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GLOBAL ATTRACTOR FOR WEAKLY DAMPED, FORCED mKdV EQUATION BELOW ENERGY SPACE

Part of: Infinite-dimensional dissipative dynamical systems Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  20 June 2019

PRASHANT GOYAL Open the ORCID record for PRASHANT GOYAL [Opens in a new window]
PRASHANT GOYAL*
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8501, Japan email goyalprashant194@gmail.com
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Abstract

We prove the existence of the global attractor in ${\dot{H}}^{s}$, $s>11/12$ for the weakly damped and forced mKdV on the one-dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space has been established. We directly apply the $I$-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

MSC classification

Primary: 35B41: Attractors
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems

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Type
Article
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Nagoya Mathematical Journal , Volume 241 , March 2021 , pp. 171 - 203
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© 2019 Foundation Nagoya Mathematical Journal

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