2019/10/09 by Claudio Muñoz, Gustavo Ponce, Muñoz, Claudio +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1910.03897
openalex publication_date 2019/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the limit infimum, as time ,t , goes to infinity, of any\nuniformly bounded in time H3/2+\∩ L1 solution to the Intermediate Long\nWave equation converge to zero locally in an increasing-in-time region of space\nof order ,t/\log(t). Also, for solutions with a mild L1-norm growth in\ntime is established that its limit infimum converge to zero, as time goes to\ninfinity. This confirms the non existence of breathers and other solutions for\nthe ILW model moving with a speed "slower" than a soliton. We also prove that\nin the far field linearly dominated region, the L2 norm of the solution also\nconverges to zero as time approaches infinity.\n In addition, we deduced several scenarios for which the initial value problem\nassociated to the generalized Benjamin-Ono and the generalized Intermediate\nLong Wave equations cannot possess time periodic solutions (breathers).\n Finally, as it was previously demonstrated in solutions of the KdV and BO\nequations, we establish the following propagation of regularity result : if the\ndatum u0\∈ H3/2+( mathbb R)\∩ Hm((x0,\∞)), for some\n ;x0\∈ mathbb R, ,m\∈ Z+, ,m\≥ 2, then the corresponding solution\nu(t,\⋅) of the Intermediate Long Wave equation belongs to\nHm(\β,\∞), for any t>0 and \β\∈ mathbb R.\n