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Register a new userOn the Jordan-Kinderlehrer-Otto scheme
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Authors
Paul W.Y. Lee
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In this paper, we prove that the Jordan-Kinderlehrer-Otto scheme for a family of linear parabolic equations on the flat torus converges uniformly in space.
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We establish a version of the Harnack inequality for the Jordan-Kinderlehrer-Otto scheme of the heat equation on the flat torus.
In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the authors last year and some of the examples were presented in an essay uploaded to the arXiv. In this paper we describe several infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects.
We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero. This lower bound is shown in dimension one and in the radially symmetric case.
We consider nonlinear hyperbolic conservation laws, posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and in which the flux is defined as a flux field of n-forms depending on a parameter (the unknown variable). We introduce a formulation of the initial and boundary value problem which is geometric in nature and is more natural than the vector field approach recently developed for Riemannian manifolds. Our main assumption on the manifold and the flux field is a global hyperbolicity condition, which provides a global time-orientation as is standard in Lorentzian geometry and general relativity. Assuming that the manifold admits a foliation by compact slices, we establish the existence of a semi-group of entropy solutions. Moreover, given any two hypersurfaces with one lying in the future of the other, we establish a contraction property which compares two entropy solutions, in a (geometrically natural) distance equivalent to the L1 distance. To carry out the proofs, we rely on a new version of the finite volume method, which only requires the knowledge of the given n-volume form structure on the (n+1)-manifold and involves the {sl total flux} across faces of the elements of the triangulations, only, rather than the product of a numerical flux times the measure of that face.
We provide a strengthening of Jordan separation, to the setting of maps from a compact topological space X into a sphere, where the source space X is not necessarily a codimension one sphere, and the map is not necessarily injective.
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