initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to the initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent space, nor uses embeddings of manifold to linear spaces of higher dimensions.
format Article
author Antoniouk, A.Val.
Antoniouk, A.Vict.
author_facet Antoniouk, A.Val.
Antoniouk, A.Vict.
author_sort Antoniouk, A.Val.
title Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
title_short Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
title_full Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
title_fullStr Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
title_full_unstemmed Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
title_sort continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
publisher Інститут математики НАН України
publishDate 2008
topic_facet Статті
url http://dspace.nbuv.gov.ua/handle/123456789/164759
citation_txt Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2008. — Т. 60, № 10. — С. 1299–1316. — Бібліогр.: 14 назв. — англ.
series Український математичний журнал
work_keys_str_mv AT antonioukaval continuitywithrespecttoinitialdataandabsolutecontinuityapproachtothefirstorderregularityofnonlineardiffusionsonnoncompactmanifolds
AT antonioukavict continuitywithrespecttoinitialdataandabsolutecontinuityapproachtothefirstorderregularityofnonlineardiffusionsonnoncompactmanifolds
first_indexed 2023-10-18T22:14:34Z
last_indexed 2023-10-18T22:14:34Z
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