Cesar J. Niche

Decay characterization of solutions to dissipative equations

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 {\em decay character}, which is a number associated to the initial datum that describes the behavior of the datum 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 of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.

On the topological entropy of an optical Hamiltonian flow

In this paper, we prove two formulae for the topological entropy of an -optical Hamiltonian flow induced by HC∞(M,), where is a Lagrangian distribution. In these formulae, 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 distribution or to a proper modification of it.

On the decay of infinite energy solutions to the Navier–Stokes equations in the plane

Abstract Infinite energy solutions to the Navier–Stokes equations in R 2 may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.

Inviscid limit for SQG equation in different dispersive regimes via relative energy inequality

Abstract We consider the dissipative surface quasigeostrophic equation with a dispersive forcing term and study the relation of solutions with vanishing viscosity to solutions of the inviscid equation with strong, constant or no dispersion. We show convergence by developing estimates based on the relative energy inequality.

A remark on unique ergodicity and the contact type condition

We prove that for a broad class of exact symplectic manifolds including

Decay characterization of solutions to Navier–Stokes–Voigt equations in terms of the initial datum

{\mathbb{R}^{2m}}

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

R2m, the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and the observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.