1,720,968 research outputs found
Absolutely continuous curves in extended Wasserstein-Orlicz spaces
Full text linkIn this paper we extend a previous result of the author [Lis07] of characterization of absolutely continuous curves in Wasserstein spaces to a more general class of spaces: the spaces of probability measures endowed with the Wasserstein-Orlicz distance constructed on extended Polish spaces (in general non separable), recently considered in [AGS14]. An application to the geodesics of this Wasserstein-Orlicz space is also given
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
No full textWe study existence and approximation of non-negative solutions of a class of nonlinear diffusion equations with variable coefficients. The results are obtained interpreting this kind of equations as "gradient flow" of a suitable energy functional with respect to a suitable Wasserstein distance. More precisely the Wasserstein distance between probability measures on the euclidean space endowed with the Riemannian distance induced by the inverse matrix of the coefficients of the equation. Long time asymptotic behavior and rate decay to stationary state for solutions of the equation are studied. A contraction property in Wasserstein distance for solutions of the equation is studied in a particular case
Characterization of absolutely continuous curves in Wasserstein spaces
No full textLet X be a separable, complete metric space and Pp(X) be the space of Borel probability measures with finite moment of order p > 1, metrized by the Wasserstein distance. In this paper we prove that every absolutely continuous curve with finite p-energy in the space Pp(X) can be represented by a Borel probability measure on C([0,T];X) concentrated on the set of absolutely continuous curves with finite p-energy in X. Moreover this measure satisfies a suitable property of minimality which entails an important relation on the energy of the curves. We apply this result to the geodesics of Pp(X) and to the continuity equation in Banach spaces
On the asymptotic behavior of the gradient flow of a polyconvex functional
No full textIn this paper, we study the asymptotic behavior of the solutions of the system of non-linear partial differential equations studied in a paper of Evans-Gangbo-Savin for the evolution of a family of diffeomorphisms. We prove existence and regularity of the asymptotic state of solutions and we find an explicit rate of convergence of the time dependent solution to the corresponding final state. We study also a system not considered in the paper of Evans-Gangbo-Savin, linked to a linear Fokker-Planck equation. For this system we show existence of solutions, of the asymptotic state, the regularity and the rate of convergence of the solution to a final state. In both cases, the final states are obtained from the composition of the limit in time of the flow map with the initial data
Gradient flows for non-smooth interaction potentials
No full textWe deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or λ-convex) potential, possibly not smooth at several points, generalizing the results of Carrillo et al. (2011). We also identify the cases in which the dynamic is still governed by the continuity equation with well-characterized nonlocal velocity field
A gradient flow approach to the porous medium equation with fractional pressure
No full textWe consider a family of porous media equations with fractional pressure, recently studied by Caffarelli and Vazquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the L^p norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case
Uniqueness for Keller-Segel-type chemotaxis models
No full textWe prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data
Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics
No full textAbstractIn this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape
On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals
Full text linkWe study a new class of distances between Radon measures similar to those studied in J. Dolbeault, B. Nazaret, G. Savaré, "A new class of transport distances between measures",Calc. Var. Partial Differential Equations, 34 (2009), pp. 193--231. These distances (more correctly pseudo-distances because can assume the value ) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in D.-N.-S.) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in with finite moments and the set of measuresdefined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness
Measure valued solutions of sub-linear diffusion equations with a drift term
No full textIn this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing and bounded function β. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass m and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called L2-Wasserstein distance.
Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass mc, which can be explicitely characterized in terms of β and of the drift term. If the initial mass is less then mc, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure.
Conversely, when the total mass m of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass m − mc is accumulated
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