Methods and Applications of Analysis

Volume 15 (2008)

Number 2

Hamilton-Jacobi Equations in the Wasserstein Space

Pages: 155 – 184

DOI: http://dx.doi.org/10.4310/MAA.2008.v15.n2.a4

Authors

Wilfrid Gangbo

Truyen Nguyen

Adrian Tudorascu

Abstract

We introduce a concept of viscosity solutions for Hamilton-Jacobi equations (HJE) in the Wasserstein space. We prove existence of solutions for the Cauchy problem for certain Hamiltonians defined on the Wasserstein space over the real line. In order to illustrate the link between HJE in the Wasserstein space and Fluid Mechanics, in the last part of the paper we focus on a special Hamiltonian. The characteristics for these HJE are solutions of physical systems in finite dimensional spaces.

Keywords

Hamilton-Jacobi equations in infinite dimension; viscosity solutions; mass transfer; Wasserstein metric

2010 Mathematics Subject Classification

47J25, 49J40, 82C40

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