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Article Dans Une Revue Journal of Dynamics and Differential Equations Année : 2015

Distribution of Energy and Convergence to Equilibria in Extended Dissipative Systems

Thierry Gallay
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Résumé

We are interested in understanding the dynamics of dissipative partial differential equations on unbounded spatial domains. We consider systems for which the energy density e ≥ 0 satisfies an evolution law of the form ∂ t e = div x f − d, where −f is the energy flux and d ≥ 0 the energy dissipation rate. We also suppose that |f | 2 ≤ b(e)d for some nonnegative function b. Under these assumptions we establish simple and universal bounds on the time-integrated energy flux, which in turn allow us to estimate the amount of energy that is dissipated in a given domain over a long interval of time. In low space dimensions N ≤ 2, we deduce that any relatively compact trajectory converges on average to the set of equilibria, in a sense that we quantify precisely. As an application, we consider the incompressible Navier-Stokes equation in the infinite cylinder R × T, and for solutions that are merely bounded we prove that the vorticity converges uniformly to zero on large subdomains, if we disregard a small subset of the time interval.
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Dates et versions

hal-01227509 , version 1 (11-11-2015)

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Thierry Gallay, Siniša Slijepčević. Distribution of Energy and Convergence to Equilibria in Extended Dissipative Systems. Journal of Dynamics and Differential Equations, 2015, ⟨10.1007/s10884-014-9376-z⟩. ⟨hal-01227509⟩
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