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In this paper, we present some unexpected answers to two problems with longstanding interest. More specifically, we show that a bounded primitive function of an asymptotically almost periodic function from R to a Banach space X is remotely almost periodic if and only if c 0 ⊄ X. This gives a natural analogy of the classical Bohl-Bohr-Amerio-Kadets theorem for asymptotically almost periodic functions. Based on this result, we succeed in extending the classical Loomis theorem in [13] to asymptotically almost periodic case without any ergodic conditions, and apply our Loomis type theorems to abstract Cauchy problems. Moreover, we give an example of inhomogeneous Schrödinger equation with asymptotically almost periodic coefficient, whose solution is not asymptotically almost periodic but remotely almost periodic. This demonstrates that remotely almost periodic functions is the natural class for solutions to some partial differential equations, which is similar to a remarkable phenomenon revealed by Shen and Yi (1998) [20] using dynamical system approach. [ABSTRACT FROM AUTHOR]

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m ≥ 2. Let H L 1 (X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group e i t L associated with L such that ‖ (I + L) − n / 2 e i t L f ‖ L 1 (X) + ‖ (I + L) − n / 2 e i t L f ‖ H L 1 (X) ≤ C (1 + | t |) n / 2 ‖ f ‖ H L 1 (X) , t ∈ R for some constant C = C (n , m) > 0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □ b on polynomial model domains treated by Nagel–Stein [41] , where e − t □ b satisfies only the second order Davies-Gaffney estimates. Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint L p -Sobolev bound for e i t L : ‖ (I + L) − s e i t L f ‖ L p (X) ≤ C (1 + | t |) s ‖ f ‖ L p (X) , t ∈ R , s ≥ n | 1 2 − 1 p | for every 1 < p < ∞ , which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space R n. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω , F , P , θ). Let K be a ϕ -invariant random set with compact ω -section K (ω) such that ϕ (ω) (K (ω)) = K (θ ω). We proved that if h top (ϕ , K) > 0 , then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system. [ABSTRACT FROM AUTHOR]

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval [ 0 , T ] for an arbitrary given T > 0 when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some W 1 , p space and L x 3 L v 1 + space for soft potential. By using the classical L 2 – L ∞ method and bootstrap argument, we extend the local solutions from small time scale to large time scale. [ABSTRACT FROM AUTHOR]

Combinatorial vector fields on simplicial complexes introduced by Robin Forman constitute a combinatorial analogue of classical flows. They have found numerous and varied applications in recent years. Yet, their formal relationship to classical dynamical systems has been less clear. In this paper we prove that for every combinatorial vector field on a finite simplicial complex X one can construct a semiflow on the underlying polytope X which exhibits the same dynamics. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information. [ABSTRACT FROM AUTHOR]