Browsing by Author "Kwak, Chulkwang"
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- ItemASYMPTOTIC DYNAMICS FOR THE SMALL DATA WEAKLY DISPERSIVE ONE-DIMENSIONAL HAMILTONIAN ABCD SYSTEM(2020) Kwak, Chulkwang; Munoz, ClaudioConsider the Hamiltonian abed system in one dimension, with data posed in the energy space H-1 x H-1. This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where a, c < 0 and b = d > 0. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this 2 x 2 system was given in [J. Math. Pure Appl. (9) 127 (2019), 121-159] in a strongly dispersive regime, i.e., under essentially the conditions
- ItemLocal well-posedness of the fifth-order KdV-type equations on the half-line(2019) Cavalcante, M.; Kwak, Chulkwang
- ItemLow regularity Cauchy problem for the fifth-order modified KdV equations on T(2018) Kwak, Chulkwang
- ItemOn the Dynamics of Zero-Speed Solutions for Camassa-Holm-Type Equations(2021) Alejo, Miguel A.; Cortez, Manuel Fernando; Kwak, Chulkwang; Munoz, ClaudioIn this paper, we consider globally defined solutions of Camassa-Holm (CH)-type equations outside the well-known nonzero-speed, peakon region. These equations include the standard CH and Degasperis-Procesi (DP) equations, as well as nonintegrable generalizations such as the b-family, elastic rod, and Benjamin-Bona-Mahony (BBM) equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as t ->+infinity on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size vertical bar x vertical bar less than or similar to t(1/2-) as t ->+infinity. As a consequence, we also show scattering and decay in CH-type equations with long-range nonlinearities. Our proof relies in the introduction of suitable virial functionals a la Martel-Merle in the spirit of the works of [74, 75] and [50] adapted to CH-, DP-, and BBM-type dynamics, one of them placed in L-x(1) and the 2nd one in the energy space H-x(1). Both functionals combined lead to local-in-space decay to zero in vertical bar x vertical bar less than or similar to t(1/2-) as t -> +infinity. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH-type equations as well.
- ItemPeriodic fourth-order cubic NLS : local well-posedness and non-squeezing property(2018) Kwak, Chulkwang
- ItemProbabilistic well-posedness of generalized KdV(2018) Hwang, G.; Kwak, Chulkwang
- ItemThe scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space(2019) Kwak, Chulkwang; Muñoz, C.; Poblete, F.; Pozo, J. C.
- ItemWell-posedness issues on the periodic modified Kawahara equation(2020) Kwak, Chulkwang