We address the study of decay rates of solutions to dissipative equations. The characterization of these rates is given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes its behavior near the origin in frequency space. We then use the decay character and the Fourier Splitting method to obtain upper and lower bounds for decay rates of solutions to the dissipative quasi-geostrophic equation and to a compressible approximation to the 3D Navier-Stokes equations.

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 . We also prove that when

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, we show that given a negatively curved manifold M , a neigbourhood UR of M in T * M , a sufficiently C 1 -small magnetic field σ and a non-trivial free homotopy class of loops α, then the magnetic flow of certain Hamiltonians supported in UR with big enough minimum, has a one-periodic orbit in α. As a consequence, we obtain estimates for the relative Hofer-Zehnder capacity and the Biran-Polterovich-Salamon capacity of a neighbourhood of M .

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