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Regular attractor for damped KdV‐Burgers equations on R
Author(s) -
Guo Yantao,
Wang Ming
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4541
Subject(s) - mathematics , compact space , attractor , korteweg–de vries equation , smoothing , burgers' equation , space (punctuation) , infinity , mathematical analysis , interpolation (computer graphics) , pure mathematics , partial differential equation , nonlinear system , motion (physics) , classical mechanics , linguistics , statistics , physics , philosophy , quantum mechanics
We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force f ( x ) ∈ L p ( R ) ⋂ L q ( R ) ( 1 < p ⩽ 2 , 2 < q < + ∞ ) on R . We establish L p − L q estimates of the evolution e − t ( ∂ x 2 − ∂ x 3 ) , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as t goes to infinity. Next, we prove the asymptotical regularity of solutions in space H 9 2 − 1 p − ( R ) and H 7 2 + 1 2 q − , q ( R ) by the smoothing effect of e − t ( ∂ x 2 − ∂ x 3 ) . The regularity and the asymptotical compactness in L 2 yields the asymptotical compactness in H 7 2 − 1 p⋂ H 3 , q ( R ) by an interpolation arguement. Finally, we conclude the existence of an ( L 2 ( R ) , H 7 2 − 1 p( R ) ⋂ H 3 , q ( R ) ) globalattractor.