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13
A Wasserstein approach to the onedimensional sticky particle system
, 2009
"... Abstract. We present a simple approach to study the one–dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space P2(R) of probability measures with finite quadratic moments. Starting from a discrete system of a finite number of “sticky ” particles, we obtain new explicit ..."
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Abstract. We present a simple approach to study the one–dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space P2(R) of probability measures with finite quadratic moments. Starting from a discrete system of a finite number of “sticky ” particles, we obtain new explicit estimates of the solution in terms of the initial mass and momentum and we are able to construct an evolution semigroup in a measuretheoretic phase space, allowing mass distributions in P2(R) and corresponding L 2velocity fields. We investigate various interesting properties of this semigroup, in particular its link with the gradient flow of the (opposite) squared Wasserstein distance. Our arguments rely on an equivalent formulation of the evolution as a gradient flow in the convex cone of nondecreasing functions in the Hilbert space L 2 (0, 1), which corresponds to the Lagrangian system of coordinates given by the canonical monotone rearrangement of the measures.
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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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.
Curves of minimal action over metric spaces
 Ann. Mat. Pura Appl
"... Abstract. Given a metric space X, we consider a class of action functionals, generalizing those considered in [10] and [3], which measure the cost of joining two given points x0 and x1, by means of an absolutely continuous curve. In the case X is given by a space of probability measures, we can thin ..."
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Cited by 4 (1 self)
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Abstract. Given a metric space X, we consider a class of action functionals, generalizing those considered in [10] and [3], which measure the cost of joining two given points x0 and x1, by means of an absolutely continuous curve. In the case X is given by a space of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus on the possibility to split the mass in its moving part and its part that (in some sense) has already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part. 1.
A Simple Proof of Global Existence for the 1D Pressureless Gas Dynamics Equations
, 2013
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Pressureless Euler/Euler–Poisson systems via adhesion dynamics and scalar conservation laws
"... 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 EulerPoisson system with zero background charge. We consider the case of finite, non ..."
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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 EulerPoisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite secondorder 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.
Lagrangian Dynamics on an infinitedimensional torus; a Weak KAM theorem
"... 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 Wea ..."
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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 actionminimizing solutions of prescribed rotation number for the onedimensional nonlinear Vlasov system with periodic potential. 1
Tudorascu,: On Lagrangian solutions for the semigeostrophic system with singular initial data
 SIAM J. Math. Anal
"... Abstract We show that weak (Eulerian) solutions for the SemiGeostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangi ..."
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Abstract We show that weak (Eulerian) solutions for the SemiGeostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We prove existence of such solutions in the case of discrete measures in dual space. We also prove that weak Lagrangian solutions in physical space determine solutions in the dual space. This implies conservation of geostrophic energy along the Lagrangian trajectories in the physical space.
A Time Discretization for the Pressureless Gas Dynamics Equations
 Preprint. Marc Sedjro, Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, D52062 Aachen, Germany Email address: sedjro@instmath.rwthaachen.de Michael Westdickenberg, Lehrstuhl für Mathematik (Analysis), RWTH Aachen Univers
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RECONSTRUCTION OF THE EARLY UNIVERSE, ZELDOVICH APPROXIMATION AND MONGEAMPÈRE GRAVITATION
"... We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [45]. By substituting the fully nonlinear MongeAmpère equation for the linear Poisson equation to model gravitation, we introduce a modified mathem ..."
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We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [45]. By substituting the fully nonlinear MongeAmpère equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model (”MongeAmpère gravitation/MAG”), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of selfdual Lagrangians developped by Ghoussoub [29]. A fully discrete algorithm is introduced for the EUR problem in one space dimension.