Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Abstract

Abstract. In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces B s 2,r in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-tosolution map provided that the initial data belongs to B s 2,r . We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B s 2,r to C([−T, T ]; B s 2,r ). Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.

Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

“…Case 2β + 1 > 0: As a preliminary observation, we note the following: The conditions (16) (14), we have also Z(s) = 0, thus the trajectory in the (Y, Z)-phase diagram passes through the origin (0, 0) and, due to (16) and (13), has as end point (Y (1), Z(1) precisely the reflection at the Z-axis of its starting point (Y (0), Z(0)).…”

“…while for Z we have the periodic continuity condition (13). We split the further analysis into two cases depending on the value of β: Case 2β + 1 ≤ 0: Equation (15) implies that Z ′ ≥ 0.…”

“…Well-posedness results, local in time, with strong solutions in Sobolev and Besov spaces have been obtained for both cases, periodic and non-periodic, in [12,13]. We cannot always expect globally defined strong solutions.…”

Wave breaking of periodic solutions to the Fornberg-Whitham equation

“…Well-posedness results for (3)(4) with spatial regularity according to Sobolev or Besov scales have been obtained in [7,8]. Here we will make use of the following simpler consequence: If s > 3/2 and u 0 ∈ H s (T), then there exists T 0 > 0 such that (3-4) possesses a…”

The stability of solutions for the Fornberg–Whitham equation in L 1 ( R ) $L^{1}(\mathbb{R})$ space

“…is found. Recently, Holmes and Thompson [3] have established the existence and uniqueness of the FW equation in the Besov space in both non-periodic and periodic cases and discussed the sharpness of continuity on the data-to-solution map. A Cauchy-Kowalevski type result, which guarantees the existence and uniqueness of real analytic solutions for Eq.…”