Displaying similar documents to “On questions of decay and existence for the viscous Camassa–Holm equations”

Decay rates of Volterra equations on ℝN

Monica Conti, Stefania Gatti, Vittorino Pata (2007)

Open Mathematics

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This note is concerned with the linear Volterra equation of hyperbolic type ∂ t t u ( t ) - α Δ u ( t ) + ∫ 0 t μ ( s ) Δ u ( t - s ) d s = 0 on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

On optimal decay rates for weak solutions to the Navier-Stokes equations in R n

Tetsuro Miyakawa, Maria Elena Schonbek (2001)

Mathematica Bohemica

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This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in ℝ n . Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound ∥ u ( t ) ∥ ≥ ( t + 1 ) - n + 4 2 .

On the instantaneous spreading for the Navier–Stokes system in the whole space

Lorenzo Brandolese, Yves Meyer (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the spatial behavior of the velocity field u ( x , t ) of a fluid filling the whole space ℝ n ( n ≥ 2 ) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions ∫ u h ( x , t ) u k ( x , t ) d x = c ( t ) δ h , k under more general assumptions on the localization of u . We also give some new examples of solutions which have a stronger spatial localization than in the generic case.