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2 results on '"Department of Nonlinear Analysis [Kyiv]"'

1. Variational principle for weighted porous media equation

Author

Alexandra V. Antoniouk, Marc Arnaudon, Dep. Nonlinear Analysis Institute of Mathematics NAS Ukraine, Department of Nonlinear Analysis [Kyiv], Institute of Mathematics of NAS of Ukraine, National Academy of Sciences of Ukraine (NASU)-National Academy of Sciences of Ukraine (NASU)-Institute of Mathematics of NAS of Ukraine, National Academy of Sciences of Ukraine (NASU)-National Academy of Sciences of Ukraine (NASU), Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Geodesic, diffusion flow, 01 natural sciences, symbols.namesake, Variational principle, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, variational principle, incompressible fluid, Probability (math.PR), 010102 general mathematics, Mathematical analysis, porous medium, General Medicine, State (functional analysis), 58J65, Manifold, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010101 applied mathematics, Luke's variational principle, Euler's formula, symbols, Compressibility, Mathematics::Differential Geometry, Porous medium, Mathematics - Probability
Abstract

International audience; In this paper we state the variational principle for the weighted porous media equation. It extends V.I. Arnold's approach to the description of Euler flows as a geodesics on some manifold, i.e. as a critical points of some energy functional.

Published
2014
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2. Generalized stochastic flows and applications to incompressible viscous fluids

Author

Marc Arnaudon, Alexandra V. Antoniouk, Ana Bela Cruzeiro, Dep. Nonlinear Analysis Institute of Mathematics NAS Ukraine, Department of Nonlinear Analysis [Kyiv], Institute of Mathematics of NAS of Ukraine, National Academy of Sciences of Ukraine (NASU)-National Academy of Sciences of Ukraine (NASU)-Institute of Mathematics of NAS of Ukraine, National Academy of Sciences of Ukraine (NASU)-National Academy of Sciences of Ukraine (NASU), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), grupo de fisica matematica, Universidade de Lisboa (ULISBOA), and ANR-09-BLAN-0364,ProbaGeo(2009)

Subjects
Physics::Fluid Dynamics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], General Mathematics, Norm (mathematics), Mathematical analysis, Compressibility, Porous medium, Kinetic energy, Infimum and supremum, Brownian motion, Mathematics
Abstract

International audience; We introduce a notion of generalized stochastic flows on manifolds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the $L^2$ norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also construct generalized flows with prescribed drift and kinetic energy smaller than the $L^2$ norm of the drift. The results are actually presented for general $L^q$ norms, thus including not only the Navier-Stokes equations but also other equations such as the porous media.

Published
2014

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