Well-posedness for a two-dimensional dispersive model arising from\n capillary-gravity flows

2020/05/18 by Oscar Riaño, Riaño, Oscar
Mathematics · Physics and Astronomy · #35A01 #35B60 #35B65 #35Q35 #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.2005.09184

openalex publication_date 2020/05/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

This paper is aimed to establish well-posedness in several settings for the\nCauchy problem associated to a model arising in the study of capillary-gravity\nflows. More precisely, we determinate local well-posedness conclusions in\nclassical Sobolev spaces and some spaces adapted to the energy of the equation.\nA key ingredient is a commutator estimate involving the Hilbert transform and\nfractional derivatives. We also study local well-posedness for the associated\nperiodic initial value problem. Additionally, by determining well-posedness in\nanisotropic weighted Sobolev spaces as well as some unique continuation\nprinciples, we characterize the spatial behavior of solutions of this model. As\na further consequence of our results, we derive new conclusions for the Shrira\nequation which appears in the context of waves in shear flows.\n

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