We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ 0 is in L 2 only, we prove that the L 2 norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ 0 in L p ∩ L 2 , with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L 2.… (More)
In this article, we prove two formulas for the topological entropy of an F-optical Hamiltonian flow induced by H ∈ C ∞ (M, R), where F is a Lagrangian distribution. In these formulas, we calculate the topological entropy as the exponential growth rate of the average of the determinant of the differential of the flow, restricted to the Lagrangian… (More)
By examining the Fourier transform of the initial datum near the origin, we define the decay character of the datum and provide a method to study the lower and upper algebraic rates of decay of solutions to a wide class of dissipative system of equations.
We prove formulae relating the topological entropy of a magnetic flow to the growth rate of the average number of trajectories connecting two points.
For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines,… (More)