Truyen Nguyen

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 Hamil- tonians 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.

Abstract: This paper uses a variational approach to establish existence of solutions (σt, vt) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ0, σT are prescribed in \({\mathcal {P}_2(\mathbb {R})}\) , 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 → σt in \({\mathcal {P}_2(\mathbb {R})}\) . When σt = δy(t) is a Dirac mass, the Euler–Poisson system reduces to \({\ddot {y} + y=0}\) . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure Appl Math, to appear) as a Hamiltonian system.

Abstract: 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.

Abstract: We investigate the global time existence of smooth solutions for the Shigesada--Kawasaki--Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global $W^{1,p}$-estimates of Calderon--Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli--Peral perturbation technique together with a new two-parameter scaling argument.

Abstract: We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.

Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Abstract: We survey old and new regularity theory for the Monge-Ampere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampere type equations arising in that context.

Abstract: We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.