Author

# Truyen Nguyen

Other affiliations: Georgia Institute of Technology, Temple University, University of Kansas

Bio: Truyen Nguyen is an academic researcher from University of Akron. The author has contributed to research in topics: Bounded function & Upper and lower bounds. The author has an hindex of 15, co-authored 48 publications receiving 615 citations. Previous affiliations of Truyen Nguyen include Georgia Institute of Technology & Temple University.

##### Papers

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TL;DR: In this article, the authors introduce a concept of viscosity solutions for Hamilton-Jacobi equations in the Wasserstein space, which are solutions of physical systems in finite dimensional spaces.

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.

93 citations

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TL;DR: In this paper, a variational approach is used to establish existence of solutions for the 1-d Euler-Poisson system by minimizing an action, where the initial and terminal points σ0, σT are prescribed in the Borel probability measures on the real line, of finite second-order moments.

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.

48 citations

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TL;DR: The time regularity of the solution for the pressureless Euler system is proved and it is obtained that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness.

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.

43 citations

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TL;DR: It is shown that the Shigesada--Kawasaki--Teramoto system of cross-diffusion equations of two competing species in population dynamics has a unique smooth solution for all time in bounded domains of any dimension.

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.

34 citations

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TL;DR: In this paper, a class of action integrals defined over probability measure-valued path space is introduced, and extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense.

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.

30 citations

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02 Jan 2013

TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.

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.

4,558 citations

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TL;DR: In this paper, the authors 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 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.

147 citations

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TL;DR: In this article, the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemanian manifold are computed. But the curvature of the Wermstein space is not known.

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

121 citations