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29
Some geometric calculations on Wasserstein space
, 2007
"... We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. ..."
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Cited by 23 (2 self)
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We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
OPTIMAL TRANSPORT FOR THE SYSTEM OF ISENTROPIC EULER EQUATIONS
"... Abstract. We introduce a new variational time discretization for the system of isentropic Euler equations. In each timestep the internal energy is reduced as much as possible, subject to a constraint imposed by a new cost functional that measures the deviation of particles from their characteristic ..."
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Cited by 11 (5 self)
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Abstract. We introduce a new variational time discretization for the system of isentropic Euler equations. In each timestep the internal energy is reduced as much as possible, subject to a constraint imposed by a new cost functional that measures the deviation of particles from their characteristic paths.
HamiltonJacobi equations in the Wasserstein space
, 2008
"... We introduce a concept of viscosity solutions for HamiltonJacobi 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 Wassers ..."
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Cited by 8 (1 self)
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We introduce a concept of viscosity solutions for HamiltonJacobi 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.
A global uniqueness result for an evolution problem arising in superconductivity
 Unione Mat. Ital. (9) II
"... Abstract. We consider an energy functional on measures in R 2 arising in superconductivity as a limit case of the wellknown Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen ..."
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Cited by 8 (1 self)
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Abstract. We consider an energy functional on measures in R 2 arising in superconductivity as a limit case of the wellknown Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen as a mean field model for the evolution of vortex densities. Improving the analysis made in [AS], we obtain a new existence and uniqueness result for the evolution problem. 1.
The fully compressible semigeostrophic system from meteorology.
 Arch. Rational Mech. Anal.
, 2003
"... Abstract The fully compressible semigeostrophic system is widely used in the modelling of largescale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alterna ..."
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Cited by 6 (2 self)
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Abstract The fully compressible semigeostrophic system is widely used in the modelling of largescale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the socalled geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.
DIFFERENTIAL FORMS ON WASSERSTEIN SPACE AND INFINITEDIMENSIONAL HAMILTONIAN SYSTEMS
"... Abstract. Let M denote the space of probability measures on RD endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced in [4]. In this paper we develop a calculus for the corresponding class of differential forms on M. In p ..."
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Cited by 4 (2 self)
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Abstract. Let M denote the space of probability measures on RD endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced in [4]. In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green’s theorem for 1forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D = 2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced in [3]. Throughout the paper we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of RD. 1.
PROJECTIONS ONTO THE CONE OF OPTIMAL TRANSPORT MAPS AND COMPRESSIBLE FLUID FLOWS
"... Abstract. The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the ..."
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Abstract. The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measurevalued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.
OPTIMAL CONTROL FOR A MIXED FLOW OF HAMILTONIAN AND GRADIENT TYPE IN SPACE OF PROBABILITY MEASURES
"... Abstract. In this paper we investigate an optimal control problem in the space of measures on R 2. The problem is motivated by a stochastic interacting particle model which gives the 2D NavierStokes equations in their vorticity formulation as meanfield equation. We prove that the associated Hamil ..."
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Abstract. In this paper we investigate an optimal control problem in the space of measures on R 2. The problem is motivated by a stochastic interacting particle model which gives the 2D NavierStokes equations in their vorticity formulation as meanfield equation. We prove that the associated HamiltonJacobiBellman equation, in the space of probability measures, is wellposed in an appropriately defined viscosity solution sense.