Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural? |
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Authors: | A Val Antonyuk A Vik Antonyuk |
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Institution: | (1) Institute of Mathematics, Ukrainian Academy of Sciences, Kiev |
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Abstract: | It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related
parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations
are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance
of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations
on manifolds and semigroups, we study C
∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions
of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a
unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional
separation of curvature.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006. |
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Keywords: | |
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