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A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

Jean-Paul Daniel (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution...

A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint

Fujie, Kentarou, Senba, Takasi (2017)

Proceedings of Equadiff 14

We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ( 8 π ) 2 , whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known 8 π problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel system...

A gradient estimate for solutions of the heat equation. II

Charles S. Kahane (2001)

Czechoslovak Mathematical Journal

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.

A heat approximation

Miroslav Dont (2000)

Applications of Mathematics

The Fourier problem on planar domains with time variable boundary is considered using integral equations. A simple numerical method for the integral equation is described and the convergence of the method is proved. It is shown how to approximate the solution of the Fourier problem and how to estimate the error. A numerical example is given.

A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

David J. Knezevic, Endre Süli (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for...

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