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.
The "sticky particles" model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler-Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier, who showed that one-dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.
This paper uses a variational approach to establish existence of solutions (σ t , v t ) for the 1-d Euler-Poisson system by minimizing an action. We assume that the initial and terminal points σ 0 , σ T are prescribed in P 2 (IR), the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer of the action when the time interval [0, T ] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σ t in P 2 (IR). When σ t = δ y(t) is a Dirac mass, the Euler-Poisson system reduces toÿ + y = 0. The kinetic version of the Euler-Poisson, i.e. the Vlasov-Poisson system was studied in [1] as a Hamiltonian system.
The space L 2 (0, 1) has a natural Riemannian structure on the basis of which we introduce an L 2 (0, 1)-infinite-dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.
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