Renormalization of stochastic continuity equations on Riemannian manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

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Ball-homogeneous and disk-homogeneous Riemannian manifolds

Mathematische Zeitschrift ◽

10.1007/bf01214716 ◽

1982 ◽

Vol 180 (4) ◽

pp. 429-444 ◽

Cited By ~ 15

Author(s):

Old?ich Kowalski ◽

Lieven Vanhecke

Keyword(s):

Riemannian Manifolds ◽

Homogeneous Riemannian Manifolds

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Impulsive Relaxation of Continuity Equations and Modeling of Colliding Ensembles

Communications in Computer and Information Science - Optimization and Applications ◽

10.1007/978-3-030-10934-9_26 ◽

2019 ◽

pp. 367-381

Author(s):

Maxim Staritsyn ◽

Nikolay Pogodaev

Keyword(s):

Continuity Equations

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A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry074 ◽

2018 ◽

Vol 40 (1) ◽

pp. 405-421 ◽

Cited By ~ 1

Author(s):

N Chatterjee ◽

U S Fjordholm

Keyword(s):

Initial Data ◽

Conservation Laws ◽

Finite Volume Method ◽

Finite Volume ◽

Numerical Examples ◽

Continuity Equations ◽

Multiple Dimensions ◽

Nonlocal Conservation Laws ◽

Volume Method ◽

Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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