Abstract. Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on R d are studied. These equations constitute gradient flows for the perturbed information functionalswith respect to the L 2 -Wasserstein metric. The value of α ranges from α = 1/2, corresponding to a simplified quantum drift diffusion model, to α = 1, corresponding to a thin film type equation.

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We analyze the large-time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years. As specific examples, we study models with fixed saving propensity introduced by A. Chakraborty and B.K. Chakrabarthi [11], as well as models involving both exchange between agents and speculative trading as considered by S. Cordier, L. Pareschi and one of the authors [14]. We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one in [11], while models with speculative trades introduced in [14] develop fat tails if the market is "risky enough". The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation [26,1,39], and from a recursive relation which allows to calculate arbitrary moments of the stationary state.

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α , then the limit is a scale mixture of ν α . Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

Abstract. The logarithmic fourth-order equationcalled the Derrida-Lebowitz-Speer-Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnegative solutions in space dimensions d ≤ 3 is shown. Furthermore, a family of entropy-entropy dissipation inequalities is derived in arbitrary space dimensions, and rates of the exponential decay of the weak solutions to the homogeneous steady state are estimated. The proofs are based on the algorithmic entropy construction method developed by the authors and on an exponential variable transformation. Finally, an example for non-uniqueness of the solution is provided.Key words. Logarithmic fourth-order equation, entropy-entropy dissipation method, existence of weak solutions, long-time behavior of solutions, decay rates, nonuniqueness of solutions.