The initial-boundary value problem for the Kawahara equation on the half-line

Low regularity for the Kawahara equation on the half‐line

“…The data ( ϕ , h 1 , h 2 ) will be in the space . The equation was first proposed by Kawahara, 1 it arises in the theory of shallow water waves, and it is regarded as a singular perturbation of KdV equation, we refer to other studies 2–6 and references therein.…”

“…Kwak 5 proved the LWP of () in on the half‐line, by using Duhamel boundary forcing operator, for more details, we can refer to Kwak 15 and Cavalcante and Kwak 16 …”

“…Remark The main goal in the paper is to show the local well‐posedness of () in the low‐regularity (including negative regularity) Sobolev space. We consider the regularity for , the compatibility conditions for high regularities are negligible, we can see Cavalcante and Kwak 5 for the details. We also note that the nonlinearities are smoother than initial data, see Lemma 3.5.…”

On the solutions for a Benney-Lin type equation

“…Moreover, in [55], the author deduced (8) to describe describing one-dimensional propagation of smal-amplitude long waves in various problems of fluid dynamics and plasma physics. Mathematical properties of (8) were studied recently in many detail, including the local and global well-posedness in Bourgain spaces [41,40,53,54,90], the local and global well-posedness in energy space [43,45,50,88,93,94], the existence of solitary wave solution [9,51], the stability of periodic traveling wave solutions see [2,1,73,87,96], the well-posedness of the initial-boundary value problem on a bounded domain [10,13,44,58], the initial-boundary value problem on the half-line [12,11], periodic solutions [6,79], and numerical solutions [80,4,42,52,72,91]. In [74], the authors prove that the solution of (8) converges to the solution of (7), while, following [26,19,20,27,66,82], in [23,24], the convergence of the solution of…”

“…Our result is twofold, first we assume (4) and prove the global wellposedness of the solution of the Cauchy problem of (1). Assuming (5) we prove the well-posedness of the solution of the Cauchy problem of (1) in the time interval [0, T ] for any T satisfying (12). Our arguments are based on the Aubin-Lions Lemma (see [37,38,39,83]).…”